Haibao Duan

Jiayu Li

Xinan Ma

Nanhua Xi

Zhen Yang

Jifeng Zhang

Alba Marina Sabogal

Robert Paluba

Elodie Vernet

Thomas Morzadec

Paul Bastide

Vincent Thouvenot

Martin Royer

Un Joyeux

Bangxian Han

Shu Shen

Yiwen Ding

Junyi Xie

Bo Xia

Songyan Xie

Taiwang Deng

Weihua He

Weichen Gao

Yueyuan Gao

Haiyan Xu

Yangqin Fang

Yi Huang

Zheng Chen

Xiaodong Wang

Zhi Jiang

Wei-Guo Foo

Dinh-Tuan Huynh

Estelle Tardy

Yeping Zhang

Bingxiao Liu

Cong Xue

Yang Cao

Disheng Xu

Quang- Huy Nguyen

Kefu Zhu

Zilolla Ashorbogeva

Norberto Vidueira

Guokuan Shao

exprimer ma reconnaissance à mes professeurs en Chine

Keywords: positive closed current, plurisubharmonic function, moderate measure, Dinh-Sibony equidistribution theorem, meromorphic transform, intermediate degree, multi-projective space, random holomorphic section, Hölder potential, big line bundle, Fubini-Study current. Classification AMS 2010: 32A60, 32L10, 32U40 courant positif fermé, fonction plurisousharmonique, mesure moderée, théorème d'équidistribution de Dinh-Sibony, transformation méromorphe, degré intermédiaire, espace multi-projectif, section holomorphe aléatoire, potentiel höldérien, fibré en droites gros, courant de, 32L10, 32U40

REMERCIEMENTS

Tout d'abord, je tiens à exprimer ma plus profonde gratitude à mes directeurs de thèse, Nessim Sibony et Viêt-Anh Nguyên. Je les remercie de m'avoir fait confiance et m'avoir amené un domaine mathématique très intéress ant et enrichissant. Je les remercie pour des discussions, des remarques et des conseils trés utiles. Cela a changé ma façon de faire des mathématiques. Je les remercie vivement pour ses encouragements et aussi pour m'avoir appris le métier de chercheur.

Je remercie sincèrement Dan Coman et Xiaonan Ma d'avoir accepté de rapporter cette thèse. Je les remercie pour le temps qu'ils ont accordé à la lecture de cette thèse et à l'élaboration de leur rapport. Merci tout particulièrement à Xiaonan Ma pour ses conseils avisés, qui ont apporté des améliorations significatives à ce manuscrit. Je remercie également Tien-Cuong Dinh, Elisha Falbel et Stéphane Nonenmacher qui me font l'honneur d'être membres du jury. Je remercie à Stéphane Charpentier, qui m'a invité à faire un exposé à Marseille.

Cette thèse à été financée par l'attribution d'un contrat doctoral de l'Univers ité Paris-Sud, et je leur remercie sincrement. Merci aussi à la Fondation de Mathématiques Jacques Hadamard, qui a soutenu ma première année d'étude en France. Merci à le département de mathématiques, qui m'a fourni de merveilleuses conditions de travail durant la préparation de cette thèse. Merci à Valérie Blandin Lavigne et Rey Florence, qui m'ont donné beaucoup d'aide durant toutes ces années.

Je remercie Xiaonan Ma qui a eu une grande influence

Introduction.

Distribution of zeros of random polynomials is a classical subject. Waring [START_REF] Todhunter | A history of the mathematical theory of probability[END_REF] used a probabilistic method to determine the number of imaginary zeros of an algebraic polynomial. More rigorous and systematic research started with the paper of Bloch-Pólya [START_REF] Bloch | On the roots of certain algebraic equations[END_REF] in 1930s. They gave an order of the expected number of real roots of certain random algebraic polynomial equations. Kac [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF] obtained an exact distribution of real roots for random polynomials with normal distribution coefficients. Results about polynomials with other distributions or settings were extended notably. We refer the reader to [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF][START_REF] Bleher | Correlation between zeros of a random polynomial[END_REF][START_REF] Bloom | Zeros of random polynomials on C m[END_REF][START_REF] Bharucha-Reid | Random polynomials, Probability and Mathematical Statistics[END_REF][START_REF] Erdős | On the distribution of roots of polynomials[END_REF][START_REF] Kac | On the average number of real roots of a random algebraic equation II[END_REF][START_REF] Shepp | The complex zeros of random polynomials[END_REF] and references therein for more results and further discussions.

A classical theorem due to Hammersley [START_REF] Hammersley | The zeros of a random polynomial[END_REF] asserts that the normalized zeros of complex Gaussian random polynomials of large degree tend to accumulate on the unit circle with uniformly distribution. An analogous result holds for random SU (2) polynomials whose expected distribution of zeros is uniform on P 1 (cf. [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF]Appendix C]). The general results about the holomorphic sections of a positive line bundle associated to the Lebesgue measures were obtained by Shiffman-Zelditch [START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF]. Note that the result covers the equidistribution property of random SU (2) polynomials when X = P 1 and L = O(1). Dinh-Sibony [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] extended the equidistribution property in the case of general measures and obtained a good estimate of the convergence speed. The potential-theoretic approach from Fornaess-Sibony [START_REF] Fornaess | Complex dynamics in higher dimensions II[END_REF] was used in the previous works.

Motivation and applications

The general measures in Dinh-Sibony's equidistribution theorems [23, Theorem 4.1, Theorem 7.3] have quite strict conditions. One explicit non-trivial example of a singular measure with real coefficients was given in [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Corollary 7.4]. But it seems difficult to provide a wide class of general measures to satisfy the equidistribution theorems. Our motivation of the first part is to construct a large family of singular moderate measures which satisfies the equidistribution property. Roughly speaking, we prove that the normalized currents defined by the zeros of random sections of high powers of a positive line bundle on a projective manifold converge weakly to the curvature form associated to certain family of singular moderate measures. Our method follows the techniques of exponential estimates for plurisubharmonic functions by Dinh-Nguyên-Sibony [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF], see Section 2.3.

In the second part, we are motivated to generalize the equidistribution property in the case of several big line bundles endowed with singular metrics. We follow the approaches of Dinh-Sibony [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF], Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] and Section 2 [START_REF] Shao | Equidistribution of zeros of random holomorphic sections for moderate measures[END_REF]. To achieve our results, we establish sharp estimates on the constants related to Alexander-Dinh-Sibony capacity on multi-projective spaces [START_REF] Shao | Equidistribution on big line bundles with singular metrics for moderate measures[END_REF], see Section 3.4.

The Alexander-Dinh-Sibony capacity is closely related to the global extremal function (cf. [23, A.2] and [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Section 5]). The L 2 -method plays also an important role in the study of equidistribution properties. For example, we apply L 2 -method to explore the asymptotic behavior of Bergman kernel functions. Hence we can obtain equidistribution theorems with good convergence speed (cf. [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF], [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF], [START_REF] Demailly | A numerical criterion for very ample line bundles[END_REF] etc). The equidistribution property provides a tool to study the Unique Ergodicity conjecture by Rudnick-Sarnak, see [START_REF] Rudnick | The behavior of eigenstates of arithmetic hyperbolic manifolds[END_REF], [START_REF] Holowinsky | Mass distribution for Hecke eigenforms[END_REF], [START_REF] Rudnick | On the asymptotic distribution of zeros of modular forms[END_REF]. It can be also applied to study quantum chaos in statistical physics (cf. [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF], [START_REF] Nonnenmacher | Chaotic eigenfunctions in phase space[END_REF] etc).

Part I

We give the basic setting in order to introduce the main theorems. For every complex vector space V of finite dimension, let ω F S be the standard Kähler form induced by the Fubini-Study metric on its projective space P(V ) normalized by P(V ) ω dim P(V ) F S = 1. Let X be a projective manifold of dimension n, L be an ample line bundle over X. Fix a Hermitian metric h on L such that the curvature form ω is Kähler on X. Let L p be the pth tensor product of L. Denote by H 0 (X, L p ) the space of all holomorphic sections of L p . Let PH 0 (X, L p ) be the associated projective space with ω F S as its normalized Fubini-Study form. Set n p := dim PH 0 (X, L p ). Let [s p = 0] be the current defined by the zero set of s p , where s p ∈ PH 0 (X, L p ). Set P X := p≥1 PH 0 (X, L p ). See Section 1.2.3 for the notion of moderate measures.

Fix some exponent 0 < ρ < 1, a function u : M → R defined on a compact metric space (M, dist) is said to be of class C ρ with modulus c if sup x,y∈M

x =y |u(x) -u(y)| dist(x, y) ρ ≤ c.

a smooth positively-curved metric. They also constructed a singular measure with real coefficients which satisfies equidistribution property (cf. [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Corollary 7.4]). When the Lebesgue measures in Shiffman-Zelditch's result are replaced by moderate measures with Hölder potentials (see Sections 2.3, 2.4), we have our main theorem as follows which gives a concrete large family of singular moderate measures that satisfies equidistribution property. It can be regarded as a perturbation of standard measures induced by Fubini-Study metric.

Theorem 0.2.1. Let L be an ample line bundle over a projective manifold X of dimension n and 0 < ρ < 1 be an exponent. Then there exists a constant c = c(X, L, ρ) > 1 with the following property. For each p ≥ 1, 1 ≤ j ≤ n p , let u p,j : PH 0 (X, L p ) → R be a function and ξ p , p > 0 two numbers such that (i) u p,j is of class C ρ with modulus ξ p , ∀1 ≤ j ≤ n p ;

(ii) u p,j is p ω F S -p.s.h., ∀1 ≤ j ≤ n p ; (iii) ξ p ≤ 1/c p n , p ≤ 1/c p n . Let σ p = (dd c u p,1 +ω F S )∧(dd c u p,2 +ω F S )∧•••∧(dd c u p,np +ω F S ) be the probability measure on PH 0 (X, L p ). Endow P X with the product measure σ = p≥1 σ p .

Then for almost every s = (s p ) ∈ P X with respect to σ, the sequence of currents { 1 p [s p = 0]} converges weakly to ω.

The following result gives a convergence speed for the equidistribution in Theorem 0.2.1.

Theorem 0.2.2. In the setting of Theorem 0.2.1, there exist subsets E p ⊂ PH 0 (X, L p ) and a positive constant C depending only on X, L such that for all p sufficiently large, we have

σ p (E p ) ≤ C p 2 and | 1 p [s p = 0] -ω, ψ | ≤ C log p p ψ C 2 ,
for any point s p ∈ PH 0 (X, L p ) \ E p and any (n -1, n -1)-form ψ of class C 2 .

Part II

Two natural questions arise after solving the first part:

1. Is it possible to consider the case of more general line bundles, e.g. big line bundles?

2. Could we extend the zeros of sections of a single line bundle to the common zeros of sections of several ones?

Fortunately, the questions are solved in some directions in the context of probability Lebesgue measures. Such equidistribution problems and convergence speeds of holomorphic sections of singular Hermitian holomorphic line bundles have been intensively explored recently. Coman-Marinescu [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF] extended the equidistribution results of [START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF] and [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] in the case of a singular holomorphic line bundle endowed with a strictly positive-curved metric. Dinh-Ma-Marinescu [START_REF] Dinh | Equidistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles[END_REF] investigated the equidistribution for big line bundles endowed with semipositive-curved metrics. Coman-Ma-Marinescu [START_REF] Coman | Equidistribution for sequences of line bundles on normal Kähler spaces[END_REF] established the equidistribution results for singular holomorphic line bundles on a compact normal Kähler complex space. Our second part has been studied by Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] in the context of probability Lebesgue measures. We refer the reader to [START_REF] Coman | Convergence of Fubini-Study currents for orbifold line bundles[END_REF][START_REF] Coman | On the approximation of positive closed currents on compact Kähler manifolds[END_REF] for a more detailed discussion.

Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] studied the equidistribution of common zeros of sections of several big line bundles. The measures of the equidistribution theorem in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] are the standard ones induced by the Fubini-Study metric. On the other hand, our work [START_REF] Shao | Equidistribution of zeros of random holomorphic sections for moderate measures[END_REF] investigated the equidistribution of zeros of sections of a single positive holomorphic line bundle associated to moderate measures. The metric on the line bundle in the latter work is smooth. In this part, the normalized currents are defined by the common zeros of m-tuples of random sections of high powers of m singular Hermitian big line bundles on a compact Kähler manifold. When the measures in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] are replaced by suitable moderate ones, we show that the above currents still distribute asymptotically to the wedge product of the curvature currents of the singular metrics. Consequently, we generalize both the main theorems in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] and [START_REF] Shao | Equidistribution of zeros of random holomorphic sections for moderate measures[END_REF].

Our method follows the approach of Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF]. Adapting their work, we prove that the intersections of currents of integration along subvarieties are well-defined almost everywhere with respect to a finite product of moderate measures (see Section 3.3). Moreover, their strategy of using Dinh-Sibony equidistribution theory leads us to obtain an estimate for the convergence speed (see Section 3.4 and Section 3.5). Then we combine the above with the technical analysis of moderate measures to achieve our results. Here our hard core work consists of estimating efficiently some constants which are intimately associated with multi-projective spaces (see Section 3.4

below).

We start with the basic settings of this part. Let X be a compact Kähler manifold of dimension n with a fixed smooth Kähler form ω. Recall that a singular Hermitian holomorphic line bundle (L, h) is a holomorphic line bundle L with a Hermitian metric which is given in any trivialization by a weight function e -ϕ such that ϕ is locally integrable (cf. Definition 1.3.13). Let c 1 (L, h) be its curvature current which represents the first Chern class. To be precise, if e L is a holomorphic frame of L on an open subset U ⊂ X, then |e L | 2 h = e -2ϕ , c 1 (L, h) = dd c ϕ on U . The case when c 1 (L, h) ≥ 0 as a current is particularly interesting. We say that a holomorphic line bundle L is big if it admits a singular metric h with that c 1 (L, h) ≥ ω for some constant > 0 (cf. Theorem 1.3.23).

Let (L k , h k ), 1 ≤ k ≤ m ≤ n, be m singular Hermitian holomorphic line bundles on X. Let L p k be the pth tensor powers of L k . Denote by H 0 (2) (X, L p k ) the Bergman space of L 2 -holomorphic sections of L p k relative to the metric h k,p := h ⊗p k induced by h k and the volume form ω n on X, endowed with the inner product S, S k,p := X h k,p (S, S )ω n , ∀S, S ∈ H 0 (2) (X, L p k ). Let PH 0 (2) (X, L p k ) be the associated projective space. Set d k,p := dim H 0

(2) (X, L p k ) -1. We have p n /C ≤ d k,p ≤ Cp n , where C > 0 is a constant independent of k and p (cf. Theorem 3.4.1). Now we consider the multi-projective space

X p := PH 0 (2) (X, L p 1 ) × ... × PH 0 (2) (X, L p m )
endowed with a probability measure σ p for every p ≥ 1. Let π k,p : X p → PH 0 (2) (X, L p k ) be the natural projections. Denote by [S = 0] the current defined by the zero set of S ∈ H 0 (X, L p k ). Set [S p = 0] := [S p1 = 0] ∧ ... ∧ [S pm = 0], ∀S p = (S p1 , ..., S pm ) ∈ X p , whenever it is well-defined. Let

P X := ∞ p=1 X p .
It is a probability space with the product measure σ = ∞ p=1 σ p . We denote by dist the distance on X induced by the fixed Kähler form ω. Let φ : U → [-∞, ∞) be a function on an open subset U ⊂ X, A ⊂ X a proper analytic subset. Following the terminology in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF], φ is called Hölder with singularities along A if there are positive constants c, δ and 0 < ν ≤ 1 satisfying that |φ(z) -φ(w)| ≤ cdist(z, w) ν min{dist(z, A), dist(w, A)} δ for all z, w ∈ U \ A. A singular metric h of L is defined to be Hölder with singularities along A if every local weight of h is Hölder with singularities along A. For motivations as well as examples of such metrics, we refer the readers to [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF].

The multi-projective space X p in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] is equipped with the probability measure σ 0 p which is the product of the Lebesgue measures induced by Fubini-Study metrics on the components. In this part, we define singular moderate measures σ p as perturbations of σ 0 p on X p . For each p ≥ 1, 1 ≤ k ≤ m, 1 ≤ j ≤ d k,p , let u k,p j : PH 0 (2) (X, L p k ) → R be an upper-semi continuous function. Fix 0 < ρ < 1 and a sequence of positive constants {c p } p≥1 . We call {u k,p j } a family of (c p , ρ)functions if all u k,p j satisfy the following two conditions:

• u k,p j is of class C ρ with modulus c p , • u k,p j is a c p ω F S -p.s.h.
Then for each p ≥ 1, there is a probability measure

σ p = m k=1 d k,p j=1 π k,p (dd c u k,p j + ω F S )
on X p . By Theorem 0.2.1 and Remark 2.3.7,

d k,p j=1 (dd c u k,p j +ω F S ) is a moderate measure on PH 0 (2) (X, L p k ) when c p ≤ 1/c p n for a suitable constant c > 1, ∀1 ≤ k ≤ m, p ≥ 1. The probability measure on P X σ = ∞ p=1 σ p = ∞ p=1 m k=1 d k,p j=1 π k,p (dd c u k,p j + ω F S ) (1)
is said to be generated by a family of (c p , ρ)-functions {u k,p j } on {PH 0 (2) (X, L p k )}. Here is our main theorem.

Theorem 0.3.1. Let (X, ω) be a compact Kähler manifold of dimension n, (L k , h k ), 1 ≤ k ≤ m ≤ n, be m singular Hermitian holomorphic big line bundles on X. The metric h k is continuous outside a proper analytic sub- set A k ⊂ X, c 1 (L k , h k ) ≥ ω
on X for some constant > 0, and A 1 , ..., A m are in general position. Let 0 < ρ < 1. Then there exists a constant c > 1 which depends only on X, L k , ρ with the following property: If σ is the probability measure on P X generated by a family of (1/c p n , ρ)-functions {u j k,p } on {PH 0

(2) (X, L p k )} defined by (2), then for almost every {S p } p≥1 ∈ P X with respect to σ, we have in the weak sense of currents as p → ∞ on X,

1 p m [S p = 0] → c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ).
Remark 0.3.2. When all u j k,p ≡ 0, then σ p are the Lebesgue measures σ 0 p on X p and we obtain [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF]Theorem 1.2]. In addition, the constant c is independent of the choices of singular metrics on the big line bundles.

When the metrics h k are all Hölder with singularities, we can also extend the result in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] about the estimate of the speed of the above convergence associated to the moderate measures defined by [START_REF] Alexander | Projective capacity, In Recent developments in several complex variables[END_REF]. This theorem is also a generalization of Theorem 0.2.2.

Theorem 0.3.3. We keep the notations and the hypotheses of Theorem 0.3.1. Suppose, moreover, that h k is Hölder with singularities along A k , 1 ≤ k ≤ m. Then there exist a positive constant ξ which depends only on m, and another positive constant C which depends on X, (L 1 , h 1 ), ..., (L m , h m ) with the following property: Given any sequence of positive numbers {λ p } ∞ p=1 with the following conditions lim inf p→∞ λ p log p > (1 + ξn)C and lim p→∞ λ p p n = 0, there exist subsets E p ⊂ X p such that for all p sufficiently large, (i)

σ p (E p ) ≤ Cp ξn exp(- λ p C ),
(ii) for any point S p ∈ X p \ E p and any

(n -m, n -m)-form φ of class C 2 , 1 p m [S p = 0] - m k=1 c 1 (L k , h k ), φ ≤ Cλ p p φ C 2 .
Remark 0. Note that in Theorem 0.3.1 and all other equiditribution theorems studied in this thesis, the limit of the considered convergence sequence cannot be an arbitrarily given positive closed current. For example, there is a condition that c 1 (L k , h k ) ≥ ω in Theorem 0.3.1. However, it is possible when X = P n , L = O(1), m = 1 with a probability measure σ chosen properly.

Theorem 0.3.5. Given any positive closed current T of bidegree (1, 1) with mass 1 on P n . Let d p = dim H 0 (P n , O(p)) -1. Then there exists a family of smooth probability measures σ p = (ω F S + dd c u p ) dp on PH 0 (P n , O(p)) for some smooth real functions u p with the following property: For almost every S = (S p ) ∈ p≥1 PH 0 (P n , O(p)) with respect to σ = p≥1 σ p , we have in the weak sense of currents as p → ∞ on P n ,

1 p [S p = 0] → T.

Introduction. (French version)

Distribution des zéros de polynômes aléatoires est un sujet classique. Waring [START_REF] Todhunter | A history of the mathematical theory of probability[END_REF] a utilisé une méthode probabiliste pour déterminer le nombre de zéros imaginaires d'un polynôme algébrique. Des recherches plus rigoureuse et systématique a commencé avec le papier de Bloch-Pólya [START_REF] Bloch | On the roots of certain algebraic equations[END_REF]. Ils ont donné un ordre du nombre espérance de racines réelles de certaines équations polynômes aléatoires. Kac [START_REF] Kac | On the average number of real roots of a random algebraic equation II[END_REF] a obtenu une distribution exacte des racines réelles pour les polynômes aléatoires avec des coefficients de distribution normaux. Résultats sur polynômes avec d'autres distributions ou paramètres ont été étendues notamment. Voir [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF][START_REF] Bleher | Correlation between zeros of a random polynomial[END_REF][START_REF] Bloom | Zeros of random polynomials on C m[END_REF][START_REF] Bharucha-Reid | Random polynomials, Probability and Mathematical Statistics[END_REF][START_REF] Erdős | On the distribution of roots of polynomials[END_REF][START_REF] Kac | On the average number of real roots of a random algebraic equation II[END_REF][START_REF] Shepp | The complex zeros of random polynomials[END_REF] et références qui y sont pour plus de résultats.

Un théoréme classique de Hammersley [START_REF] Hammersley | The zeros of a random polynomial[END_REF] affirme que les zéros normalisés de gaussien complexe polynômes aléatoires de degré élevé ont tendance à accumuler sur le cercle unité avec une distribution uniforme. Un résultat analogue est valable pour SU (2) polynômes aléatoire dont la distribution espérance de zéros est uniforme sur P 1 (cf. [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF]Appendix C]). Les résultats généraux sur les sections holomorphes de fibrés en droites positive associée aux mesures de Lebesgue ont été obtenues par Shiffman-Zelditch [START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF]. Grosso modo, presque partout séquence de courants normalisés définis par sections holomorphes converge faiblement vers la forme Kählerienne par rapport aux mesures de Lebesgue. Notez que le résultat couvre la propriété équidistribution de SU (2) polynômes aléatoire lorsque X = P 1 et L = O(1). Dinh-Sibony [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] étendu la propriété équidistribution dans le cas de mesures générales et obtenu une bonne estimation de la vitesse de convergence. L'approche de potentiel théorique de Fornaess-Sibony [START_REF] Fornaess | Complex dynamics in higher dimensions II[END_REF] a été utilisé dans les travaux précédent.

Motivation et applications

Les mesures générales dans les théorèmes de équidistribution de Notre motivation de la premiére partie est de construire une grande famille des mesures modérées singulières qui satisfait la propriété équidistribution. Nous montrons que les courants normalisés définis par les zéros de sections aléatoires de puissances élevées de un fibré en droites positive sur une variété projective convergent faiblement à la forme de courbure associée à certaines familles de mesures modérées singulières. Notre méthode suit les techniques d'estimations exponentielles pour les fonctions plurisousharmoniques par Dinh-Nguyên-Sibony [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF], voir la section 2.3.

Dans la deuxième partie, nous sommes motivés à généraliser la propriété équidistribution dans le cas de plusieurs fibrés en droites gros dotées de métriques singulières. Nous suivons les approches de Dinh-Sibony [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF], Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] et [54, section 2]. Pour atteindre nos résultats, nous établissons des estimations pointues sur les constantes liées à la capacité Alexander-Dinh-Sibony sur les espaces multi-projectives [START_REF] Shao | Equidistribution on big line bundles with singular metrics for moderate measures[END_REF], voir la section 3.4.

La capacité Alexander-Dinh-Sibony est étroitement liée à la fonction globale extrémal (cf. [23, A.2] et [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Section 5]). Le L 2 -méthode joue aussi un rôle important dans l'étude des propriétés équidistribution. Par exemple, nous appliquons L 2 -méthode pour explorer le comportement asymptotique des fonctions du noyau de Bergman. Par conséquent, nous pouvons obtenir des théorèmes de équidistribution avec une bonne vitesse de convergence (cf. [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF], [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF], [START_REF] Demailly | A numerical criterion for very ample line bundles[END_REF] etc). La propriété équidistribution fournit un outil pour étudier la conjecture Unique Ergodicité par Rudnick-Sarnak, voir [START_REF] Rudnick | The behavior of eigenstates of arithmetic hyperbolic manifolds[END_REF], [START_REF] Holowinsky | Mass distribution for Hecke eigenforms[END_REF], [START_REF] Rudnick | On the asymptotic distribution of zeros of modular forms[END_REF]. Il peut aussi être appliquée à l'étude du chaos quantique en physique statistique (cf. [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF], [START_REF] Nonnenmacher | Chaotic eigenfunctions in phase space[END_REF] etc).

Partie I

Nous donnons le fond de base, afin d'introduire les principaux théorèmes. Pour chaque espace vectoriel complexe V de dimension finie, soit ω F S la forme Kählerienne standard induite par la métrique de Fubini-Study sur son espace projectif P(V ) normalisé par P(V ) ω Soit L p le p-produit tensoriel de L avec la métrique h p := h ⊗p . Désignons par H 0 (X, L p ) l'espace de toutes les sections holomorphes de L p . Soit PH 0 (X, L p ) l'espace projectif associé à ω F S comme sa forme Fubini-Study normalisée. Laissez n p := dim PH 0 (X, L p ). Définissez un produit intérieur sur H 0 (X, L p ) de la façon suivante s, s := X h p (s, s )ω n , ∀s, s ∈ H 0 (X, L p ). Nous pouvons choisir une base orthonormée {s pj } np j=0 par rapport au produit intérieur. Pour toute s p ∈ PH 0 (X, L p ), écrirez

s p := np j=0 a pj s pj .
Ainsi, la section s p peut être identifié avec les coefficients (a p0 , ..., a pnp ). La notion de sections holomorphes aléatoires découle des variables aléatoires [a p0 , ..., a pnp ] avec une distribution induites par la mesure de probabilité ω np F S sur PH 0 (X, L p ). Soit [s p = 0] le courant défini par l'ensemble zéro de s p , où s p ∈ PH 0 (X, L p ). Laissez P X := p≥1 PH 0 (X, L p ).

Nous avons besoin de la notion de mesures modérées localement qui a été introduite par Dinh-Sibony [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF]. La notion découle d'un résultat classique [START_REF] Hörmander | An introduction to complex Anaysis in Several Variables[END_REF]Theorem 4.4.5].

Définition. Soit X une variété complexe. Une mesure positive mu est modérée localement si pour tout ouvert U ⊂ X, un sous-ensemble compact K ⊂ U et toute la famille compacte U de q.p.s.h. fonctions sur U , il y a des constantes α > 0, c > 0 tel que

K exp(-αφ)dµ ≤ c, ∀φ ∈ U.
Dinh-Nguyên-Sibony [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF] ont prouvé que la mesure de l'quilibre de endomorphisme de P n est modérées localement. Dans la thèse, nous traitons des mesures modérées dans une variété compacte Kählerienne X de dimension n avec une forme Kählerienne ω fixe tel que ω n est la forme de volume de probabilité. Considérons une mesure positive µ sur X, µ est dit PLB si tout les q.p.s.h. fonctions sont µ-intégrable. Lorsque dim X = 1, µ est PLB si et seulement si elle admet un potentiel borné locale [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF]. Laisser F := {φ q.p.s.h. on X :

dd c φ ≥ -ω, max X φ = 0}.
L'ensemble F est compact dans L p (X) et bornée dans L 1 (µ) quand µ est une mesure PLB par Théorème 1.2.33 et Proposition 1.2.47. Rappelons que toute q.p.s.h. fonctions sur X sont cω-p.s.h. pour une constante c ≥ 0 et bornée uniformément d'en haut. Ensuite, l'étude de q.p.s.h. fonctions peuvent être réduits à l'étude de la famille F. Soit σ p la mesure probabilité de Lebesgue sur PH 0 (X, L p ) et σ la mesure prodiut de celles-ci sur P X . Shiffman-Zelditch [START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF] a prouvé que la séquence des courants Théorème. Dans le cadre du théorème de Shiffman-Zelditch, les mesures de probabilité ω np F S sur PH 0 (X, L p ) sont remplacés par des mesures de probabilité PLB σ p avec les conditions suivantes:

{ 1 p [s p = 0]} converge faiblement vers ω pour σ-presque partout (s p ) ∈ P X . C'est-à-dire, 1 p [s p = 0] → ω pour σ-presque partout (s p ) ∈ P X .
∞ p=1 ∆ p (pt) < ∞, ∀t > 0, R p p → 0. Alors la convergence faible 1 p [s p = 0] → ω est toujours vrai pour σ-presque partout (s p ) ∈ P X .
Les ingrédients clés pour prouver le théorème sont quelques méthodes pluripotentes, notion de transformations méromorphes et dd c -méthode. Ils ont aussi construit une mesure singulière à coefficients réels qui satisfait la propriété équidistribution (cf. [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Corollaire 7.4]).

Lorsque les mesures de Lebesgue dans le résultat de Shiffman-Zelditch sont remplacées par des mesures modérées avec Hölder potentiels (voir sections 2.3, 2.4), nous avons notre théorème principal de la manière suivante qui donne une grande famille de mesures modérées singulières qui satisfait la propriété équidistribution. Il peut être considéré comme une perturbation des mesures standard induites par la métrique Fubini-Study.

Théorème. Soit L un fibré en droites ample sur une variété projective X de dimension n et 0 < ρ < 1 un exposant. Alors il existe une constante c = c(X, L, ρ) > 1 avec la propriété suivante. Pour chaque p ≥ 1, 1 ≤ j ≤ n p , soit u p,j : PH 0 (X, L p ) → R une fonction et ξ p , p > 0 deux nombres tels que: (i) u p,j est de classe C ρ avec module ξ p , ∀1 ≤ j ≤ n p ; (ii) u p,j est p ω F S -p.s.h., ∀1 ≤ j ≤ n p ; (iii)

ξ p ≤ 1/c p n , p ≤ 1/c p n . Soit σ p = (dd c u p,1 + ω F S ) ∧ (dd c u p,2 + ω F S ) ∧ • • • ∧ (dd c u p,np + ω F S )
la mesure de probabilité sur PH 0 (X, L p ). Doter P X avec la mesure produit σ = p≥1 σ p . Ensuite, pour presque partout s = (s p ) ∈ P X par rapport à σ, la séquence des courants { 1 p [s p = 0]} converge faiblement vers ω.

Le résultat suivant donne une vitesse de convergence pour l'équidistribution dans le théorème 0.2.1.

Théorème. Dans le cadre du théorème 0.2.1, il existe des sous-ensembles E p ⊂ PH 0 (X, L p ) et une constante positive C dépendant seulement X, L tel que pour tout p suffisamment grand, nous avons

σ p (E p ) ≤ C p 2 | 1 p [s p = 0] -ω, ψ | ≤ C log p p ψ C 2 , pour tout point s p ∈ PH 0 (X, L p ) \ E p et des (n -1, n -1)-forme ψ de la classe C 2 .

Partie II

Deux questions se posent naturelles après avoir résolu la première partie:

1. Il est possible de considérer le cas de plusieurs fibrés en droites générales, par exemple fibrés en droites gros?

2. On peut étendre les zéros de sections d'un seul fibré en droites aux zéros communs des sections de plusieurs ceux?

Heureusement, les questions sont résolues dans certaines directions dans le contexte de mesures de probabilité de Lebesgue. Ces problèmes déquidistribution et des vitesses de convergence des sections holomorphes de fibrés en droites singuliers holomorphes hermitiennes ont été intensivement exploré récemment. métrique sur le fibré en droites dans ce dernier ouvrage est lisse.

Dans cette partie, les courants normalisés sont définis par les zéros communs de m tuples des sections aléatoires de puissances élevées de m fibrés en droites gros hermitiens singulier sur une variété compacte Kählerienne. Nous montrons qu'ils distribuent asymptotiquement vers le produit des courants de courbure des métriques singulières associées à certaines mesures modérées. Par conséquent, nous généralisons les deux théorèmes principaux [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] et [START_REF] Shao | Equidistribution of zeros of random holomorphic sections for moderate measures[END_REF].

Notre méthode suit l'approche de Coman-Marinescu-Nguyên [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF]. Adapter leur travail, nous montrons que les intersections des courants d'intégration le long de sous-variétés sont bien définis presque partout par rapport à un produit fini de mesures modérées (voir Section 3.3). De plus, leur stratégie d'utilisation de la théorie déquidistribution Dinh-Sibony nous conduit à obtenir une estimation de la vitesse de convergence (voir Section 3.4 et Section 3.5). Ensuite, nous combinons ce qui précède à l'analyse technique des mesures modérées pour atteindre nos résultats, dans lequel notre travail de base est dans les estimations des constantes sur les espaces multi-projectives (voir section 3.4).

Nous commençons avec les fonds de base de cette partie. Soit X une variété compacte Kählerienne de dimension n avec une forme Kählerienne lisse ω. Rappelons qu'un fibré en droites singulier hermitienne holomorphe (L, h) est un fibré en droites holomorphe L avec une métrique hermitienne qui est donnée dans toute trivialisation par un poids fonction e -ϕ tels que ϕ est localement intégrable (cf. Définition 1.3.13). Soit c 1 (L, h) son courant de courbure qui représente la première classe de Chern. Pour être précis, si e L est un cadre holomorphe de L sur un sous-ensemble ouvert U ⊂ X,

alors |e L | 2 h = e -2ϕ , c 1 (L, h) = dd c ϕ sur U . Ici d = ∂ + ∂, d c = 1 2πi (∂ -∂).
Le cas où c 1 (L, h) ≥ 0 comme un courant est particulièrement intéressant. On dit qu'un fibré en droites holomorphe L est gros si elle admet une métrique singulière h avec c 1 (L, h) ≥ ω pour une constante > 0 (cf. Théorème 1.3.23).

Soit (L

k , h k ), 1 ≤ k ≤ m ≤ n,
d k,p = O(p n ).
Maintenant, nous considérons l'espace multi-projective

X p := PH 0 (2) (X, L p 1 ) × ... × PH 0 (2) (X, L p m )
doté d'une mesure de probabilité σ p pour chaque p ≥ 1. Soit π k,p : X p → PH 0 (2) (X, L p k ) les projections naturelles. Désignons par [S = 0] le courant défini par l'ensemble zéro de S ∈ H 0 (X, L p k ). Soit [S p = 0] := [S p1 = 0] ∧ ... ∧ [S pm = 0], ∀S p = (S p1 , ..., S pm ) ∈ X p , chaque fois qu'il est bien définie. Laisser

P X := ∞ p=1 X p .
Il est un espace de probabilité avec la mesure produit σ = ∞ p=1 σ p .

On note dist la distance sur X induite par la forme Kählerienne ω. Soit φ : U → [-∞, ∞) une fonction sur un sous-ensemble ouvert U ⊂ X, A ⊂ X un sous-ensemble analytique approprié. Conformément à la terminologie dans [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF], φ est appelé Hölder avec singularités le long A s'il y a des constantes positives c, δ et 0 < ν ≤ 1 satisfaisant qui

|φ(z) -φ(w)| ≤ cdist(z, w) ν min{dist(z, A), dist(w, A)} δ
pour tout z, w ∈ U \ A. Une métrique singulière h de L est définie comme Hölder avec singularités le long A si chaque poids local h est Hölder avec singularités le long de A. Pour motivations ainsi que des exemples de ces mesures, voir [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF].

Le multi-espace projectif X p dans [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] est équipé de la mesure de probabilité σ 0 p qui est le produit des mesures de Lebesgue induites par les mesures Fubini-Study sur les composants. Dans cette partie, nous définissons des mesures modérées singulières σ p comme des perturbations de σ 0 p sur X p . Pour chaque

p ≥ 1, 1 ≤ k ≤ m, 1 ≤ j ≤ d k,p , laissez u k,p j : PH 0 (2) (X, L p k ) → R
(2) (X, L p k ) quand c p ≤ 1/c p n pour un constant approprié c > 1, ∀1 ≤ k ≤ m, p ≥ 1. La mesure de probabilité sur P X σ = ∞ p=1 σ p = ∞ p=1 m k=1 d k,p j=1 π k,p (dd c u k,p j + ω F S )
est dit être généré par une famille de (c p , ρ)-functions {u k,p j } sur {PH 0 (2) (X, L p k )}.

Voici notre théorème principal.

Théorème. Soit (X, ω) une variété compacte Kählerienne de dimension n,

(L k , h k ), 1 ≤ k ≤ m ≤ n,
p → ∞ sur X, 1 p m [S p = 0] → c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ).
Remarque. Quand tout u j k,p ≡ 0, alors σ p sont les mesures de Lebesgue σ 0 p sur X p et nous obtenons [14, Théorème 1.2]. En outre, la constante c est indépendante des choix de métriques singulières sur les fibré en droites gros.

Lorsque la métrique h k sont tous Hölder avec singularités, nous pouvons aussi étendre le résultat dans [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] sur l'estimation de la vitesse de la convergence ci-dessus associée aux mesures modérées définies par [START_REF] Alexander | Projective capacity, In Recent developments in several complex variables[END_REF]

σ p (E p ) ≤ Cp ξn exp(- λ p C ), (ii) pour tout point S p ∈ X p \ E p et des (n -m, n -m)-forme φ de classe C 2 , 1 p m [S p = 0] - m k=1 c 1 (L k , h k ), φ ≤ Cλ p p φ C 2 .
Remarque. La principale différence entre Théorème 0. Notez que dans Théorème 0.3.1 et toutes les autres théorèmes équiditribution étudiés dans cette thèse, la limite de la séquence de convergence ne peut pas être un courant positif fermé arbitrairement donné. Par exemple, il y a une condition que c 1 (L k , h k ) ≥ ω dans Théorème 0.3.1. Cependant, il est possible lorsque X = P n , L = O(1), m = 1 avec une mesure de probabilité σ choisi correctement.

Théorème. Compte tenu de tout courant positif fermé T de bidegré (1, 1) de la masse 1 sur P n . Soit d p = dim H 0 (P n , O(p)) -1. Alors il existe une famille de mesures de probabilité lisses σ p = (ω F S + dd c u p ) dp sur PH 0 (P n , O(p)) pour certaines fonctions réelles lisses u p avec la propriété suivante: Pour presque partout S = (S p ) ∈ p≥1 PH 0 (P n , O(p)) par rapport à σ = p≥1 σ p , nous avons dans le sens faible des courants comme p → ∞ sur P n ,

1 p [S p = 0] → T.

Chapter 1

Preliminaries.

This chapter supplies the definitions, notations and background needed in the thesis. We mainly introduce currents, plurisubharmonic functions and Hermitian holomorphic line bundles. This chapter follows basically Demailly's book [START_REF] Demailly | Complex analytic and differential geometry[END_REF]. See [START_REF] Demailly | Complex analytic and differential geometry[END_REF], [START_REF] Dinh | Introduction to the theory of currents[END_REF], [START_REF] Griffiths | Principles of algebraic geometry[END_REF], [START_REF] Huybrechts | Complex geometry: an introduction[END_REF], [START_REF] Klimek | Pluripotential theory[END_REF] and [START_REF] Ma | Holomorphic Morse inequalities and Bergman kernels[END_REF] for complete references.

1.1 Currents.

The notion of currents was introduced by Georges de Rham. It generalizes the notion of distributions.

Currents on complex manifolds

Let X be a smooth complex manifold of dimension n. A differential form φ of bidegree (p, q) is a section of p,q T X. In a local coordinate, we denote by (dz 1 , ..., dz n ) the corresponding basis of the cotangent space at a point. Let (x 1 , ...x 2n ) be the local real coordinate of (z 1 , ..., z n ). We can write

φ(z) = |I|=p,|J|=q φ I,J dz I ∧ dz J ,
where φ I,J are complex smooth functions, dz I = dz i 1 ∧ ... ∧ dz ip when I = (i 1 , ..., i p ) and dz J = dz j 1 ∧ ... ∧ dz jq when J = (j 1 , ..., j q ). Let D p,q (X) be the set of all differential forms of bidegree (p, q) with compact support. We introduce a topology on D p,q (X). If Ω is a subset of X, we denote D p,q (Ω) the space of all elements φ ∈ D p,q (X) with compact support in Ω. Let {Ω j } ∞ j=1 be a sequence of relatively compact open subsets of X with Ωj ⊂ Ω j+1 for every j, and ∪ ∞ j=1 Ω j = X. To every compact subset K in a local coordinate and every integer l ∈ N, we define a semi-norm

φ l,K := sup z∈K max |I|=p,|J|=q |α|≤l D α φ I,J (z) ,
where α = (α 1 , ..., α 2n ) run over N 2n and D α = ∂ |α| /∂x α 1 1 ...∂x α 2n 2n is a derivation of order |α|. Since X is assumed to be separable, we can equip the space D p,q ( Ω) with the topology induced by a countable set of semi-norm • l,K with K contained in Ω. Such topology is called the topology of the uniform convergence of coefficients and all the derivatives. Then D p,q ( Ω) is a Fréchet space with the topology. Hence we equip D p,q (X) with the topology of the strict inductive limit of the spaces D p,q ( Ωj ). Definition 1.1.1. A current of bidegree (p, q) (or (p, q)-current) on X is a linear continuous form T : D n-p,n-q (X) → C. Let φ be a form in D n-p,n-q (X), the value of T at φ is denoted by T, φ . The form φ is called a test form.

A (p, q)-current can also be called a current of bidimension (n -p, n -q). Definition 1.1.2. A sequence of (p, q)-currents {T j } converges weakly (or converges in the sense of currents) to a current T if T j , φ → T, φ for any test form φ ∈ D n-p,n-q (X).

Denote by D p,q (X) the set of all (p, q)-currents. It is the dual of D p,q (X). Note that a (p, q)-current T can be regarded as a form with distribution coefficients:

T = |I|=p,|J|=q T I,J dz I ∧ dz J ,
where T I,J are distributions.

Example 1.1.3. A form ψ of bidegree (p, q) is a (p, q)-current T ψ via exterior product as follows.

T ψ , φ = X ψ ∧ φ, ∀φ ∈ D n-p,n-q (X).

Example 1.1.4. Let Y be a closed complex submanifold of X of dimension p, then we can define the current of integration over Y by

[Y ], φ = Y φ, ∀φ ∈ D p,p (X). If X = C n and Y = {z 1 = ... = z n-p = 0}, then we have [Y ] = ( i p 2 ) p δ 0 (z 1 , ..., z n-p ) ⊗ 1(z I )dz I ∧ dz I ,
where I = (n -p + 1, ..., n), δ 0 is the Dirac measure at the origin of the space {z I = 0}.

The above examples explain the terminology of dimension and degree used for a current.

Since a current is a continuous form in the topology defined above, we have the following property.

Proposition 1.1.5. Let T be a (p, q)-current on X and K ⊂ X be a compact subset. Then there exist a positive integer l and a positive constant

C such that | T, φ | ≤ C φ l,K ,
for every φ ∈ D n-p,n-q (X) with supp(φ) ⊂ K.

Definition 1.1.6. If the integer l in Proposition 1.1.5 can be chosen independently of K, the current T is called of finite order. The smallest integer l satisfying the property is called the order of T .

It is easy to see that the currents in the above two examples are both of order 0. Definition 1.1.7. The support of a (p, q)-current T is defined to be the smallest closed subset supp(T ) of X such that T vanishes on X \ supp(T ). That is to say, T, φ = 0 for every test form φ ∈ D n-p,n-q (X \ supp(T )).

Note that the current

[Y ] in Example 1.1.4 has support Y .

Operators on currents

There are several operators on currents which are similar to those on differential forms. Recall that d = ∂ + ∂. In a local coordinate, we have

∂φ : = I,J k i=1 ∂φ I,J ∂z i dz i ∧ dz I ∧ d∂dz J , ∂φ : = I,J k i=1 ∂φ I,J ∂z i dz i ∧ dz I ∧ d∂dz J . Define d c := 1 2πi (∂ -∂). It is a real operator. It follows that dd c = i π ∂ ∂.
The normalization of d c is convenient for many purposes. For example, it simplifies the Lelong-Poincaré formula. We define the exterior derivative of currents as follows. Let T be a (p, q)-current, the (p + 1, q)-current ∂T and the (p, q + 1)-current ∂T is defined by ∂T, φ : = (-1) p+q+1 T, ∂φ , ∂T, φ : = (-1) p+q+1 T, ∂φ .

The current dT can be defined similarly. The maps T → dT , T → ∂T and T → ∂T are continuous for the topology of currents. T is called closed if dT = 0. In particular, we have dd c T, φ = T, dd c φ . An application of the classical Stokes' formula yields

d[Y ] = (-1) n-p+1 [∂Y ],
where ∂Y denotes the boundary of the complex manifold Y . Definition 1.1.8. Let T be a (p, q)-current and ψ be a form of bidegree (p 1 , q 1 ). We define the wedge product T ∧ ψ by

T ∧ ψ, φ := T, φ ∧ ψ , ∀φ ∈ D n-p-p 1 ,n-q-q 1 (X).
Let X be another smooth complex manifold of dimension n . Let f : X → X be a holomorphic map which is proper on the support of T . We introduce the notions of direct image and inverse image of a current. Definition 1.1.9. The direct image of T by f is defined by

f (T ), φ := T, f (φ) , ∀φ ∈ D n-p,n-q (X ).
The current f (T ) is of bidegree (n -n + p, n -n + q) and of bidimension (n -p, n -q). So the operator f preserves the dimension of currents.

Proposition 1.1.10. The direct image operator f is continuous. The support of f (T ) is contained in f (suppT ). Moreover, it commutes with the exterior derivative, i.e.

d(f (T )) = f (dT ).
Assume in addition that f is a submersion. Let φ be a form of bidegree (p, q) (even with L 1 loc coefficients) with the condition that f is proper. Then f φ is a form of bidegree (n -n + p, n -n + q). Moreover, f φ is calculated by integration over the fibers of f , i.e.

f φ(w) = z∈f -1 (w) φ(z).
So we can define the inverse image of a current. Definition 1.1.11. Let T be a (p, q)-current on X . The inverse image of T by f is defined by

f (T ), φ := T, f (φ) , ∀φ ∈ D n-p,n-q (X).
The operator f preserves the degree of currents. For example, we have

f [Y ] = [f -1 (Y )].
Assume that the support of T or a smooth function g on X is compact, We can define the convolution T g by

T g := I,J T I,J g,
where T I,J are the distribution coefficients of T . A convolution of a distribution and a smooth function with compact support is a smooth function, which is a direct consequence of Taylor's formula with compactness condition. So T g is a smooth form. Hence we can approximate currents by smooth forms in the weak sense. Refer to [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] for more information about convolutions of distributions or currents.

De Rham theorem

By an analogous argument, we can define the following cohomology group associated to currents

H p c (X, R) := real valued closed p-currents on X real valued exact p-currents on X .
When X is paracompact, it follows from the theory of sheaf cohomology [16, Chapter IV] that the de Rham cohomology group

H p (X, R) is isomorphic to H p c (X, R).
In particular every closed p-current is cohomologous to a closed smooth p-form.

Denote by H p,q (X, C) the Dolbeault cohomology group. If X is a complex projective space P n of dimension n, by Hodge decomposition theorem [16, Chapter VI], we deduce that H p,q (P n , C) = 0, for p = q, H p,p (P n , C) C.

We will see later that the generator of H p,p (P n , C) is the p times tensor of the Fubini-Study form.

It is easy to calculate the Dolbeault cohomology groups of product spaces of several complex projective spaces by Künneth formula. The above results will be used in cohomological arguments in the proofs of our main theorems.

Positive closed currents

The notion of positive closed currents was introduced by Pierre Lelong [START_REF] Lelong | Fonctions plurisousharmoniques et formes différentielles positives[END_REF] in 1957. It generalized analytic subsets in complex manifolds, since every analytic subset can be associated to a current by integration over its set of regular points and all such currents are proved to be positive closed. It has many applications in complex analysis (especially pluripotential theory) and dynamical systems in higher dimensions. Definition 1.1.12. A (p, p)-form φ is called positive if it is equal to a finite combination of forms (iα 1 ∧ ᾱ1 ) ∧ ... ∧ (iα p ∧ ᾱp ) at each point, where α j are (1, 0)-forms. The form φ is said to be weakly positive if φ ∧ ψ is positive for any positive (n -p, n -p)-form ψ. A (p, p)-current T is said to be positive (resp. weakly positive) if T, φ ≥ 0 for every weakly positive (resp. positive) test form of bidegree (n -p, n -p).

Note that all positive closed currents are real by duality. The two notions of positivity and weakly positivity coincide when p = 0, 1, n -1, n. Moreover, the operators f and f preserve the positivity. Proposition 1.1.13. Let T be a positive (p, p)-current. Then T is of order zero. The coefficients T I,J are distributions of order zero, i.e. complex measures.

Proof. We give a sketch proof here. If T is a distribution and it is positive, then it can be extended to a positive linear functional on the space of complex continuous functions. Hence T is a positive measure. Note that the real vector space p,p (C n , R) admits a basis consisting of positive forms. We can choose such a basis {ψ J } for n-p,n-p (C n , R). Then we set {φ I } to be the basis for p,p (C n , R) which is dual to {ψ J }. So T = T I φ I . If g is a non-negative test function, we have

T I (g) = T (gφ I ) ≥ 0.
By the previous argument, T I is a positive measure. Then the coefficients of T are complex measures if it is expressed in terms of an arbitrary basis. Hence T is of order 0.

Definition 1.1.14. Let β = i∂ ∂ z 2 , define σ T := 1 2 n-p (n -p)! T ∧ β n-p .
Then σ T is said to be the trace measure of T .

Recall that a Hermitian metric on X is a smooth positive Hermitian form h = h jk dz j ⊗ dz k . The form ω = i h jk dz j ∧ dz k is the associated positive (1, 1)-form. The following is a famous theorem due to Wirtinger. Then |α| ≤ 1 and the equality holds if and only if Y is a complex analytic submanifold of X.

We have defined currents of integration over a complex submanifold. This notion can be also defined over analytic subsets. The following Lelong's theorem claims that such currents are positive closed ones.

[Y ], φ = reg(Y ) φ, ∀φ ∈ D n-p,n-p (X).
Then the current [Y ] is well-defined and positive closed.

The theorem shows that positive closed currents extend the notion of analytic subsets in complex manifolds. Moreover, the volume of reg(Y ) near singular points is locally bounded. Now we introduce a support theorem which is useful in latter chapters. Definition 1.1.17. A current T is called normal if T and dT are both of order 0.

Every positive closed current is normal.

Theorem 1.1.18. Let T be a normal (p, p)-current. If the support of T is contained in an analytic subset V of dimension less than p, then T = 0.

Another useful support result is the following Theorem 1.1.19. Let V be an analytic subset of X with global irreducible components V j of pure dimension p. Assume that T is a closed (p, p)-current of order 0 with support contained in V . Then T can be written as the form λ j [V j ] where λ j ∈ C.

Plurisubharmonic functions

In this section, we introduce the notions of plurisubharmic functions and their basis properties. The positive closed (1, 1)-currents can be studied by plurisubharmic functions locally. Then we define Monge-Ampère operators. Some continuity properties of Monge-Ampère operators are needed in our thesis. The quasi-plurisubharmic functions can be defined in compact complex manifolds. We conclude this section with the notion of moderate measures with respect to a compact family of quasi-plurisubharmic functions.

Plurisubharmonic functions

Plurisubharmonic (p.s.h. for short) functions were introduced by Lelong and Oka in 1942. They play a central role in the study of complex analysis. For example, there are analogies between plurisubharmonicity and pseudoconvexity. The p.s.h. functions are the natural counterpart of the subharmonic functions of one variable. Note that, to some extend, the class of subharmonic functions of several complex variables is quite large. The p.s.h. functions are precisely the ones among subharmonic functions which are invariant under compositions with biholomorphic maps. That is why p.s.h. functions make sense on complex manifolds.

Definition 1.2.1. Let Ω be an open subset of C n . A function u : Ω → [-∞, ∞) is said to be plurisubharmonic function if it is upper semicontinuous, not identically ∞ on each connected component of Ω and satisfy u(z) ≤ 1 2π 2π 0 u(z + we iθ )dθ, for each z ∈ Ω and w ∈ C n such that {z + wλ : λ ∈ C, |λ| ≤ 1} ⊂ Ω.
The set of p.s.h. functions (resp. subharmonic functions) on Ω is denoted by P sh(Ω) (resp. Sh(Ω)). If in addition u ∈ C 2 (Ω), then it is easy to see that u ∈ Sh(Ω). Most of properties for subharmonic functions in R 2n can carry over to the case of plurisubharmonic functions in C n . Note that the semicontinuity implies that p.s.h. functions are locally bounded from above. Denote by {χ } the standard smoothing kernels [16, 2.D.3]. The following result is the approximation theorem for p.s.h. functions.

Theorem 1.2.2. Let u ∈ P sh(Ω). Set Ω := {z ∈ Ω : dist(z, ∂Ω) > } for > 0. Then u χ ∈ C ∞ (Ω ) ∩ P sh(Ω ).
Moreover, the family of {u χ } is non decreasing, and for each z ∈ Ω, we have

lim →0 u χ (z) = u(z).
In general, we can not find a decreasing sequence {u j } ⊂ C ∞ (Ω) ∩ P sh(Ω) which converges pointwise to u. Proposition 1.2.5. If u ∈ P sh(C n ) is bounded above, then u is constant.

Theorem 1.2.6. [42, Theorem 2.9.12] Let Ω (resp. Ω ) be an open subset of

C n (resp. C n ). If u ∈ P sh(Ω) and f : Ω → Ω is a holomorphic map, then the composition u • f is a p.s.h. function in Ω . Conversely, u ∈ P sh(Ω) if and only if u • g is subharmonic in g -1 (Ω) for every complex linear isomorphism g : Ω → Ω. Proposition 1.2.7. Let Ω be an open subset of C n . (i)
The set P sh(Ω) is a convex cone. That is to say, if a > 0, b > 0 and u, v ∈ P sh(Ω), then au + bv ∈ P sh(Ω).

(ii) If {u j } is a decreasing sequence of p.s.h. functions in a connected open subset Ω, then u := lim j→∞ u j is p.s.h. or u ≡ -∞.

(iii) If u : Ω → R and the sequence of p.s.h. functions in Ω converges uniformly to u on compact subsets of Ω, then u ∈ P sh(Ω).

(iv) Let u 1 , ..., u m ∈ P sh(Ω). Let χ : R m → R be a convex function and χ(t 1 , ..., t m ) is non decreasing in each variable t j . Then χ(u 1 , ..., u m ) ∈ P sh(Ω).

In particular u 1 + ... + u m , max{u 1 , ..., u m }, log(e u 1 + ... + e um ) are p.s.h. functions.

Example 1.2.8. Since log |z| is a subharmonic function on C, log |f | ∈ P sh(X), for any holomorphic function f on X. For any holomorphic functions f j and

α j ≥ 0, 1 ≤ j ≤ m, log(|f 1 | α 1 + ... + |f m | αm ) ∈ P sh(X).
Let {u α } α∈Λ be a family of upper semicontinuous functions from Ω to [-∞, ∞). Assume that {u α } are locally uniformly bounded from above. Then we can define the upper envelope by u := sup u α . Note that u may not be upper semicontinuous, so we consider its upper semicontinuous regularization as follows,

u (z) = lim →0 sup B(z, ) u ≥ u(z).
It is easy to see that u is upper semicontinuous. The following property is called Choquet's lemma.

Lemma 1.2.9. Every family {u α } admits a countable subfamily

{v j } = {u α j } with its upper envelope v satisfying v ≤ u ≤ u = v .
Proposition 1.2.10. Let {u α } ⊂ P sh(Ω) be locally uniformly bounded from above and u be the upper envelope. Then its upper semicontinuous regularization u is also p.s.h. and is equal to u almost everywhere.

Proposition 1.2.11. The set P sh(Ω) is closed in L 1 loc (Ω) and every bounded subset is relatively compact. More precisely, if {u j } ⊂ P sh(Ω) is a sequence which is locally bounded from above, then either it converges locally uniformly to ∞, or there exists a subsequence {u j k } which converges to a p.s.h. function in L p loc (Ω) for any 1 ≤ p < ∞. Definition 1.2.12. A function u is called pluriharmonic if u and -u are both plurisubharmonic.

A pluriharmonic function u verifies the condition that dd c u = 0. It is easy to check that the real part and image part of a holomorphic function are both pluriharmonic. Conversely, a pluriharmonic function is locally the real part of a holomorphic function. We have the following theorem in the global case. Theorem 1.2.13. If the first De Rham cohomology group of X is zero, then every plurisubharmonic function is the real part of some holomorphic function on X.

Pluriharmonic functions are the counterpart of harmonic functions in several complex variables. But when we define harmonic functions by using the following way: A subharmonic function u is said to be harmonic if for every relatively compact open subset V of Ω and every upper semicontinuous function v in V , we have

v ∈ P sh(Ω) and v ≤ u in ∂Ω imply v ≤ u in Ω.
The counterpart of harmonic functions following this definition is called maximal plurisubharmonic functions. Note that harmonic functions are solutions of homogeneous Laplace equations. We will see later that the corresponding equations for maximal plurisubharmonic functions is called Monge-Ampère equations.

Definition 1.2.14. A subset V of X is said to be pluripolar if for every point z ∈ X there is a neighbourhood U of z and a p.s.h. function u on U such that V ∩ U ⊂ {u = -∞}.

We know that if u is a p.s.h. function on a connected subset, then either u ≡ -∞ or u is locally integrable. Hence a pluripolar set is of zero Lebesgue measure.

Proposition 1.2.15. Any proper analytic subset of X is pluripolar. The Hausdorff dimension of a pluripolar set is smaller or equal to 2n -2.

Theorem 1.2.16. Let V be a closed pluripolar subset of X and u be a p.s.h. function on X \ V . If u is locally bounded from above near V . Then there is a unique extension ũ ∈ P sh(X) such that ũ = u on X \ V .

There are close relationships between p.s.h. functions and positive closed (1, 1)-currents. If u ∈ P sh(Ω), then dd c u is a positive closed (1, 1)-current. This can be deduced by the approximation theorem of currents and dominated convergence theorem. Conversely, we have the following result, Proposition 1.2.17. Any positive closed (1, 1)-current T can be locally written as dd c u, where u is a p.s.h. function. The function u is called the local potential of T .

Note that two local potentials differ by a pluriharmonic function, so the study of the singularities of positive closed (1, 1)-currents can be reduced to study the local potentials. The Lelong-Poincaré formula offers a good example of potentials of currents by integration over hypersurfaces. Theorem 1.2.18. Let f be a holomorphic function on X which does not vanish identically on any connected component of X. Then the function log |f | is p.s.h. and verify the equation

dd c log |f | = m j [Z j ],
where m j Z j is the divisor of f .

The above equation holds even when f is meromorphic, then log |f | is locally integrable and m j can be negative numbers. We recall here an useful result, the so-called dd c -lemma. Proposition 1.2.19. Let X be a compact Kähler manifold and T be a closed (p, q)-current on X. Then T is dd c -exact if and only if it is exact (or ∂-exact or ∂-exact).

Intersection of currents and Monge-Ampère operators

Let u be a p.s.h. function and T be a positive closed (p, p)-current, p ≤ n-1.

Recall that σ T is its trace measure. Due to the classical result by Bedford-Taylor [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF], we can define

dd c u ∧ T := dd c (uT ),
when u is locally σ T -integrable, in particular when u is continuous or locally bounded. It follows from the approximation theorem that dd c u ∧ T is a positive closed current. When u 1 , ..., u q are all locally bounded, we can define inductively

dd c u 1 ∧ dd c u 2 ∧ ... ∧ dd c u q ∧ T = dd c (u 1 dd c u 2 ∧ ... ∧ dd c u q ∧ T ).
It is a positive closed current.

Definition 1.2.20. If u is a locally bounded p.s.h. function, then (dd c u) n is called Monge-Ampère operator.

Sometimes the map

(u 1 , ..., u n ) → dd c u 1 ∧ dd c u 2 ∧ ... ∧ dd c u n .
is also called Monge-Ampère operator. We have the following Chern-Levine-Nirenberg inequality.

Theorem 1.2.21. Let L and K are compact subsets of X. Let L is relatively compact in K • . Assume that v is a locally σ T -integrable p.s.h. function. Then there exists a constant C LK independent of T, u j and v such that

dd c u 1 ∧ ... ∧ dd c u q ∧ T L ≤ C LK T K u 1 L ∞ (K) ... u q L ∞ (K) , vdd c u 1 ∧ ... ∧ dd c u q ∧ T L ≤ C LK vT K u 1 L ∞ (K) ... u q L ∞ (K) .
One of the most properties of Monge-Ampère operator is its continuity on decreasing sequences of p.s.h. functions. Theorem 1.2.22. Let u 1 , ..., u q be locally bounded p.s.h. functions. Let {u j 1 },..., {u j q } be decreasing sequences of p.s.h. functions which converge pointwise to u 1 ,...,u q respectively. Then we have in the weak sense of currents

u j 1 dd c u j 2 ∧ ... ∧ dd c u j q ∧ T → u 1 dd c u 2 ∧ ... ∧ dd c u q ∧ T, dd c u j 1 ∧ ... ∧ dd c u j q ∧ T → dd c u 1 ∧ ... ∧ dd c u q ∧ T.
In the following chapters we will deal with the intersections of several currents by integration over analytic subvarieties. So we introduce Monge-Ampère operators on unbounded p.s.h. functions. Definition 1.2.23. Let u be a p.s.h. function. The unbounded locus L(u) is defined to be the set of points z ∈ X such that u is unbounded in any neighborhood of z.

We can define Monge-Ampère operators on unbounded p.s.h. functions when the intersections of unbounded loci are sufficiently small measured by Hausdorff dimensions.

Theorem 1.2.24. Let T be a (p, p)-current and u 1 , ..., u q be p.s.h. functions on X, q ≤ n -p. If the (2n -2p -2k + 1)-Hausdorff dimension of the set L(u j 1 )∩...∩L(u j k )∩suppT is equal to 0 for all indices j 1 < ... < j k in {1, ..., q}, then the currents u 1 dd c u 2 ∧ ... ∧ dd c u q ∧ T and dd c u 1 ∧ ... ∧ dd c u q ∧ T are well defined with locally finite mass.

The continuity property of Monge-Ampère operators is also valid for unbounded p.s.h. functions.

Definition 1.2.25. The analytic subsets V 1 , ..., V q of X is said to be in general position if codim V j 1 ∩ ... ∩ V j k ≥ k for all indices j 1 < ... < j k in {1, ..., q}.
When T is of bidegree (0, 0), we have the following useful criterion Corollary 1.2.26. If the unbounded locus L(u j ) is contained in an analytic subset V j such that V 1 , ..., V q are in general position, then dd c u 1 ∧ ... ∧ dd c u q is well defined.

In particular, the current

[V 1 ] ∧ ... ∧ [V q ]
is well defined when V 1 , ..., V q are in general position. Fornaess and Sibony defined Monge-Ampère operators and obtained continuity properties in other assumptions on u 1 , ..., u q and T . We refer the reader to [27, Section 3] for a more detailed discussion. Definition 1.2.27. Let Ω be an open subset in C n and V be a closed subset in Ω. We say that V is in the envelope of p-pseudoconvexity of Ω \ V with respect to Ω if every point in V can be reached by pushing polydiscs of dimension (n -p) by using biholomorphic images of (n -p, p) Hartogs figures with hulls in Ω.

Theorem 1.2.28. Let T be a positive closed (p, p)-current and u 1 , ..., u q be p.s.h. functions on Ω, q ≤ n -p. If u j ≤ 0, ∀1 ≤ j ≤ q, and L(u j 1 ) ∩ ... ∩ L(u j k ) ∩ suppT is in the envelope of (n -p -k + 1)-pseudoconvexity of the complement for all indices j 1 < ... < j k in {1, ..., q}, then the currents u 1 dd c u 2 ∧ ... ∧ dd c u q ∧ T and dd c u 1 ∧ ... ∧ dd c u q ∧ T are well defined with locally finite mass. Moreover, if u l j converges to u j in L 1 loc (Ω) and u l j ≥ u j , we have

u j 1 dd c u j 2 ∧ ... ∧ dd c u j q ∧ T → u 1 dd c u 2 ∧ ... ∧ dd c u q ∧ T, dd c u j 1 ∧ ... ∧ dd c u j q ∧ T → dd c u 1 ∧ ... ∧ dd c u q ∧ T.

Quai-plurisubharmonic functions and moderate measures

The notion of quai-plurisubharmonic (q.p.s.h. for short) functions was introduced by Yau. It plays an important role in the following chapters. P.s.h. functions have local properties. By the maximum principle we know that p.s.h. functions in compact complex manifolds are always constants. But q.p.s.h. function is a global notion which can be defined in compact complex manifolds.

It can describe the Hermitian metrics on complex line bundles. In this subsection, we assume that X is a compact Kähler manifold of dimension n with Kähler form ω such that ω n is a probability measure. The case when X is only a complex manifold will be specified. Definition 1.2.29. A q.p.s.h. function on X is locally the difference of a p.s.h. function and a smooth one.

Definition 1.2.30. Let γ be a closed real (1, 1)-current on X. An upper-semi continuous function u : X → [-∞, ∞) in L 1 loc (X) is said to be γ-p.s.h. if dd c u + γ ≥ 0.
Note that a function u on X is q.p.s.h. if and only if it is cω-p.s.h. for some constant c > 0. Denote by P sh(X, γ) the set of all γ-p.s.h. functions. Observe that P sh(X, γ) is nonempty if and only if there is a positive closed (1, 1)-current which is cohomologous to γ. There are some special cases when all γ-p.s.h. functions are constants.

Example 1.2.31. Let E be the exceptional divisor of a smooth blow-up X. Then P sh(X, [E]) ∼ = R. See [START_REF] Huybrechts | Complex geometry: an introduction[END_REF]Chapter 2.5] for the definition of a blowup. To see this, let π : X → X 1 be a blow-up. The smooth center of π is Y of codimension ≥ 2. The exceptional divisor E is the subset π -1 (Y ). Let u ∈ P sh(X, [E]). Since π : X \ E → X 1 \ Y is biholomorphic, by considering all test forms with support in X \ E, we obtain that dd c (u

• π -1 ) ≥ 0 in X 1 \ Y .
Since codim Y ≥ 2, we can extend u • π -1 trivially through Y to a global p.s.h. function on X 1 . The maximum principle implies that u • π -1 is constant, so is u. By dd c -lemma, there are no other positive closed (1, 1)-currents which are cohomologous to [E].

We can easily some properties of q.p.s.h. functions from those of p.s.h. functions.

Proposition 1.2.32. Let u and {u j } be q.p.s.h. functions, we have (i) u belongs to L p (X) for every 1 ≤ p < ∞. (ii) If {u j } is a decreasing sequence satisfying dd c u j ≥ -ω, then the limit of {u j } is also a q.p.s.h. function. (iii) If {u j } is uniformly bounded from above and dd c u j ≥ -ω, then either it converges uniformly to ∞, or there exists a subsequence u j k which converges to a q.p.s.h. function v in L p (X) for every 1 ≤ p < ∞ and dd c v ≥ -ω. (iv) The functions u 1 + ... + u m , max{u 1 , ..., u m }, log(e u 1 + ... + e um ) are also q.p.s.h. functions.

The third result of the above property shows also the compactness of q.p.s.h. functions. More generally we have Theorem 1.2.33. The family of the q.p.s.h. functions which satisfy dd c u ≥ -ω and one of the following three normalization conditions

(i) max X u = 0, (ii) X uω n = 0, (iii) X |u|ω n ≤ C, where C is a constant is compact in L p (X) for every 1 ≤ p < ∞.
Moreover, the family of these q.p.s.h. functions is uniformly bounded from above .

Proof. The result with condition (i) follows from (iii) of the above proposition. Let u j verify the condition (ii). Since a q.p.s.h. function is bounded from above in compact manifolds, set a j := sup X u j . Then there are no subsequences of {u j -a j } which converges uniformly to -∞. Hence the sequence {u j -a j } is bounded in L p (X). Otherwise, if there is a subsequence {u j k -a j k } which converges to a q.p.s.h. function v and u j k -a j k L p → ∞, then it yields contradiction with v ∈ L p (X). Note that

a j = a j X ω n = - X (u j -a j )ω n .
So {a j } is bounded. The family of q.p.s.h. function u j is bounded in L p (X) and there is a convergent subsequence. The result with condition (ii) holds. Let u j verify the condition (iii). Let X (u j -b j )ω n = 0. Then b j is bounded. Then the result is a consequence of the result with condition (ii).

By the dd c -lemma on compact Kähler manifolds, we can easily deduce the following result [23, Proposition 2.2]. Proposition 1.2.34. There exists a constant r > 0 such that for any positive closed current T of bidegree (1, 1) with mass 1 on (X, ω), there is a smooth (1, 1)-form α which depends only on the cohomology class of T and a q.p.s.h. function u satisfying that

-rω ≤ α ≤ rω, dd c u -T = α.
Denote by r(X, ω) the smallest number of r which satisfy the above property.

We have an analogous regularization result for q.p.s.h. functions on compact Kähler manifolds. The following theorem is due to Demailly.

Theorem 1.2.35. Let u be a q.p.s.h. function on X. Then there is a decreasing sequence of smooth functions u j satisfying dd c u j ≥ -ω for some positive constant , which converges pointwise to u. Q.p.s.h. functions can be also defined on any complex manifolds. B locki-Ko lodziej obtained a generalization of regularization result.

Theorem 1.2.36. Let X be a complex manifold with a fixed Hermitian form ω and K ⊂ X be a compact subset. Assume that γ is a real closed form of bidegree (1, 1) and u is a locally bounded γ-p.s.h. function. Then for any open neighborhood U of K, there exists a decreasing sequence of smooth functions u j on U such that (i) u j converges pointwise to u, (ii) dd c u j + γ ≥j ω, where j tends to 0. Remark 1.2.37. In the above theorem, if γ is positive and u is any γ-p.s.h. function, then the same conclusion holds. Since γ is positive, constant functions are γ-p.s.h. functions. Then we consider the locally bounded γ-p.s.h. functions max{u, -j}, j ≥ 1 and use diagonal arguments.

In general the global regularization for q.p.s.h. functions on a complex manifold fails.

Our core work through the thesis lies in some estimates of constants on complex projective spaces. The complex projective space P n of dimension n is a compact Kähler manifold. Definition 1.2.38. A projective manifold is a complex submanifold of some complex projective space.

Projective manifolds are algebraic by Chow's theorem. Let π : C n+1 \ {0} → P n be the canonical projection and [z 0 , ..., z n ] be the homogeneous coordinate of P n . Denote by U i the set of points [z 0 , ..., z n ] such that z i = 0. It is a local chart on P n . All the charts can cover P n . Definition 1.2.39. The Fubini-Study form is defined to be

ω F S := dd c log( n j=0 |z j /z i | 2 ) 1/2 in the local chart U i .
In other words, we have

π (ω F S ) = dd c log( n j=0 |z j | 2 ) 1/2
In the local chart U 0 , assume that z 0 = 1. The corresponding Fubini-Study metric has the following Hermitian components

h i j = 1 π (1 + |z| 2 )δ i j -z i zj (1 + |z| 2 ) 2 ,
where

|z| 2 = |z 1 | 2 + ... + |z n | 2 .
Note that ω n F S is a probability measure on P n . This also enlighten the role of normalization in the definition of the operator d c .

The cohomology class of ω p F S is a generator of the group H p,p (P n , R). Any positive closed (1, 1)-current T of mass 1 is cohomologous to ω F S . Then we obtain a q.p.s.h. function u on P n such that dd c u = T -ω F S . We will write ω F S to be the Fubini-Study form of the complex projective space P N of any dimension N . We give two examples of q.p.s.h. functions on

P n . Example 1.2.40. Set |z| 2 = |z 0 | 2 + ... + |z n | 2 . Let µ be a probability measure on P n . (i) v 1 (z) = max(log |z 0 | |z| , ..., log |z n | |z| ) is a ω F S -p.s.h. function. (ii) v 2 (z) = P n log z ∧ w |z||w| dµ(w)
is a ω F S -p.s.h. function. This function can be used to define some capacities on P n [START_REF] Molzon | Average growth estimates for hyperplane sections of entire analytic sets[END_REF]. Positive closed (1, 1)-currents on P n which are cohomologous to ω F S can be characterized by p.s.h. functions on C n [32, Example 2.2]. To be precise, there is a one-to-one correspondence between P sh(P n , ω F S ) and the Lelong class: We have a generalization for any compact family of p.s.h. functions. Corollary 1.2.43. Let U be a compact family of p.s.h. functions in a complex manifold X of dimension n with a Hermitian form ω. For any compact subset K of X, there exist constants c > 0, α > 0 such that

L(C n ) := {v ∈ P sh(C n ) : v(z) ≤ 1 2 log(1 + |z| 2 ) + C v , C v is a constant}.
K exp(-αu)ω k ≤ c, for all u ∈ U.
Dinh-Sibony [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF] introduced the notion of locally moderate measures. Definition 1.2.44. Let X be a complex manifold. A positive measure µ is locally moderate if for any open set U ⊂ X, any compact subset K ⊂ U and any compact family U of q.p.s.h. functions on U , there are constants α > 0, c > 0 such that

K exp(-αφ)dµ ≤ c, ∀φ ∈ U.
Dinh-Nguyên-Sibony [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF] proved that the equilibrium measure of endomorphism of P n is locally moderate.

In the thesis, we deal with the moderate measures in a compact Kähler manifold X of dimension n with a fixed Kähler form ω such that ω n is the standard probability volume form. Consider a positive measure µ on X, µ is said to be PLB if all the q.p.s.h. functions are µ-integrable. When dim X = 1, µ is PLB if and only if it admits a local bounded potential [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF]. Let

F := {φ q.p.s.h. on X : dd c φ ≥ -ω, max X φ = 0}. (1.1)
The set F is compact in L p (X) and bounded in L 1 (µ) when µ is a PLB measure by Theorem 1.2.33 and Proposition 1.2.47.

Recall that any q.p.s.h. functions on X are cω-p.s.h. for some constant c ≥ 0 and uniformly bounded from above. Then the study of q.p.s.h. functions can be reduced to the study of the family F. Observe that there is a large family of positive singular measures which are PLB.

Example 1.2.45. Let µ 0 be a smooth probability measure and T be a positive (n -1, n -1)-current on X. Set µ := µ 0 + dd c T . Then µ is a PLB measure. Consider a smooth function φ ∈ F, we have

0 ≤ X (-φ)dµ = X (-φ)dµ 0 + X (-φ)dd c T ≤ C φ L 1 + (T, -dd c φ) ≤ C φ L 1 + (T, ω) < ∞.
Then the result follows from the regularization of q.p.s.h. functions. Definition 1.2.46. Let µ be a PLB measure on X. We say that µ is (c, α)moderate for some constants c > 0, α > 0 if X exp(-αφ)dµ ≤ c for all φ ∈ F. The measure µ is called moderate if there exist constants c > 0, α > 0 such that it is (c, α)-moderate.

For example, the standard volume form ω n is moderate by Theorem 1.2.42. Let us recall the following proposition in [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF].

Proposition 1.2.47. Let µ be a PLB measure on X. The family of the q.p.s.h. functions which satisfy dd c u ≥ -ω and one of the following three normalization conditions

(i) max X u = 0, (ii) X udµ = 0, (iii) X |u|dµ ≤ C, where C is a constant
is bounded in L 1 (µ) and bounded from above. In particular, there exists a constant c > 0 independent of u such that µ(u < -t) ≤ c t for any t > 0.

When X = P n , we recall the following proposition [23, Corollary A.5] which plays a crucial role in the following chapters, Proposition 1.2.48. There are constants c 0 > 0 and α 0 > 0 independent of n such that

P n exp(-α 0 φ)ω n F S ≤ c 0 n, ∀φ ∈ F.
The following lemma gives an alternative definition of moderate measures [START_REF] Dinh | Characterization of Monge-Ampère measures with Hölder continuous potentials[END_REF].

Lemma 1.2.49. A PLB measure µ is moderate if and only if there exist two constants c > 0, α > 0 such that

µ{z ∈ K : φ(z) < -M } ≤ c e -α M
for any M ≥ 0 and φ ∈ F. Remark 1.2.50. We can take c = c, α = α when c, α are given and take c = 2c , α = α /2 when c , α are given.

Holomorphic line bundles

We introduce some basic notions and properties about holomorphic line bundles. There are closed relationships between holomorphic line bundles and divisors. The Kodaira maps are needed which can be regarded as meromorphic transforms in the following chapters. The Kodaira embedding theorem assets that the notions of positive line bundles and ample line bundles are equivalent on compact Kähler manifolds. Then we introduce big line bundles with several criterions and Nadel vanishing theorm. See [START_REF] Demailly | Complex analytic and differential geometry[END_REF] for the knowledge of connections, analytic sets and sheaf cohomology.

Holomorphic line bundles and first Chern class

Recall that a complex line bundle is a complex vector bundle of rank 1. Let X be a complex manifold of dimension n and π : L → X be a complex line bundle. Denote by L z := π -1 (z) the fiber at a point z. There exists an open covering (U α ) of X and smooth diffeomorphisms

θ α : L Uα → U α × C,
such that for every z ∈ U α the restriction map θ α : L z → {z} × C is a linear isomorphism. Such smooth diffeomorphisms are called local trivializations. Let g αβ be the transition functions satisfying

θ α • θ -1 β (z, ξ) = (z, g αβ (z)ξ), (z, ξ) ∈ (U α ∩ U β ) × C.
The collection of the transition functions {g αβ } defines a Čech 1-cocycle with values in the multiplicative sheaf E of invertible smooth functions on X, i.e.

{g αβ } ∈ H 1 (X, E ). Theorem 1.3.1.
There is a one-to-one correspondence between the group of isomorphism classes of complex line bundles and the Čech cohomology group H 1 (X, E ). The group structure of the former group is the tensor product of line bundles.

Definition 1.3.2. A complex line bundle L is said to be Hermitian if there is a positive definite Hermitian metric h on each fiber L z varying smoothly with z ∈ X.

Let D be a connection on L and Θ(D) be the corresponding curvature. Θ(D) is a closed form of degree 2. The De Rham cohomology class {Θ(D)} is of complex coefficients and independent of the choice of D. When L is Hermitian, we can choose a connection D such that iΘ(D) is a real form, i.e. {iΘ(D)} ∈ H 2 (X, R). Such connection is called Hermitian connection. The exponential exact sequence of sheaves

0 → Z → E → E → 1
gives a coboundary map in the long exact sequence of Cěch cohomology

δ : H 1 (X, E ) → H 2 (X, Z).
Since H 1 (X, E ) = H 2 (X, E ) = 0, the map δ is an isomorphism. It follows from the definition that a complex line bundle is determined up to smooth isomorphism by its first Chern class. As usual the first Chern class is defined to be the image of c 1 (L) under the following natural map

H 2 (X, Z) → H 2 (X, R). Theorem 1.3.4. The first Chern class c 1 (L) ∈ H 2 (X, R) is equal to the De Rham cohomology class { i 2π Θ(D)}.
Definition 1.3.5. A holomorphic line bundle L is a complex line bundle with a holomorphic map π : L → X and biholomorphic trivializations.

It follows that the transition functions are also holomorphic. Let H 0 (X, L) be the space of holomorphic sections of L. Denote by O the multiplicative sheaf of invertible holomorphic functions on X. Similarly there is a one-to-one correspondence between the group of isomorphism classes of holomorphic line bundles and the Čech cohomology group H 1 (X, O ). The group H 1 (X, O ) is called the Picard group of X.

One notable difference between smooth line bundles and holomorphic ones is that the operator ∂ can be well-defined globally on holomorphic line bundles. We are interested in Hermitian holomorphic line bundles. There exists a unique Hermitian connection D such that its (0, 1)-connection is ∂ in a Hermitian holomorphic line bundle.

Divisors and Lelong-Poincaré formula

Since hypersurfaces are given by the zeros of a global holomorphic sections of a holomorphic line bundle, there are close relationships between divisors and holomorphic line bundles. Recall that an analytic hypersurface of X is an analytic subvariety of codimension one. Definition 1.3.7. A divisor D on X is a locally finite formal linear combination D = a i Y i , where Y i are irreducible analytic hypersurfaces and a i are integers.

Denote by Div(X) the set of all divisors on X. The sum is finite if X is a compact complex manifold. A divisor is said to be effective if all a i are nonnegative. Every hypersurface defines an effective divisor Y i . Recall that the order ord Y,z (f ) of a meromorphic function f along Y at z ∈ Y is the largest integer a such that f = g a h with h ∈ O X,z . If Y is irreducible, then the order is independent of z ∈ Y . Hence we can define ord Y (f ) along an irreducible hypersurface.

Definition 1.3.8. The divisor associated to a meromorphic function f is

div(f ) := ord Y (f )Y,
where the index of the sum runs over all irreducible hypersurfaces in X. Such divisor is called principle.

Denote by M the multiplicative sheaf of invertible meromorphic functions on X.

Proposition 1.3.9. There exist the following group homomorphisms

H 0 (X, M /O ) ≡ Div(X) → P ic(X).
A divisor D ∈ Div(X) is assigned to a holomorphic line bundle O(D). In fact D corresponds to an element f ∈ H 0 (X, M /O ), which is represented by a family of functions f α ∈ M (U α ) with respect to an open covering {U α }. Then the transition functions g αβ = f α f -1 β defines the line bundle O(D). Note that the line bundle O(D) can be identified to the sheaf of germs of meromorphic functions f satisfying div(f ) + D ≥ 0. Definition 1.3.10. A meromorphic section of L is a section s defined by a collection of meromorphic functions f α = θ α (s) ∈ M (U α ).

Since {f α } corresponds to an element in H 0 (X, M /O ), then the meromorphic section s defines a divisor div(s) which is written locally as div(f α ). The divisor div(s) is effective if and only if s is a holomorphic section.

The current defined by div(s) is written as [div(s)] or [s = 0] when s is holomorphic. The definition of div(s) implies that O(div(s)) is isomorphic to L for any nonzero meromorphic section s. It follows from Lelong In particular, the current dd c ϕ represents the first Chern class c 1 (L).

Following the above arguments, we obtain

dd c log |s| 2 h = div(s) - i 2π Θ(L).
The following theorem is the Lelong-Poincaré formula for meromorphic sections of line bundles.

Theorem 1.3.12. Let L be a Hermitian holomorphic bundle and s be a meromorphic section of L which does not vanish identically on any component of

X. Then O(div(s)) ∼ = L, c 1 (L) = {[div(s)]}. Consequently we have c 1 (O(div(s))) = {[div(s)]}.
We extend the notion of Hermitian line bundle to the singular case.

Definition 1.3.13. Let L be a holomorphic line bundle on a complex manifold X. A singular Hermitian metric h on L is a sesquilinear Hermitian-symmetric form on each fiber such that for any local holomorphic frame e L of L on U ⊂ X, Note that i 2π Θ is just the associated (1, 1)-form of the Fubini-Study metric, which is positive.

we have |e L | 2 h = e -2ϕ ∈ [0, ∞], where ϕ ∈ L 1 loc (U )

Kodaira embedding theorem

We assume X is a compact Kähler manifold and L is a holomorphic line bundle on X in this subsection. Recall that H 0 (X, L) denotes the space of holomorphic sections of L. The kth tensor product of L is L k . It follows from Hodge theory that the dimension of H 0 (X, L) is finite. Let N = dim H 0 (X, L). Definition 1.3.17. A point x ∈ X is said to be a base point if s(x) = 0 for all s ∈ H 0 (X, L). The base locus Bs(L) is the set of all base points. Let s 0 , ..., s N be a basis of H 0 (X, L), then Bs(L) = (s 0 = 0) ∩ ... ∩ (s N = 0) which is an analytic subvariety. Denote by H 0 (X, L) the dual space of H 0 (X, L). Definition 1.3.18. The Kodaira map associated to L is defined by

Φ : X \ Bs(L) → P(H 0 (X, L) ), Φ(x) = {s ∈ H 0 (X, L) : s(x) = 0}.
We give a local analytic description of the Kodaira map. Let s 0 , ..., s N be a basis of H 0 (X, L) and e L be a local holomorphic frame of L on U , where U can be chosen as a contractible Stein open subset. Then there exist holomorphic functions f j such that s j = f j e L . By an identification via the basis, the Kodaira map can be expressed locally as

Φ : X → P N , Φ(x) = [f 0 (x), ..., f N (x)].
This map does depend on the choice of the basis. But two such maps differ only by a linear transformation of P N for two different choices of basis. The Kodaira map is a meromorphic map on X, but it is holomorphic on X \Bs(L). Definition 1.3.19. A holomorphic line bundle is called ample if the Kodaira map associated to L k is an embedding for some integer k > 0.

A compact Kähler manifold is projective if and only if it admits an ample line bundle. A natural question is when the Kodaira map defines an embedding of X. If it is an embedding map, the X is a projective manifold. We have the following important theorem called Kodaira embedding theorem.

Theorem 1.3.20. Let L be a holomorphic line bundle over a compact Kähler manifold X. Then L is positive if and only if L is ample.

Big line bundles

We will use Kodaira maps associated to high tensor powers of a holomorphic line bundle as follows, Φ p : X \ Bs(L p ) → P(H 0 (X, L p ) ) Define p := {rank x Φ p : x ∈ X \ Bs(L p )}. If H 0 (X, L p ) is equal to {0}, we take the convention that p = -∞. Theorem 1.3.23. Let L be a holomorphic line bundle over a connected compact complex manifold X of dimension n. The following are equivalent: (i) L is big; (ii) lim sup p→∞ p -n dim H 0 (X, L p ) > 0; (iii) L admits a singular Hermitian metric such that the curvature current is strictly positive; (iv) L admits a singular Hermitian metric which is smooth outside a proper analytic set of X such that the curvature current is strictly positive.

If X is compact Kähler manifold with a fixed Kähler form ω, then the big line bundle L admits a singular Hermitian metric h such that c 1 (L, h) ≥ ω for some constant > 0.

The asymptotic expansion of dimensions of H 0 (X, L p ) is crucial through our thesis. Let d p = dim H 0 (X, L p ). If L is positive, then the Hirzebruch-Riemann-Roch theorem and Kodaira-Serre vanishing theorem show that d p is given by a Hilbert polynomial whose dominate term is p n /n! X c 1 (L) n , see [START_REF] Hirzebruch | Topological methods in algebraic geometry[END_REF]Section 20] Let L be a holomorphic line bundle over a connected compact complex manifold X. There there exists a constant C > 0 satisfying dim H 0 (X, L p ) ≤ Cp p ≤ Cp κ(L) , ∀p ≥ 1.

The Kodaira-Iitaka dimension κ(L) is the optimal constant independent of p. Theorem 1.3.25. If X is a projective manifold, then the natural group homomorphism Div(X) → P ic(X) is surjective.

Theorem 1.3.25 [START_REF] Huybrechts | Complex geometry: an introduction[END_REF]Corollary 5.3.7] shows that any holomorphic line bundle in a projective manifold is isomorphic to a line bundle associated to a divisor. More generally, this theorem holds also for Moishezon manifolds, see [START_REF] Ma | Holomorphic Morse inequalities and Bergman kernels[END_REF]Theorem 2.2.20] for more information.

We will use the notion of multiplier ideal sheaf introduced by Nadel [START_REF] Nadel | Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature[END_REF]. It can be applied to the lower estimate of dimensions associated to big line bundles.

Definition 1.3.26. Given a real function ϕ which is locally integrable on X, the multiplier ideal sheaf I(ϕ) is the ideal subsheaf of germs of holomorphic functions f ∈ O X,x such that |f | 2 e -2ϕ is integrable with respect to the Lebesgue measure in a local coordinate near x.

The zero variety of I(ϕ) is the set of points x such that e -2ϕ is not integrable in a neighborhood of x. Let h be a singular Hermitian metric on L and ϕ be the local weight of h in an open subset. Then the multiplier ideal sheaf of h is defined by I(h) := I(ϕ). This definition does not depend on the choice of local trivializations. Recall that K X is the canonical line bundle of X. We have the following Nadel vanishing theorem, see also [START_REF] Demailly | A numerical criterion for very ample line bundles[END_REF]Theorem 4.5].

Theorem 1.3.27. Let L be a big line bundle over a compact Kähler manifold (X, ω). Assume that L is endowed with a singular Hermitian metric h such that c 1 (L, h) ≥ ω for some constant > 0. Then we have H q (X, O(K X + L) ⊗ I(h)) = 0 for all q ≥ 1.

Chapter 2

Equidistribution of zeros of random holomorphic sections for moderate measures.

In this chapter, we establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle. The equidistribution is associated to a family of singular moderate measures. We also give a convergence speed for the equidistribution which shows an alternative proof of the main theorem.

Dinh-Sibony equidistribution theory.

The Dinh-Sibony equidistribution theory [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] is the cornerstone in the thesis. Our main theorems are proved based on the techniques and theorems from this theory. They studied the equidistribution problem associated to PLB measures, which is a generalization of the result by Shiffman-Zelditch [START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF]. Appropriate conditions are posed on PLB measures to satisfy the equidistribution property.

Constants associated to PLB measures

Let (X, ω) be a compact Kähler manifold of dimension n and µ be a PLB measure on X. Recall that r(X, ω) is the smallest number of r in Proposition 1.2.34. Following Proposition 1.2.47, we can define the notations Q(X, ω) : = {ϕ q.p.s.h. on X, dd c ϕ ≥ -r(X, ω)ω}, R(X, ω, µ) : = sup{max

X ϕ : ϕ ∈ Q(X, ω), X ϕdµ = 0} = sup{- X ϕdµ : ϕ ∈ Q(X, ω), max X ϕ = 0}, S(X, ω, µ) : = sup{ ϕdµ : ϕ ∈ Q(X, ω), X ϕω n = 0}, ∆(X, ω, µ, t) : = sup{µ(ϕ < -t) : ϕ ∈ Q(X, ω), X ϕdµ = 0}.
Here t > 0. When µ = ω n , let R 0 (X, ω) := R(X, ω, µ).

S(X, ω, µ) ≤ R(X, ω, µ) + R 0 (X, ω).
The following estimate is very fundamental in Dinh-Sibony equidistribution theory, see [23,

Proposition A.3]. Proposition 2.1.2. R 0 (P k , ω F S ) ≤ 1 2 (1 + log k).
Proof. Let π : C k+1 \ {0} → P k (or π : S 2k+1 → P k ) be the natural projection. Let dλ 2k+1 be the probability spherical measure on S 2k+1 . Given ϕ ∈ F, then dd c ϕ + ω F S is a positive closed current of mass 1. By Proposition 1.2.41, there exists a corresponding p.s.h. function v on C k+1 such that

v(λz) = log |λ| + v(z), λ ∈ C \ {0}.
Note that v(z) = ϕ • π + log z and v(0) = -∞. Moreover, the Fubini-Study volume form ω k F S and dλ 2k+1 are probability Haar measures on P k and S 2k+1 respectively. By the uniqueness of Haar measures, the following equality holds

ω k F S (A) = dλ 2k+1 (π -1 (A)),
where A is a measurable set in P k . Hence

P k ϕω k F S = S 2k+1 ϕ • πdλ 2k+1 = S 2k+1 vdλ 2k+1 .
Since max S 2k+1 v = max P k ϕ = 0, by a theorem due to Alexander [1, Theorem 2.2], we have

S 2k+1 ϕ • πdλ 2k+1 ≥ max S 2k+1 v + S 2k+1 log |z 1 |dλ 2k+1 = - 1 2 j=k j=1 1 j ≥ - 1 2 (1 + log k).
Then the proof is completed.

Meromorphic transforms

The meromorphic transforms play a crucial role in Dinh-Sibony equidistribution theory. Roughly speaking, a meromorphic transform between two complex manifolds is a surjective multivalued map with an analytic graph.

Definition 2.1.3. Let (X 1 , ω 1 ), (X 2 , ω 2 ) be two compact Kähler manifolds of dimension n 1 and n 2 respectively, a meromorphic transform F :

X 1 → X 2 is the data of an analytic subset Γ ⊂ X 1 × X 2 of pure dimension n 2 + l such that the natural projections π 1 : X 1 × X 2 → X 1 and π 2 : X 1 × X 2 → X 2 restricted to each irreducible component of Γ are surjective. Γ is called the graph of F .
Note that a meromorphic transform is determined by its graph. We write

F = π 2 •(π 1 | Γ ) -1 . The dimension of the fiber F -1 (x 2 ) := π 1 (π -1 2 | Γ (x 2 )
) is equal to l for a point x 2 ∈ X 2 generic. This is the codimension of the meromorphic transform F . If T is a current of bidegree (m, m) on X 2 , n 2 + l -n 1 ≤ m ≤ n 2 , we define F (T ) := (π 1 ) (π 2 (T ) ∧ [Γ]), where [Γ] is the current of integration over Γ. The intermediate degree of order m of a meromorphic transform F :

X 1 → X 2 is defined by λ m (F ) = X 1 F (ω m 2 ) ∧ ω n 2 +l-m 1 = X 2 ω m 2 ∧ F (ω n 2 +l-m 1
).

Set d(F ) := λ n 2 (F ) and δ(F ) := λ n 2 -1 (F ). These two notations are useful in the following arguments. We give some examples of meromorphic transforms. (i) Let G(k -l + 1, k + 1) be the Grassmannian manifold which parameterizes all the projective subspace of dimension k -l of P k . For a point ŝ ∈ G(k -l + 1, k + 1), there is a corresponding projective subspace P k-l ŝ of dimension k -l. Set

Γ 1 := {(z, ŝ) ∈ P k × G(k -l + 1, k + 1), z ∈ P k-l ŝ }.
The meromorphic transform F 1 :

P k → G(k -l + 1, k + 1) is defined by the graph Γ 1 . Since F -1 1 (ŝ) = P k-l ŝ , it is of codimension k -l.
We can give another description of F 1 by duality. Note that P k := G(k, k + 1) is the dual of P k . Denote by G (l, k + 1) the Grassmannian manifold which parameterizes all the projective subspace of dimension l -1 of P k . This Grassmannian manifold is biholomorphic to G(k -l + 1, k + 1). For a point s ∈ G (l, k + 1), there is a corresponding projective subspace P (l-1) s of dimension l -1. Choose l points s 1 , ..., s l which generate P (l-1) s

. Let P (k-1) s j be the corresponding hyperplane in P k of the point s j . Note that P k-l s := ∩ j=l j=1 P

(k-1) s j is independent of the choices of s j . Set

Γ 2 := {(z, s) ∈ P k × G (l, k + 1), z ∈ P k-l s }.
The meromorphic transform F 2 :

P k → G (l, k + 1) is defined by the graph Γ 2 . Since F -1 2 (s) = P k-l s , it is of codimension k -l.
(ii)Let P k,l := P k × ... × P k (l times) be a multi-projective space. Write s = (s 1 , ..., s l ) ∈ P k,l . Set

Γ 3 := {(s, s) ∈ P k,l × G (l, k + 1), P k-l s ⊂ P (k-1) s j , f orj = 1, .., l}.
The meromorphic transform Ψ l : P k,l → G (l, k +1) is defined by the graph Γ 3 . Denote by Ψl the adjoint of Ψ l . The composition F 3 := Ψl • F 2 : P k → P k,l is a meromorphic transform. For a point s ∈ P k,l generic, F -1 2 (s) is the subspace

P k-l s := ∩ j=l j=1 P (k-1) s j
. Then F 3 is of codimension k -l.

We will see later the graph induced by a Kodaira map defines a meromorphic transform. For more information about meromorphic transforms, refer to [23, Section 3].

General equidistribution theorem

Let (X, ω), (X p , ω p ) be compact Kähler manifolds of dimension n and n p respectively. In addition, Xp ω np p = 1. Let X := ∞ p=1 X p . It follows from Tychonoff's theorem that X is also a compact space with respect to the product topology. A point x ∈ X is denoted by (x 1 , x 2 , ...) = (x p ). The measure δ xp is the Dirac measure at the point x p on X p .

Consider a probability PLB measure σ p on each X p . Endow X with the product measure σ := ∞ p=1 σ p . Let F p : X → X p be a family of meromorphic transforms with the same codimension l < n. By the definition of the pullback of F p on currents, we see that F p (δ xp ) and F p (σ p ) are positive closed currents of bidimension (l, l) on X. Moreover, F p (δ xp ) is well defined for x p ∈ X p generic. To simplify the notations, set

R p := R(X p , ω p , σ p ), S p := S(X p , ω p , σ p ), ∆ p (t) := ∆(X p , ω p , σ p , t), δ p := δ(F p ), d p := d(F p ).
The following is the main theorem in Dinh-Sibony equidistribution theory [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Theorem 4.1].

Theorem 2.1.5. Suppose that the sequence {R p δ p d -1

p } tends to 0 and one of the following two conditions is verified

p≥1 S p δ p d -1 p < ∞, p≥1 ∆ p (δ -1 p d p t) < ∞, ∀t > 0.
Then for almost every x = (x p ) ∈ X with respect to σ, the sequence d -1 p (F p (δ xp )-F p (σ p )), ψ converges to 0 uniformly on the bounded set of (l, l)-forms on X of class C 2 . Theorem 2.1.6. Suppose that the sequence {S p δ p d -1 p } tends to 0. Then

d -1 p (F p (σ p ) -F p (ω np p
)), ψ converges to 0 uniformly on the bounded set of (l, l)-forms on X of class C 2 . To be more precise, we have

d -1 p (F p (σ p ) -F p (ω np p )), ψ ≤ 2S p δ p d -1 p φ C 2 for any (l, l)-form of class C 2 on X.
There is a crucial estimate that we mention as follows. First we define for every p > 0, > 0,

E p ( ) := φ C 2 ≤1 {x p ∈ X p : F p (δ xp ) -F p (σ p ), φ ≥ d p }. Theorem 2.1.7. Let η ,p := δ -1 p d p -3R p , then σ p (E p ( )) ≤ ∆ p (η ,p ).

Estimate for moderate measures on P k

In this section, we give an estimate for moderate measures on P k . Our work is inspired by the techniques of exponential estimates for p.s.h. functions from [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF]. Such estimate, combined with Dinh-Sibony equidistribution theory, implies the main theorems. Let (X, ω) be a compact Kähler manifold of dimension k and ω k be its standard volume form such that ω k is a probability measure on X.

Locally moderate currents

Let S be a positive closed current of bidegree (p, p) on X, the trace measure is σ S = S ∧ ω k-p for a fixed Hermitian form ω on X. Here X may not be compact. S is said to be locally moderate if its trace measure is locally Lemma 2.2.4. Let K 0 be as above. For any point z ∈ P k , there exists a neighbourhood K z of z which is similar to K 0 . Denote by N k the minimum number of such neighbourhoods K 0 that cover P k . Then N k = O(8 k ).

Proof. Since π(S 0 ) ⊂ K 0 , then M k ≥ N k . So it remains to prove that M k = O(8 k ). We endow S 2k+1 with the great-circle distance. S 0 can be regarded as an open ball with central point [1, 0, ..., 0] of radius π 4 . Denote S 0 by B([1, 0, ..., 0], π 4 ). Let

S 1 = B([1, 0, ..., 0], π 8 ) = {(x 0 , y 0 , ..., x k , y k ) ∈ S 2k+1 , x 0 > √ 2+ √ 2 2
}. We first consider the open balls of radius π 8 . All of them are similar to each other. We put the maximal number of balls

B(z 1 , π 8 ), ..., B(z m k , π 8
) in S 2k+1 such that all of them are disjoint mutually. Then S 2k+1 = j=m k j=1 B(z j , π 4 ). If there exists a point w ∈ S 2k+1 \ j=m k j=1 B(z j , π 4 ), then the great-circle distance between w and z j is larger than or equal to π 4 for all 1

≤ j ≤ m k . Hence B(w, π 8 ) ⊂ S 2k+1 \ j=m k j=1 B(z j , π 8
), contradicts with the maximality. Then M k ≤ m k ≤ V ol(S 2k+1 )/V ol(S 1 ), the last inequality is due to the mutual disjointedness. It means that N k = O(V ol(S 2k+1 )/V ol(S 1 )).

We now use the spherical coordinate for S 2k+1 . Let

x 0 = cos θ 1 , y 0 = sin θ 1 cos θ 2 , ..., x k = sin θ 1 sin θ 2 • • • sin θ 2k cos θ 2k+1 , y k = sin θ 1 sin θ 2 • • • sin θ 2k sin θ 2k+1 . Then the volume element of S 2k+1 is d S 2k+1 V = sin 2k θ 1 sin 2k-1 θ 2 • • • sin θ 2k dθ 1 dθ 2 • • • dθ 2k+1 . V ol(S 2k+1 ) = π 0 sin 2k θ 1 dθ 1 π 0 sin 2k-1 θ 2 dθ 2 • • • π 0 sin θ 2k dθ 2k 2π 0 dθ 2k+1 V ol(S 1 ) = π 8 0 sin 2k θ 1 dθ 1 π 0 sin 2k-1 θ 2 dθ 2 • • • π 0 sin θ 2k dθ 2k 2π 0 dθ 2k+1 . This yields O(V ol(S 2k+1 )/V ol(S 1 )) = O( π 0 sin 2k θ 1 dθ 1 / π 8 0 sin 2k θ 1 dθ 1 ).
Then it suffices to show that π 0 sin 2k θ 1 dθ 1 / π 8 0 sin 2k θ 1 dθ 1 ≤ 8 k+1 , ∀k ≥ 7. When k = 7, the inequality is right. By induction on k and the following integrals

sin 2k θ 1 dθ 1 = - sin 2k-1 θ 1 cos θ 1 2k + 2k -1 2k sin 2k-2 θ 1 dθ 1 ,
the proof is reduced to show that

π 8 0 sin 2k θ 1 dθ 1 ≥ 8 7 1 2k+1 √ 2 4 ( 2- √ 2
4 ) k . By the relation between π 8 0 sin 2k θ 1 dθ 1 and π 8 0 sin 2k+6 θ 1 dθ 1 , we have

π 8 0 sin 2k θ 1 dθ 1 ≥ 1 2k + 1 √ 2 4 ( 2 - √ 2 4
)

k (1 + 2 - √ 2 4 2k + 2 2k + 3 + ( 2 - √ 2 4
) 2 (2k + 2)(2k + 4) (2k + 3)(2k + 5) )

Then the proof is completed.

Local estimate for moderate measures

Let F be defined in (1.1) when X = P k and θ defined in (2.1). The following lemma is crucial for the main proposition in this section. Lemma 2.2.5. Let u be of class C ρ with modulus on a neighbourhood U of B 1 in C k with dd c u ≥ 0 in the sense of currents, 0 < ρ < 1. Set ω :=

1 2 dd c log(1 + z 2 ). Let F 0 = {φ • θ -1 on U : φ ∈ F} and T a positive closed (k -1, k -1)-current. If T ∧ ω is (c, α)-moderate on U , then B 1 exp(- αρ 4 φ)dd c (uT ) ≤ ck (c 1 e α + c 2 α )
where c 1 , c 2 are positive constants independent of k, ρ and T .

Proof. We modify the function u on U . Subtracting a constant, we assume that u ≤ -/2 on B 1 . Consider the function v(z) = max(u(z), A log |z|) for a constant A > 0 large enough such that v coincides with u near the origin and v(z) = A log |z| near the boundary of B 1 . For example, A = 1 2 log 1 1-4r . A is independent of the choice of u. Fix 0 < r < 1/16, we are allowed to assume that u = A log |z| on B 1 \ B 1-4r . For the smooth function χ defined in Lemma 2.3.3, we can assume that χ C 2 < c 3 for some constant c 3 > 1 large enough independent of k, since the terms in the definition of the norm

• C 2 are smooth on the compact subset B1-r \ B 1-2r . Set σ T = T ∧ ω, σ T = dd c (uT ), φ M = max(φ, -M ), ψ M = φ M -1 -φ M , for φ ∈ F 0 , M ≥ 0.
To prove the lemma, we need to estimate the mass of dd c (uT ) on {φ < -M }. Since suppχ ⊂ B 1-r , hence

σ T {φ < -M } ≤ χψ M dd c (uT ). Since T is (c, α)-moderate, then σ T {z ∈ B 1-r , φ(z) ≤ -M + 1} ≤ ce α e -αM .

By Lemma 2.3.3, we have

B 1 χψ M dd c (uT ) = - B 1-r \B 1-3r dd c χ ∧ ψ M uT - B 1-r \B 1-3r dχ ∧ ψ M d c u ∧ T + B 1-r \B 1-3r d c χ ∧ ψ M du ∧ T + B 1-r χudd c ψ M ∧ T (2.2) We know that ω = 1 2 dd c log(1+ z 2 ) = i 2π k j,l=1 ( dz j ∧dz l 1+ z 2 -zj z l dz j ∧dz l (1+ z 2 ) 2 )
. By simple computations, the eigenvalues of the corresponding Hermitian matrix of ω are 1 π(1+ z 2 ) 2 and 1 π(1+ z 2 ) (k -1 times). On the other hand, the eigenvalues of the corresponding Hermitian matrix of i k j,l=1 dz j ∧dz l are k and 0 (k-1 times). So there exists a constant m 1 > 0 small enough such that ω-

m 1 k i k j,l=1 dz j ∧dz l > 0 on B 1 . Hence |dd c χ∧uT | ≤ |uc 3 i k j,l=1 dz j ∧dz l ∧T | ≤ A| log(1-3r)|c 3 k m 1 σ T . Observing that 0 ≤ ψ M ≤ 1, suppψ M ⊂ {φ < -M + 1}, we obtain B 1-r \B 1-3r dd c χ ∧ ψ M uT ≤ A| log(1 -3r)|c 3 k m 1 ce α e -αM .
Since we know u explicitly on supp(dχ), we obtain

B 1-r \B 1-3r dχ ∧ ψ M d c u ∧ T ≤ A 1 -3r c 3 km 2 ce α e -αM , B 1-r \B 1-3r d c χ ∧ ψ M du ∧ T ≤ A 1 -3r c 3 km 2 ce α e -αM .
for a constant m 2 > 0 large enough independent of k. The sum of the first three terms is less than

c 4 kce α e -αM (2.3)
where

c 4 = Ac 3 ( | log(1-3r)| m 1 + 2m 2 1-3r
) is independent of k and ρ. For the last integral in (2.2), we use a regularization procedure and the condition of ρ-Hölder continuity of u. Let {u δ } be the smooth approximation of u obtained by convolution. For some fixed 0 < δ < 1 small enough, u δ is defined in a neighborhood of B1-r . There exists a suitable function u δ satisfying that u δ C 2 ≤ δ -(2-ρ) and u -u δ ∞ ≤ δ ρ , where the latter inequality follows from that u is of class C ρ with modulus . The above two inequalities are independent of k. We write

B 1 χudd c ψ M ∧ T = χdd c ψ M ∧ T u δ + χ(dd c φ M -1 -dd c φ M ) ∧ T (u -u δ ). Since χdd c (φT ) = dd c χ ∧ φT ≤ k χ C 2 B 1-r |φ|dσ T ,
We obtain

χ(dd c φ M -1 -dd c φ M ) ∧ T ≤ 2k χ C 2 B 1-r |φ|dσ T ≤ 2k χ C 2 1 α B 1-r exp(-αφ)dσ T ≤ 2c 3 k c α .
Then

χ(dd c φ M -1 -dd c φ M ) ∧ T (u -u δ ) ≤ 2c 3 k c α δ ρ (2.4)
Using Lemma 2.3.3 again, we obtain

χdd c ψ M ∧ T u δ = B 1-r \B 1-3r dd c χ ∧ ψ M T u δ + B 1-r \B 1-3r dχ ∧ ψ M T ∧ d c u δ - B 1-r \B 1-3r d c χ ∧ ψ M T ∧ du δ + B 1-r χψ M T ∧ dd c u δ .
By the same argument, the first three integrals have the same dominant constant c 4 kce α e -αM .

(2.5)

The final term

χψ M T ∧ dd c u δ ≤ ce α e -αM u δ C 2 ≤ ce α e -αM δ -(2-ρ) .
(2.6)

Let δ = e -αM/2 small enough, since it is sufficient to consider M big. Then e -αM ρ/2 = e -αM e αM (2-ρ)/2 . Combining (2.3), (2.4), (2.5), (2.6), we have

σ T {z ∈ B 1 , φ < -M } ≤ ck(2c 4 e α + e α k + 2 c 3 α )e -αM 2 ρ .
So by Remark 1.2.50 we have

B 1 exp(- αρ 4 φ)dd c (uT ) ≤ 2 ck(2c 4 e α + e α k + 2 c 3 α ) ≤ ck(c 1 e α + c 2 α ),
where c 1 = 4c 4 + 2, c 2 = 4c 3 .

Main result and its proof

The following proposition is our main result about the estimate for moderate measures on P k . Proposition 2.2.6. Suppose that u j is of class C ρ with modulus on P k for some 0 < ρ < 1, 0 < < 1, and that u j is an ω F S -p.s.h. function for all 1 ≤ j ≤ k. Assume that < β 0 k -3 ( ρ 12 ) 2k , where β 0 is a positive constant independent of k and ρ. Then there exists a positive constant c 5 independent of k and ρ, such that

P k exp(-α 0 ( ρ 4 ) k φ)(∧ j=k j=1 (dd c u j + ω F S + ω F S ) -ω k F S ) ≤ c 5 ( ρ 4 ) k (2.7)
for all φ ∈ F, where α 0 is the constant in Proposition 1.2.48. In other words,

(∧ j=k j=1 (dd c u j + ω F S + ω F S ) -ω k F S ) is (c 5 ( ρ 4 ) k , α 0 ( ρ 4 ) k )-moderate.
Proof. We pull back the integral (2.7) locally to that on C k . There is a potential v = 2 log(1 + z 2 ) on C k such that (θ -1 ) ( ω F S ) = dd c v, where the map θ is defined in (2.1). Set ũj := u j • θ -1 + v. Note that u j is ω F S -p.s.h., then dd c ũj ≥ 0. Since u j is of class C ρ with modulus on P k , log(1 + z 2 ) is of class C a on C k for all 0 < a < 1, then we may assume that ũj is of class C ρ with modulus on B 1 . Hence ũj C ρ (B 1 ) ≤ . Let ω = 1 2 dd c log(1 + z 2 ), we have

K 0 exp(-αφ) ∧ j=k j=1 (dd c u j + ω F S + ω F S ) = B 1 exp(-αφ • θ -1 )(θ -1 ) ∧ j=k j=1 (dd c u j + ω F S + ω F S ) = B 1 exp(-αφ • θ -1 )(dd c ũ1 + ω) ∧ • • • ∧ (dd c ũk + ω) (2.8)
We replace ũj (resp. φ • θ -1 ) by u j (resp. φ) in the sequel. Since there are two constants c 0 > 0, α 0 > 0 independent of k and ρ, such that

P k exp(-α 0 φ)ω k F S ≤ c 0 k, (2.9)
by pulling back the integral in B 1 with Lemma 2.3.5, we have

B 1 exp(-α 0 ρ 4 φ)(dd c u j ) ∧ ω k-1 ≤ c 0 k 2 (c 1 e α 0 + c 2 α 0 ).
By induction we can show that

B 1 exp(-α 0 ( ρ 4 ) j φ)dd c u l 1 ∧ • • • ∧ dd c u l j ∧ ω k-j ≤ c 0 k( k) j j-1 l=0 (c 1 e α 0 ( ρ 4 ) l + c 2 α 0 ( ρ 4 ) l ) for all 1 ≤ l 1 < • • • < l j ≤ k. Let β 0 = 1/(c 1 e α 0 + c 2 α 0 ), 0 = β 0 k -3 ( 1 8 ) k ( ρ 4 ) 3k-1 2 > , 0 = 1 2 , 2 = ( ρ 4 ) k . Here β 0 is independent of k and ρ. Let 1 = 3 /( ρ 4 ) k+1 2 , then 3 = β 0 ( ρ 32 ) k /k 3 . Hence B 1 exp(-α 0 ( ρ 4 ) k φ)((dd c u 1 + ω) ∧ • • • ∧ (dd c u k + ω) -ω k ) = k j=1 k j B 1 exp(-α 0 ( ρ 4 ) j φ)dd c u 1 ∧ • • • ∧ dd c u j ∧ ω k-j ≤ k j=1 k j c 0 k( 1 k) j ( 1 β 0 ) j ( ρ 4 ) k+k-1+•••+k-(j-1) ≤ k j=1 k j c 0 k( 1 k) j ( 1 β 0 ) j ( ρ 4 ) k+1 2 j ≤ c 0 k k j=1 k j ( 3 k β 0 ) j ≤ c 0 ( ρ 32 ) k ( k-1 j=0 1 k j ) ≤ 2c 0 ( ρ 32 ) k .
(2.10) This is equivalent to

K 0 exp(-α 0 ( ρ 4 ) k φ)((dd c u 1 + ω F S +ω F S )∧•••∧(dd c u k + ω F S +ω F S )-ω k F S ) ≤ 2c 0 ( ρ 32 ) k .
By Lemma 2.3.4, there is a positive constant N independent of k and ρ such that N k ≤ N 8 k . Let c 5 = 2c 0 N . Due to the homogeneity of P k , we have

P k exp(-α 0 ( ρ 4 ) k φ)((dd c u 1 + ω F S +ω F S )∧•••∧(dd c u k + ω F S +ω F S )-ω k F S ) ≤ c 5 ( ρ 4 ) k .
The proof is completed.

Remark 2.2.7. Since (dd c u j + ω F S ) k ≤ (dd c u j + ω F S + ω F S ) k , the above proposition, combined with (2.9), gives the following estimate

P k exp(-α 0 ( ρ 4 ) k φ)(dd c u 1 + ω F S ) ∧ • • • ∧ (dd c u k + ω F S ) ≤ c 0 k + c 5 ( ρ 4 ) k ≤ c 0 k + c 5
for all φ ∈ F. In other words, (dd

c u 1 + ω F S ) ∧ • • • ∧ (dd c u k + ω F S ) is (c 0 k + c 5 , α 0 ( ρ 4 ) k )-moderate.

Equidistribution on positive line bundles for moderate measures

In this section we prove Theorem 0.2.1 and 0.2.2. Consider the projective manifold X of dimension n and the ample line bundle L on X in Theorem 0.2.1. By Proposition 1.3.11, there exists a smooth Hermitian metric h such that

c 1 (L, h) = -dd c log h(e L , e L ) 1 2
is a strictly positive (1, 1)-form, where e L is a local holomorphic section on L. As we know,

c 1 (L, h) represents the Chern class c 1 (L) ∈ H 2 (X, Z). Let ω = c 1 (L, h) be the Kähler form, X ω n = c 1 (L) n ∈ Z + .
The line bundle L p of the pth tensor power of L has a natural Hermitian metric h p induced by h. The space H 0 (X, L p ) of holomorphic sections of L p has the following inner product,

s 1 , s 2 := 1 c 1 (L) n X h p (s 1 , s 2 )ω n ∀s 1 , s 2 ∈ H 0 (X, L p ).

Meromorphic transforms induced by Kodaira maps

First we consider a meromorphic map f : X 1 → X 2 between two complex manifolds X 1 and X 2 of dimension n 1 and n 2 respectively. It follows from [START_REF] Remmert | Holomorphe und meromorphe Abbildungen komplexer Räume[END_REF] that there exists an analytic subset I of X 1 such that f is holomorphic on X 1 \I and the closure of the graph of f over X 1 \ I is an irreducible analytic subset of dimension n 1 of X 1 × X 2 . The smallest set I is called the indeterminacy set of f . Note that I has codimension at least 2. Now we consider the meromorphic transforms from X to PH 0 (X, L p ) induced by the Kodaira maps. The meromorphic transform F p : X → PH 0 (X, L p ) has the following graph

Γ p = {(x, s) ∈ X × PH 0 (X, L p ) : s(x) = 0}.
Since L is ample, dim H 0 (X, L p ) ≥ 2 (cf. Theorem 3.4.1), for every point x ∈ X, there exists a point s ∈ PH 0 (X, L p ) such that s(x) = 0. Hence the projection from Γ p to X is surjective. Since L p is not trivial, there are no nowhere vanishing sections. That is to say, every point s ∈ PH 0 (X, L p ) must vanish at some point x ∈ X. Hence the projection from Γ p to PH 0 (X, L p ) is surjective. Then F p is indeed a meromorphic transform of codimension n -1.

Recall that ω F S denotes the Fubini-Study form of P np ∼ = PH 0 (X, L p ). The Fubini-Study current Φ p (ω F S ) of H 0 (X, L p ) is defined to be the pullback of ω F S by the Kodaira map Φ p . It is actually a L 1 -form, which is smooth outside the indeterminacy set of Φ p by the following lemma.

Lemma 2.3.1. Let f : X 1 → X 2 be a meromorphic map between two complex compact manifolds X 1 and X 2 of dimension n 1 and n 2 respectively. If φ is a smooth form of bidegree (p, q) on X 2 , then f (φ) is a L 1 -form, which is smooth outside the indeterminacy set of f . Proof. Let π 1 , π 2 be the natural projections from X 1 × X 2 to X 1 and X 2 respectively. Denote by Γ the graph of f . The regular part of Γ is reg(Γ). By definition, we have

f (φ) := (π 1 ) (π 2 (φ) ∧ [Γ]).
Then for any test form ψ on X 1 ,

f (φ), ψ = reg(Γ) π 2 (φ) ∧ π 1 (ψ).
Then f (φ) can be extended to act on the space of continuous forms. Hence it is a current of order 0. If V is a proper analytic subset of X 1 , then π -1 1 (V ) ∩ Γ is also a proper analytic subset of Γ and π -1 1 (x 1 ) ∩ Γ is of measure 0 for x 1 generic. So f (φ) has no mass on V , hence on the indeterminacy set I.

Note that π 1 : Γ \ π -1 1 (I) → X 1 \ I is a biholomorphic, therefore, we obtain

f (φ), ψ = X 1 \I π 1 π 2 (φ) ∧ ψ = X 1 \I g (φ) ∧ ψ,
where g is the restriction map of f on X 1 \ I, which is holomorphic. Then f (φ) = g (φ) is a smooth form on X 1 \ I. In addition, f (φ) has measure coefficients and has no mass on I. Hence g (φ) has L 1 -coefficients and is equal to f (φ) in the sense of currents. The proof is completed.

Note that δ p := λ np-1 (F p ) (resp. d p := λ np (F p )) is the intermediate degree of order n p -1 (resp. n p ) of F p . Lemma 2.3.2. In the above setting, δ p is bounded and

d p = pc 1 (L) n . More- over, F p (ω np F S ) = Φ p (ω F S ).
Proof. The first assertion is proved in [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Lemma 7.1] by using cohomological arguments. We prove the second one with the definition of F p . For any test (n -1, n -1)-form ψ, we have

F p (ω np F S ), ψ = Γp π 1 (ψ) ∧ π 2 (ω np F S ) = PH 0 (X,L p ) π 2 π 1 (ψ) ∧ ω np F S = PH 0 (X,L p ) π -1 2 (sp)∩Γp π 1 (ψ)ω np F S (s p ) = PH 0 (X,L p ) {x∈X:sp(x)=0} ψω np F S (s p ) = PH 0 (X,L p ) [s p = 0], ψ ω np F S (s p ) = Φ p (ω F S ), ψ .
The last equality follows from [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF]Proposition 4.2]. This completes the proof.

Proof of Theorem 0.2.1

From now on we recall the notations from Section 2.2.1 in the special case. Suppose that µ is a PLB probability measure on P k . F is defined in (1.1) when

X = P k . Let Q(P k , ω F S ) = {φ q.p.s.h. on P k : dd c φ ≥ -ω F S }, R(P k , ω F S , µ) = sup φ -φdµ, φ ∈ F , S(P k , ω F S , µ) = sup φ φdµ , φ ∈ Q(P k , ω F S ), φω k F S = 0 , ∆(P k , ω F S , µ, t) = sup φ µ(φ < -t), φ ∈ Q(P k , ω F S ), φdµ = 0
for any t > 0. When µ = ω k F S , we write R 0 (P k , ω F S ) = R(P k , ω F S , µ). Let σ p be a PLB probability measure on PH 0 (X, L p ). To simplify the notations, let

R p := R(PH 0 (X, L p ), ω F S , σ p ), R 0 p := R(PH 0 (X, L p ), ω F S , ω np F S ), S p := S(PH 0 (X, L p ), ω F S , σ p ), ∆ p (t) := ∆(PH 0 (X, L p ), ω F S , σ p , t).

2.3.4.

The second one is valid when S p = o(p) by Theorem 2.3.5. By applying Proposition 2.1.1 and Proposition 2.1.2, the proof is reduced to the estimates of R p /p and p≥1 ∆(pt) for any t > 0.

End of the proof of Theorem 0.2.1. We have

F p (ω np F S ) = (Φ p ) ω F S by Lemma 2.3.2. It follows from Theorem 2.3.6 that p -1 F p (ω np F S ) → ω (2.11)
in the weak sense of currents. We write

µ 1,p = ω np F S , µ 2,p = ∧ np j=1 (dd c u p,j + p ω F S + ω F S ) -µ 1,p . Then σ p ≤ µ 1,p + µ 2,p . Note that n p = c 1 (L) n p n /n! + O(p n-1 ). Let c > ( 12 ρ ) 2c 1 (L) n /n! > 1 such that c p n ≥ 1 β 0 n 3 p ( 12 ρ ) 2np
, then c depends only on X, L and ρ. Hence µ 2,p is a positive moderate measure satisfying Proposition 2.2.6. To estimate ∆ p , we consider any q.p.s.h. function φ on P np such that dd c φ ≥ -ω F S and φdσ p = 0. Set ϕ := φ -max P np φ. It is obvious that ϕ ∈ F by definition in (1.1). Since φdσ p = 0, max P np φ ≥ 0. Hence ϕ ≤ φ. Then we have

σ p (φ < -pt) ≤ σ p (ϕ < -pt) ≤ µ 1,p (ϕ < -pt) + µ 2,p (ϕ < -pt) ≤ exp(α 0 (-pt -ϕ))dµ 1,p + exp(α 0 ( ρ 4 ) np (-pt -ϕ))dµ 2,p ≤ c 0 n p exp(-α 0 pt) + c 5 ( ρ 4 ) np exp(-α 0 ( ρ 4 ) np pt).
The last inequality follows from Proposition 1.2.48 and Proposition 2.2.6. Then by the definition of ∆ p , we have

p≥1 ∆ p (pt) ≤ p≥1 c 0 n p exp(-α 0 pt) + p≥1 c 5 ( ρ 4 ) np exp(-α 0 ( ρ 4 ) np pt). (2.12)
It is obvious that p≥1 p n exp(-pt) < ∞ and that exp(-( ρ 4 ) np pt) tends to 1 when p tends to infinity, ∀t > 0. This yields p≥1 ∆ p (pt) < ∞. By Proposition

-1 F p (σ p ) -p -1 F p (ω np F S ) → 0 (2.15)
in the weak sense of currents. We know that F p (δ sp ) = [s p = 0] by Lemma 2.3.3. Combined with (2.12) and (2.14), Theorem 2.3.4 implies that for σalmost everywhere s ∈ P X , the following sequence

p -1 [s p = 0] -p -1 F p (σ p ) → 0 (2.16)
in the weak sense of currents. Then we deduce from (2.11), (2.15) and (2.16) that for σ-almost everywhere s ∈ P X ,

| p -1 [s p = 0] -ω, ψ | ≤ | p -1 [s p = 0] -p -1 F p (σ p ), ψ | + | p -1 F p (σ p ) -p -1 F p (ω np F S ), ψ | + | p -1 F p (ω np F S ) -ω, ψ | → 0,
for any test form ψ of bidegree (n -1, n -1) on X when p tends to ∞. That is to say, p -1 [s p = 0] converges weakly to ω. The proof is completed.

Nontrivial examples of moderate measures

Now given X and L in Theorem 0.2.1, we construct a concrete example of a sequence of functions (u p,j ) satisfying the conditions of the theorem. We require that u p,1 = • • • = u p,np = u p . Notice that we can perturbate u p so that the constants ξ p , p do not change and the perturbed functions still satisfy the conditions in Theorem 0.2.1. d F S (z,w) ρ . We will show that

d k ≤ √ πk (2.18)
at the end of the example.

For each p, we obtain a corresponding function v p by using (2.17) and identifying PH 0 (X, L p ) with P np . Consider the functions u p = c p v p with suitable constants

c p = O( 1 p n c p n ) < 1/c p n , where c = (145) c 1 (L) n /n! . Let n := c n . Since k n = O(n k ), it follows from (2.18) that d kn = O(n k ).
Consequently, u n is of class C ρ with modulus 1/c p n . Moreover, since v p is ω F S -p.s.h., we infer that u p is p ω F S -p.s.h.. So {u n } satisfy the three conditions in Theorem 0.2.1. From the above proof, we see that σ = p≥1 σ p = (dd c u p + ω F S ) np satisfies the equidistribution property.

Finally we prove (2.18). It is sufficient to consider the special case when

|z 0 | ≥ max{|z 1 |, ..., |z k |}, |w 0 | ≥ max{|w 1 |, ..., |w k |}. Then d k = 1 2 sup z,w∈K z =w log(1 + |z| 2 ) -log(1 + |w| 2 ) d F S (z, w) ρ where z = ( z 1 z 0 , ..., z 1 z 0 ), w = ( w 1 w 0 , ..., w 1 w 0 ) ∈ C k and K = {z ∈ C k : |z i | ≤ 1, 1 ≤ i ≤ k}. Let g = 2k i,j=1 g ij dx i ⊗ dx j be the associated Riemannian metric with g 11 = 1 π 1+|z| 2 -|z 1 | 2 (1+|z| 2 ) 2 . When r 1 = |z|, r 2 = |w| are fixed, d F S (
z, w) takes its minimum only when z and w are at the same line through the origin in R 2k . The distance is invariant with respect to the orthogonal group O(2k) in this case since the Fubini-Study metric is invariant with respect to the unitary group U (k) on P k . So we take the simple case when z = (r 1 , 0, ..., 0), w = (r 2 , 0, ..., 0). Hence

d k = √ π 2 sup 0≤r 1 <r 2 ≤k log(1 + r 2 2 ) -log(1 + r 2 1 ) (arctan r 2 -arctan r 1 ) ρ = √ π 2 sup 0≤s 1 <s 2 ≤arctan k log(1 + tan 2 s 2 ) -log(1 + tan 2 s 1 ) (s 2 -s 1 ) ρ ≤ √ π 2 max log(1 + k 2 ), sup s 2 -s 1 <1 0≤s 1 <s 2 ≤arctan k log(1 + tan 2 s 2 ) -log(1 + tan 2 s 1 ) s 2 -s 1 .
The function y = log(1 + tan 2 x) is increasing and convex on [0, ∞). So the second term in the last inequality is equal to (log(1 + tan 2 s))

s=arctan k = 2k.
This completes the proof of (2.18).

Proof of

| p -1 F p (σ p ) -p -1 F p (ω np F S ), ψ | ≤ C 2 log p p ψ C 2 (2.20)
for some positive constant C 2 depending only on X, L. Set

E p ( 0 ) := ψ C 2 ≤1 {s p ∈ PH 0 (X, L p ) : | p -1 [s p = 0] -p -1 F p (σ p ), ψ | ≥ 0 }
for any 0 > 0. We define E p := E p ( C 3 log p p ), where C 3 is some positive constant depending only on X, L. Note that R p = O(log p) from inequalities (2.13) and (2.14). By applying Theorem 2.1.7, we deduce that

σ p (E p ) ≤ ∆ p (C 4 log p).
Here C 4 is a positive constant depending only on X, L. Moreover, C 4 is sufficiently large such that α 0 C 4 > k + 2 since C 3 can be chosen sufficiently large. Recall that α 0 is the constant defined in Proposition 1.2.48. Then by (2.12), we obtain

σ p (E p ) ≤ ∆ p (C 4 log p) ≤ c 0 n p exp(-α 0 C 4 log p) + c 5 ( ρ 4 ) np exp(-α 0 C 4 ( ρ 4
) np log p)

≤ (c 0 + c 5 )n p 1 p α 0 C 4 ≤ C p 2 . (2.21)
Here C is a positive constant sufficiently large which depends only on X, L. Note that the third inequality of (2.21) follows from a direct calculation when p is big enough. The fact that n p = O(p n ) yields the last inequality of (2.21). By definition of E p , we obtain for any point The proof is completed.

s p ∈ PH 0 (X, L p ) \ E p , | p -1 [s p = 0] -p -1 F p (σ p ), ψ | ≤ C 3 log p p ψ C 2 . ( 2
Remark 2.3.8. Since ∞ p=1 σ p (E p ) < ∞, Theorem 0.2.2 gives an alternative proof of Theorem 0.2.1. This is a standard proof which is analogous to that of Borel-Cantelli lemma. Note that

∞ p=1 σ p (E p ) < ∞. Define E := {s = {s p } ∞
p=1 ∈ P X : s p ∈ E p for infinitely many p}. It is easy to see that E is contained in the following set ẼN := {s = {s p } ∞ p=1 ∈ P X : s p ∈ E p for at least one p ≥ N } for each integer N ≥ 1. Hence we have

σ(E) ≤ σ( ẼN ) ≤ ∞ p=N σ p (E p ) ≤ C ∞ p=N 1 p 2 .
The proof is completed by letting N tend to ∞.

Chapter 3

Equidistribution on big line bundles for moderate measures.

In this chapter, we establish an equidistribution theorem for the common zeros of random sections of high powers of several singular Hermitian big line bundles associated to moderate measures [START_REF] Shao | Equidistribution on big line bundles with singular metrics for moderate measures[END_REF]. The basic setting is taken from Part II in Chapter 0.

Intersection of Fubini-Study currents

In this section, we introduce some results about the intersection of the Fubini-Study currents associated to m line bundles. We will see that the current c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) in Theorem 0.3.1 is well-defined. Based on the elementary techniques in [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF], we also show that for almost all the zerodivisors of sections of high powers of the bundles with respect to a moderate measure are in general position. Then it follows from Corollary 1.2.26 (see also [START_REF] Demailly | Monge-Ampère operators, Lelong numbers and intersection theory[END_REF]) that the currents [S p = 0] are well-defined for almost all S p with respect to moderate measures σ in Theorem 0.3.1.

We keep the notations and hypotheses in Section 0.

It implies that dd c u k,p = 1 p γ k,p and 1 p γ k,p = c 1 (L k , h k ) + 1 2p dd c log P k,p .

Since log P k,p is a global function which belongs to L 1 (X, ω n ), 1 p γ k,p has the same cohomology class as c 1 (L k , h k ). Define the base locus of H 0 (2) (X, L p k ) as [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF]Theorem 5.1] and its proof, we know that there exist constants C > 0, p 0 ∈ N, such that

A k,p = {x ∈ X : S j k,p = 0, ∀0 ≤ j ≤ d k,p }, 1 ≤ k ≤ m. Note that A k,p is an analytic subset, A k,p ∩ U = {u k,p = -∞} and {u k = -∞} ⊂ A k ∩ U . By
P k,p (x) ≥ C, ∀p ≥ p 0 , 1 ≤ k ≤ m, x ∈ X \ A k .
Then we have

u k,p (x) ≥ u k (x) + log C 2p , ∀p ≥ p 0 , 1 ≤ k ≤ m, x ∈ U. (3.3)
Hence A k,p ⊂ A k for p ≥ p 0 . Moreover,

1 p log P k,p (x) → 0 (3.4) in L 1 (X, ω n ).
We have the following result [14, Proposition 3.1].

Proposition 3.1.1. We keep the notations and hypotheses of Theorem 0.3.1, then (i) For every J ⊂ {1, ..., m}, J = {1, ..., m} \ J, the analytic subsets A k,p and A l , for k ∈ J, l ∈ J , are in general position, when p is sufficiently large.

(ii) The currents

k∈J γ k,p ∧ l∈J c 1 (L l , h l )
are well defined for every J ⊂ {1, ..., m}, when p is sufficiently large.

Proof. Since the analytic subsets A 1 , ..., A m are in general position, the fact that A k,p ⊂ A k for all p sufficiently large yields (i). Hence the currents

k∈J γ k,p ∧ l∈J c 1 (L l , h l )
are well defined by Corollary 1.2.26. Then (ii) is proved.

Bertini theorem associated to moderate measures

Lemma 3.2.1. Let σ be a moderate measure on P N . Then every proper analytic subset of P N has measure zero with respect to σ.

Proof. By the homogeneity of P N , it is sufficient to prove that σ([z 0 ]) = 0 for some homogeneous coordinate [z 0 , ..., z N ]. Recall that for a moderate measure σ, there exist constants c > 0, α > 0 such that

P N exp(-αφ)dσ ≤ c,
∀φ ∈ F, where F is defined in (1.1). It follows that

P N |φ|dσ < ∞. Let φ = log |z 0 | |z| , where |z| 2 = N j=0 |z j | 2 .
This function is well defined on P N . Note that φ is ω F S -p.s.h., max z∈P N φ(z) = 0. Then φ ∈ F. So we have that

[z 0 =0] log |z 0 | |z| dσ < ∞.
Hence σ([z 0 ]) = 0. The proof is completed.

Adapting the proof of [14, Proposition 3.2], we obtain the following Bertini type theorem in the context of moderate measures. Now we can assume that σ k (U k ) = 1 for any U k defined as above. Set

U k+1 = {(t 1 , ..., t k+1 ) ∈ P d 1 × ... × P d k+1 : dim Z(t 1 ) ∩ ... ∩ Z(t k+1 ) ∩ Ṽj ≤ n -k -1 -j}, U = {(t 1 , ..., t k ) ∈ P d 1 × ... × P d k : dim Z(t 1 ) ∩ ... ∩ Z(t k ) ∩ Ṽj ≤ n -k -j}, U = {(t 1 , ..., t k ) ∈ P d 1 × ... × P d k : dim Z(t 1 ) ∩ ... ∩ Z(t k ) ∩ Ṽj ∩ BsV k+1 ≤ n -k -1 -j}, where 0 ≤ j ≤ m -k -1, Ṽj = BsV i 1 ∩ ... ∩ BsV i j for k + 2 ≤ i 1 < ... < i j ≤ m. Let U = U ∩ U . By using the induction on k, we know that σ k (U ) = σ k (U ) = 1, thus σ k (U ) = 1.
We need to prove that σ k+1 (U k+1 ) = 1.

It is enough to prove that

σ k+1 (W ) = 0, W := (U × P d k+1 ) \ U k+1 .
Given some t = (t 1 , ..., t k ) ∈ U , set

Z(t) : = Z(t 1 ) ∩ ... ∩ Z(t k ), W (t) : = {t k+1 ∈ P d k+1 : dim Z(t) ∩ Ṽj ∩ Z(t k+1 ) ≥ n -k -j},
then it is sufficient to show that σ k+1 (W (t)) = 0. Let Z(t)∩ Ṽj = N l=1 D l ∪B, where D l are irreducible components of Z(t)∩ Ṽj of dimension n -k -j, dim B ≤ n -k -1 -j as t ∈ U ⊂ U . By the same argument in the above, if t k+1 ∈ W (t), then Z(t) ∩ Ṽj ∩ Z(t k+1 ) is an analytic subset of Z(t) ∩ Ṽj of dimension n -k -j, hence there exists some l such that D l ⊂ Z(t) ∩ Ṽj ∩ Z(t k+1 ). We obtain that

W (t) = N l=1 F l (t), F l (t) := {t k+1 ∈ P d k+1 : D l ⊂ Z(t k+1 )}.
We claim that not all the sections of V k+1 can vanish on D l . If not, that is to say, D l ⊂ BsV k+1 , this implies that dim Z t ∩ Ṽj ∩ BsV k+1 = n -k -j, which contradicts the fact that t ∈ U . Hence we can suppose that S k+1,d k+1 ≡ 0 on D l . So

F l (t) ⊂ {t k+1,0 = 0} ∪ G l (t), G l (t) := {[1 : t k+1,1 : ... : t k+1,d k+1 ] ∈ P d k+1 : D l ⊂ Z([1 : t k+1,1 : ... : t k+1,d k+1 ])}.
There exists at most one η ∈ C such that [1 : t k+1,1 : ... : t k+1,d k+1 -1 : η] ∈ G l (t) for any (t k+1,1 , ..., t k+1,d k+1 -1 ) ∈ C d k+1 -1 . Otherwise, if there exist two complex numbers η = η , which satisfy the property, then we have on D l , S k+1,0 + t k+1,1 S k+1,1 + ... + t k+1,d k+1 -1 S k+1,d k+1 -1 + ηS k+1,d k+1 ≡ 0, S k+1,0 + t k+1,1 S k+1,1 + ... + t k+1,d k+1 -1 S k+1,d k+1 -1 + η S k+1,d k+1 ≡ 0.

Then we have a contradiction with that S k+1,d k+1 ≡ 0 on D l . This implies that σ k+1 (G l (t)) = 0. Moreover, σ k+1 (F l (t)) = 0. It follows that σ k+1 (W (t)) = 0. This completes the proof.

In the setting of Theorem 0. (ii)

Z(t i 1 ) ∩ ... ∩ Z(t i k ) is of pure dimension n -k for each 1 ≤ k ≤ m, 1 ≤ i 1 < ... < i k ≤ m.
[Z(t k ) = 0] = dd c log | d k,p j=0 t k,j s k,p j | = pc 1 (L k , h k ) + dd c log | d k,p j=0 t k,j S j k,p | h k,p . Since log | d k,p j=0 t k,j S j k,p | h k,p is a global p.s.h. function, d c log | d k,p j=0 t k,j S j k,p | h k,p is a current of order at most 1. Then dd c log | d k,p j=0 t k,j S j k,p | h k,p is closed. Hence [Z(t k ) = 0]
has the same cohomology class as pc 1 (L k , h k ). By (i), the current [Z(t i 1 ) = 0]∧...∧[Z(t i k ) = 0] is well defiend and supported in Z(t i 1 )∩...∩Z(t i k ), for each 1 ≤ k ≤ m, 1 ≤ i 1 < ... < i k ≤ m and allmost all (t 1 , ..., t m ) ∈ P d 1,p × ... × P dm,p with respect to σ p . Since L k is a big line bundle and c 1 (L k , h k ) ≥ ω for some constant > 0, then we have

Z(t i 1 )∩...∩Z(t i k ) ω n-k = X [Z(t i 1 ) = 0] ∧ ... ∧ [Z(t i k ) = 0] ∧ ω n-k = p k X c 1 (L i 1 , h i 1 ) ∧ ... ∧ c 1 (L i k , h i k ) ∧ ω n-k > p k k X ω n > 0.
It follows that Z(t i 1 ) ∩ ... ∩ Z(t i k ) = ∅. By Chapter III in [START_REF] Lojasiewicz | Introduction to complex analytic geometry[END_REF],

codim (Z(t i 1 ) ∩ ... ∩ Z(t i k )) ≤ k j=1 codim Z(t i j ) = k.
Moreover, Z(t i 1 ), ..., Z(t i k ) are in general position. Hence it is of pure dimension n -k. (ii) is then proved.

Estimate on multi-projective spaces

In this section we give our core work about some estimates on multiprojective spaces. This leads to our main theorem and convergence speed for equidistributions.

Meromorphic transforms for several line bundles

We recall some results in Dinh-Sibony equidistribution theory in the setting of Theorem 0.3.1. Let Φ p be a sequence of meromorphic transforms from a projective manifold (X, ω) into the compact Kähler manifolds (X p , ω p ) of the same codimension k. Let d 0,p = d 1,p + ... + d m,p be the dimension of X p . Consider a PLB probability measure µ p on X p , for every p > 0, > 0, we define

E p ( ) := φ C 2 ≤1 {x p ∈ X p : Φ p (δ xp ) -Φ p (µ p ), φ ≥ d(Φ p ) },
where δ xp is the Dirac measure at the point x p . Let us restate Theorem 2.2.7 in the setting of Part II.

Theorem 3.3.1. Let η ,p := δ(Φ p ) -1 d(Φ p ) -3R(X p , ω p , µ p ), then µ p (E p ( )) ≤ ∆(X p , ω p , µ p , η ,p ).
Another one is the estimate from Theorem 2.2.6. Theorem 3.3.2. We have

d(Φ p ) -1 (Φ p (µ p ) -Φ p (ω d 0,p p )), φ ≤ 2S(X p , ω p , µ p )δ(Φ p )d(Φ p ) -1 φ C 2 for any (k, k)-form φ of class C 2 on X.
From now on, we study the special case when the meromorphic transforms are induced by Kodaira maps. We already know that the Kodaira map in (3.1) is a meromorphic transform with the graph

Γ k,p = {(x, S) ∈ X × PH 0 (2) (X, L p k ) : S(x) = 0}.
Refer to Section 2.4.1. Note that

Φ k,p (x) = {S ∈ PH 0 (2) (X, L p k ) : S(x) = 0}, Φ -1 k,p (S) = {x ∈ X : S(x) = 0}. Let Φ p : X → X p
be the product map of Φ 1,p , ..., Φ m,p . We claim that Φ p is also a meromorphic transform with the graph

Γ p = {(x, S p1 , ..., S pm ) ∈ X × X p : S p1 (x) = ... = S pm (x) = 0}.
It is obvious that the projection Π 1 : Γ p → X is surjective. The projection Π 2 : Γ p → X p is proper, then the image Π 2 (Γ p ) is an analytic subvariety of X p by Remmert's proper mapping theorem [START_REF] Gunning | Introduction to holomorphic functions of several variables II[END_REF]. Note that the zero set of every S pk ∈ H 0 (2) (X, L p k ) is represented by Z(t k ) for some t k defined in Proposition 3.2.2 (ii). Then by Proposition 3.2.3 for almost every (S p1 , ..., S pm ) ∈ X p with respect to σ p , the common zero set of S p1 , ..., S pm is of pure dimension n-m ≥ 0. Then there exists some point x ∈ X such that (x, S p1 , ..., S pm ) ∈ Γ p . So σ p (Π 2 (Γ p )) = 1. Hence Π 2 is surjective. Indeed, if Π 2 is not surjective, then Π 2 (Γ p ) is a proper analytic subvariety of X p , Lemma 3.2.1 implies that σ p (Π 2 (Γ p )) = 0, a contradiction. Hence Φ p is a meromorphic transform of codimension n -m with fibers for every S p = (S p1 , ..., S pm ) ∈ X p , Φ -1 p (S p ) = {x ∈ X : S p1 (x) = ... = S pm (x) = 0}.

Considering the product map of any Φ i 1 ,p , ..., Φ i k ,p , 1 ≤ i 1 < ... < i k ≤ m, it follows from Proposition 3.2.3 that, the analytic subsets (S p1 = 0), ..., (S pm = 0) are in general position for S p = (S p1 , ..., S pm ) ∈ X p generic. Then by Corollary 1.2.26, the current [S p = 0] = [S p1 = 0] ∧ ... ∧ [S pm = 0] of bidegree (m, m) is well defined for allmost all S p ∈ X p with respect to σ p .

Intermediate degrees

It follows from Lemma 2.3.3 that Φ p (δ Sp ) = [S p = 0] for a point S p ∈ X p generic. Remark 3.3.3. By the same argument, note that Φ k,p (δ S pk ) = [S pk = 0] for each 1 ≤ k ≤ m. This yields Φ p (δ Sp ) = [S p = 0] = [S p1 = 0] ∧ ... ∧ [S pm = 0] = Φ 1,p (δ S p1 ) ∧ ... ∧ Φ m,p (δ Spm ). Recall that π k,p : X p → PH 0 (2) (X, L p k ) is the natural projection. Set ω p := c 0,p (π 1,p ω F S + ... + π m,p ω F S ).
We always assume that ω d 0,p p is a probability measure on X p . Then c 0,p satisfies the following condition, (c 0,p ) -d 0,p = d 0,p ! d 1,p !...d m,p ! .

The sequence {c 0,p } has a lower bound by using Stirling's formula (cf. [18, p9] and [START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF]Lemma 4.3]).

Lemma 3.3.4. There exists a positive constant c 0 such that c 0,p ≥ c 0 for all p ≥ 1.

To simplify the notations, we write

d p = d(Φ p ) := X Φ p (ω d 0,p p ) ∧ ω n-m , δ p = δ(Φ p ) := X Φ p (ω d 0,p -1 p ) ∧ ω n-m+1 .
Using the classical cohomological arguments, d p and δ p can be calculated as follows.

Proposition 3.3.5. In the above setting, we have

d p = p m c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) , δ p = p m-1 c 0,p m k=1 d k,p d 0,p m l=1,l =k c 1 (L l , h l ) .
Proof. We replace ω d 0,p p by a Dirac measure, since

H 2d 0,p (X p , C) ∼ = C.
Choose a Dirac measure δ S , where S = (S 1 , ..., S m ) ∈ X p , such that the analytic subsets (S 1 = 0), ..., (S 1 = 0) are in general position. By Remark 3.

3.3, the current Φ p (δ S ) = [S 1 = 0] ∧ ... ∧ [S m = 0]
is well defined. By the same argument in the proof of Proposition 3.2.3, we see that [S k = 0] has the same cohomology class as pc 1 (L k , h k ). By proposition 3.1.1, the current

c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m )
is well defined. Thus by Stokes' theorem,

X Φ p (δ S ) ∧ ω n-m = p m X c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) ∧ ω n-m . Hence d p = X Φ p (ω d 0,p p ) ∧ ω n-m = X Φ p (δ S ) ∧ ω n-m = p m c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) .
A direct computation gives the following equality

ω d 0,p -1 p = m k=1 c d 0,p -1 p (d 0,p -1)! d 1,p !...(d k,p -1)!...d m,p ! π 1,p (ω d 1,p F S )∧...∧π k,p (ω d k,p -1 F S )∧...∧π m,p (ω dm,p F S ).
Repeating the cohomological argument, we replace ω

d k,p F S (resp. ω d 1,p -1 F S
) by a generic point S k (resp. a generic complex line

D k ) in PH 0 (2) (X, L p k ). By the definition of Φ k,p , Φ k,p ([D k ]) = [Π 1k ({(x, S k ) ∈ X × D k , S k (x) = 0})].
Here Π 1k is the natural projection from X × PH 0

(2) (X, L p k ) to X. We show that Φ k,p ([D k ]) = [X]. Otherwise, if there exists a point x 0 ∈ X such that S k (x 0 ) = 0 for all sections S k ∈ D k , then by the genericity of D k , take S = S 2k (x 0 )S 1k -S 1k (x 0 )S 2k for some S 1k , S 2k ∈ D k . But S(x 0 ) = 0, we get a contradiction. So we have

Φ p ([{S 1 } × ... × D k × ... × {S m }]) = m l=1,l =k [S l = 0]. Hence Φ p ([{S 1 } × ... × D k × ... × {S m }]) = p m-1 m l=1,l =k c 1 (L l , h l ) .

Now we can replace ω

T := m k=1 d k,p c p d 0,p [{S 1 } × ... × D k × ... × {S m }]. So δ p = X Φ p (ω d 0,p -1 p ) ∧ ω n-m+1 = X Φ p (T ) ∧ ω n-m+1 = p m-1 c p m k=1 d k,p d 0,p m l=1,l =k c 1 (L l , h l ) .
This completes the proof.

(-α 0 ( ρ 4 ) N φ)dσ N ≤ β 0 N (3.5)
for all φ ∈ F, where α 0 < 1, β 0 are universal positive constants. That is to say, σ N is (β 0 N, α 0 ( ρ 4 ) N )-moderate. Proposition 3.3.9. Under the above hypotheses, there exist universal positive constants β 1 , β 2 , β 3 such that ∀t ∈ R,

R(P N , ω F S , σ N ) ≤ β 2 + 1 2 log N, ∆(P N , ω F S , σ N , t) ≤ β 0 N exp(-α 0 t) + β 1 ( ρ 4 ) N exp(-α 0 ( ρ 4 ) N t). When t ≤ 1 α 0 (log N + N log 4 ρ ), we have ∆(P N , ω F S , σ N , t) ≤ β 3 N exp(-α 0 t). Proof. By Proposition 2.1.2, R(P N , ω F S , ω N F S ) ≤ 1 2 (1 + log N ).
We write

µ 1,N = ω N F S , µ 2,N = N j=1 (dd c u j + c N ω F S + ω F S ) -ω N F S .
The measure µ 2,N is positive since u j is c N ω F S -p.s.h.. By Proposition 2.2.6, there exists a universal positive constant β 1 such that for all φ ∈ F,

P N exp(-α 0 ( ρ 4 ) N φ)dµ 2,N ≤ β 1 ( ρ 4 ) N . (3.6)
By applying (3.6) and Proposition 1.2.48, we obtain

R(P N , ω F S , σ N ) ≤ sup φ∈F -φdµ 1,N -φdµ 2,N ≤ R(P N , ω F S , ω N F S ) + sup φ∈F exp(-α 0 ( ρ 4 ) N φ)dµ 2,N /(α 0 ( ρ 4 ) N ) ≤ 1 2 (1 + log N ) + β 1 α 0 ≤ β 2 + 1 2 log N, It follows from (2.12) that ∆(P N , ω F S , σ N , t) ≤ β 0 N exp(-α 0 t) + β 1 ( ρ 4 ) N exp(-α 0 ( ρ 4 ) N t). Let N exp(-α 0 t) = ( ρ 4 ) N exp(-α 0 ( ρ 4 ) N t), then t = log N + N log 4 ρ α 0 (1 -( ρ 4 ) N ) . Hence ∆(P N , ω F S , σ N , t) ≤ β 3 N exp(-α 0 t), when t ≤ 1 α 0 (log N + N log 4 ρ )
. This completes the proof.

Now we study the estimates on multi-projective spaces. Let P 1 , ..., P m be m projective spaces. Let π k : P 1 × ... × P m → P k be the natural projection map. Let σ k be a probability moderate measure with respect to a family of (c k , ρ)-functions {u k,j } k j=1 on P k . In the sequel of this section, c k is always chosen such that the probability measure σ k satisfies the property of Proposition 3.3.8 (hence Proposition 3.3.9). Let = 1 + ... + m and

ω M P := c 1m (π 1 (ω F S ) + ... + π m (ω F S )), c - 1m = ! 1 !... m ! .
It is equivalent to that ω M P is a probability measure. Recall that the notation r(P

H 2 (P 1 × ... × P m , C) ∼ = C ⊕m .
Then T is in the cohomology class of λ = a 1 π 1 (ω F S ) + ... + a m π m (ω F S ), for some constatns a k ≥ 0, 1 ≤ k ≤ m. Then we have

0 ≤ λ ≤ max 1≤k≤m a k c 1m ω M P
in the sense of currents. Since

1 = T = P 1 ×...×P m λ ∧ ω -1 M P = c -1 1m m k=1 a k P 1 ×...×P m π k (ω F S ) ∧ (π 1 (ω F S ) + ... + π m (ω F S )) -1 = c -1 1m m k=1 a k ( -1)! 1 !...( k -1)!... m ! = m k=1 a k k c 1m , Then a k /c 1m ≤ / k , 1 ≤ k ≤ m. This implies that r(P 1 × ... × P m , ω M P ) ≤ max 1≤k≤m k .
The proof is completed.

We first consider the case when m = 2. The corresponding result of estimates in a simpler case was proved in [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF]Proposition A.8]. Set ω 12 := ω M P as the the Kähler form on P 1 × P 2 . Denote by σ the product of σ 1 and σ 2 . Write r := r(P 1 × P 2 , ω 12 ). Lemma 3.3.10 guarantees the existence of sufficiently large 1 , 2 such that r log( 1 + 2 ) min( 1 , 2 ) 1.

Proposition 3.3.11. In the above setting, let P 1 (resp. P 2 ) be a projective space endowed with a probability moderate measure σ 1 (resp. σ 2 ) satisfying Proposition 3.3.9. Suppose that 1 , 2 are chosen sufficiently large such that

r log( 1 + 2 ) min( 1 , 2 ) 1, ( ρ 4 ) min( 1 , 2 ) ( 1 + 2 ) 1.
(3.7)

Then there exist universal positive constants β 4 , β 5 such that for 0 ≤ t ≤ min( 1 , 2 ), we have

∆(P 1 × P 2 , ω 12 , σ, t) ≤ β 4 ( 1 + 2 ) 1+ α 0 2 exp(- α 0 2r t), R(P 1 × P 2 , ω 12 , σ) ≤ β 5 r(1 + log( 1 + 2 )). (3.8)
Proof. To simplify the notations, let X 1 = P 1 , X 2 = P 2 , X = X 1 ×X 2 . Denote by ω 1 (resp. ω 2 ) the normalized Fubini-Study form ω F S in P 1 (resp. P 2 ). Consider a function ψ on X with the conditions that max X ψ = 0, dd c ψ ≥ -rω 12 . Fix a point (a, b) such that ψ(a, b) = 0. Let E be the set of all points with ψ < -t for t ≥ 0. We write E = (ψ < -t). Set

F : = {x 2 ∈ X 2 , ψ(a, x 2 ) < -t/2}, E x 2 : = {x 1 ∈ X 1 , ψ(x 1 , x 2 ) < -t}. Define E := x 2 ∈X 2 \F (E x 2 × {x 2 }). Note that E ⊂ π -1 2 (F ) ∪ E . We first estimate the measure of π -1 2 (F ). Let ψ 1 (x 2 ) := ψ(a, x 2 ), then max X 2 ψ 1 = ψ 1 (b) = 0. Define a new function ψ 2 := ψ 1 -ψ 1 dσ 2 . Note that ψ 2 dσ 2 = 0, ψ 2 ≥ ψ 1 dd c ψ 2 ≥ -rω 2 .
Let R := β 2 + 1 2 log( 1 + 2 ). Since r(X 2 , ω 2 ) = 1, then by Proposition 3.3.9, we have

-ψ 1 dσ 2 = max X 2 ψ 2 ≤ rR(X 2 , ω 2 , σ 2 ) ≤ rR.
To estimate R(X, ω 12 , σ), we consider a function ψ on X with the conditions that max X ψ = 0, dd c ψ ≥ -rω 12 . For any 0 ≤ t ≤ t 0 ,

-ψdσ = ∞ 0 σ(ψ ≤ -t)dt = t 0 σ(ψ ≤ -t)dt + t 0 t σ(ψ ≤ -t)dt + ∞ t 0 σ(ψ ≤ -t)dt ≤ t 0 dt + ∞ t (β 0 + β 3 )( 1 + 2 ) exp(α 0 R) exp(- α 0 2r t)dt + 2 j=1 β 1 ∞ t 0 ( ρ 4 ) j exp(α 0 ( ρ 4 ) j R) exp(- α 0 2r ( ρ 4 ) j t)dt = t + 2r α 0 (β 0 + β 3 )( 1 + 2 ) exp(α 0 R) exp(- α 0 2r t) + 2 j=1 2r α 0 β 1 exp(α 0 ( ρ 4 ) j R) exp(- α 0 2r ( ρ 4 ) j t 0 ). (3.19)
The above inequality follows from (3.17) and (3.18). By the hypotheses in (3.7), the last term in the last equality is less than 5r α 0 β 1 for 1 , 2 sufficiently large. Hence

-ψdσ ≤ t + 2r α 0 (β 0 + β 3 )( 1 + 2 ) exp(α 0 R) exp(- α 0 2r t) + 5r α 0 β 1 . Take t = 2rR + 2r α 0 log((β 0 + β 3 )( 1 + 2 ))
. By the hypotheses in (3.7), t ≤ t 0 for 1 , 2 sufficiently large. We deduce that

-ψdσ ≤ 2rR + 2rR + 2r α 0 log((β 0 + β 3 )( 1 + 2 )) + 2r α 0 + 5r α 0 β 1 ≤ β 5 r(1 + log( 1 + 2 )),
where β 5 is a universal positive constant. This completes the proof.

The following proposition shows the main estimates in this section.

Proposition 3.3.12. In the above setting, let P k be a projective space endowed with a probability moderate measure σ k satisfying Proposition 3.

Proof of main theorems

In this section we will prove the main theorems.

Lower bound of dimensions

First we give an estimate of the dimension d k,p . The lower estimate is proved by construction of a new metric on the line bundle with only one singularity and application of vanishing theorem relative to multiplier ideal sheaves. Theorem 3.4.1. Let (X, ω) be a compact Kähler manifold of dimension n. Suppose that (L, h) is a singular Hermitian holomorphic line bundle on X such that c 1 (L, h) ≥ ω for some positive constant . Moreover, h is continuous outside a proper analytic subset A of X. Then there exist a constant C > 1 and p 0 ∈ N such that for all p ≥ p 0 p n /C ≤ dim H 0 (2) (X, L p ) ≤ Cp n . Proof. By Proposition 1.3.24, there exist a positive constant C 0 and p 0 ∈ N such that for all p ≥ p 0 dim H 0 (2) (X, L p ) ≤ C 0 p n . It is sufficient to show that there exist a positive constant C 1 and p 0 ∈ N such that for all p ≥ p 0 dim H 0 (2) (X, L p ) ≥ C 1 p n . Now we fix a point x 0 ∈ X \ A and r > 0 such that B(x 0 , 2r) ∩ A = ∅. Consider a smooth cut-off function 0 ≤ χ ≤ 1 such that it is equal to 1 on the closed set B(x 0 , r) and supported in B(x 0 , 2r). Define a new function

ψ : X → [-∞, ∞), ψ(x) = ηχ(x) log |x -x 0 |.
Here η is some positive constant. This function has only one singular point. Moreover, we consider a new metric on L as follows h 0 = h exp(-ψ).

η can be chosen sufficiently small such that on X, c 1 (L, h 0 ) ≥ 2 ω.

Indeed it suffices to show the following inequality

dd c (χ(x) log |x -x 0 |) + 2η ω ≥ 0.
Since dd c (log |x -x 0 |) = [x = x 0 ] is positive, we only prove the inequality in a small neighborhood of B(x 0 , 2r) \ B(x 0 , r), which is a simple conclusion by straightforward computations. Denote by I(h p ) the multiplier ideal sheaf associated to the metric h p . We have H 0

(2) (X, L p ) = H 0 (X, L p ⊗ I(h p )), I(h p 0 ) = I(h p ) ⊗ I(pψ). There is an identification of sheaves L p ⊗ I(h p 0 ) ∼ = K X ⊗ K X ⊗ L p ⊗ I(h p 0 ). Recall that the first Chern class of K X is Ric(ω) = -∂ ∂ log det(ω) which is smooth on X. Then c 1 (K X ⊗ L p ) = Ric(ω) + pc 1 (L, h 0 ) ≥ ω for some > 0 when p is sufficiently large. Hence it follows from the Nadel vanishing theorem (cf. Theorem 1.3.27) that there exists p 0 ∈ N such that for all p ≥ p 0 H 1 (X, L p ⊗ I(h p 0 )) = 0. Recall that γ k,p is the Fubini-Study current defined in Section 3.1. With techniques from [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF] and [START_REF] Fornaess | Oka's inequality for currents and applications[END_REF], we can show that the sequence of wedge products of these Fubini-Study currents converges weakly to the wedge product of the curvature currents of the line bundles in Theorem 0. ).

Here we can see that C 1 is some constant depending only on X, (L 1 , h 1 ), ..., (L m , h m ). Let Here C 2 is some constant depending only on X, (L This completes the proof. Therefore T corresponds to an entire plurisubharmonic function ψ in C n which belongs to the Lelong class(cf. [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Example 2.2]). By Theorem 3.4.7, there exists a sequence {p -1 log |f p |} which converges to ψ. Since holomorphic functions in C n can be approximated by polynomials, by using diagonal argument, we can choose a sequence of polynomials g p of degree ≤ p such that {p -1 log |g p |} converges to ψ. It is possible since all such p -1 log |g p | and ψ belong to the Lelong class. Since g p can be regarded as a homogeneous polynomial of degree p in C n+1 , it induces a global section S p ∈ H 0 (P n , O(p)). Hence by Lelong-Poincaré formula, 1 p [S p = 0] converges weakly to T . We obtain the following result due to Oka.

Further problems

We continue studying the equidistribution property and some approximations of currents by intersection of divisors. Several questions are posed as follows.

1) We will investigate the equidistribution of zeros of random holomorphic sections over more general spaces, such as non-compact manifolds [START_REF] Dinh | Asymptotic distribution of zeros of holomorphic sections in the non compact setting[END_REF] and normal Kähler complex spaces [START_REF] Coman | Equidistribution for sequences of line bundles on normal Kähler spaces[END_REF]. With emphasis on certain conditions of curvature forms of line bundles, I believe the equidistribution property would still hold, even in the context of moderate measures. We will also study some equidistribution properties with a limit of currents related to weighted global extremal functions (cf. [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]) and convergence results for Bergman kernels with equilibrium measures (cf. [START_REF] Berman | Bergman kernels and equilibrium measures for line bundles over projective manifolds[END_REF] and therein).

2) Approximation of currents by divisors is an interesting subject which was explored by many mathematicians, for example, Demailly [START_REF] Demailly | A numerical criterion for very ample line bundles[END_REF], Duval-Sibony [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF], Guedj [START_REF] Guedj | Approximation of currents on complex manifolds[END_REF] and Coman-Marinescu [START_REF] Coman | Convergence of Fubini-Study currents for orbifold line bundles[END_REF] etc. We would like to show that some positive closed current of bidegree (m, m) could be approximated by a sequence of intersections of divisors defined by m line bundles. It could be regarded as a generalization of results obtained in [START_REF] Coman | Convergence of Fubini-Study currents for orbifold line bundles[END_REF].

3) Following Theorem 0.3.5, we consider the following question. In the setting of Theorem 0.3.3, given any positive closed current T of bidegree (m, m), could we construct a family of diffuse measures σ p such that for σa.e. {S p } p≥1 ∈ P X , 1 p m [S p = 0] → T ? The key problem is the existence of approximation of any current by a sequence of currents (e.g. those currents defined by integration on varieties of codimension m).

4) We will study the equidistribution property of random sections with real coefficients, which is linked to the recent work by Gayet-Welschinger [START_REF] Gayet | What is the total Betti number of a random real hypersurface[END_REF].

  Dinh-Sibony [23, Theorem 4.1, Theorem 7.3] avoir des conditions très strictes. Un exemple explicite non-trivial d'une mesure singuliére avec des coefficients réels a été donnée dans [23, Corollary 7.4]. Mais il semble difficile de fournir une large classe de mesures générales pour satisfaire les théorèmes équidistribution.

Définition.

  Soit µ un PLB mesure sur X. Nous disons que µ est (c, α)modérée pour certains constantes c > 0, α > 0 si X exp(-αφ)dµ ≤ c pour tous les φ ∈ F. La mesure µ est appelé modérée s'il existe des constantes c > 0, α > 0 tel qu'il est (c, α)-modérée. Par exemple, la forme de volume ω n est modérée par Théorème 1.2.42. Voir Section 1.2 pour la notion de mesures modérées en détail. Afin d'exprimer notre théorème, nous devons aussi les terminologies suivantes. Fixez de quelque exposant 0 < ρ < 1, une fonction u : M → R définie sur un espace métrique compact (M, dist) est dit être de la classe C ρ avec module c si sup x,y∈M x =y |u(x) -u(y)| dist(x, y) ρ ≤ c. Considérons une variété complexe M avec une forme de volume fixe, soit γ un courant réel clos de bidegré (1, 1) sur M . Une fonction semi-continue supérieurement u : M → [-∞, ∞) en L 1 loc (M ) est dit γ-p.s.h. si dd c u + γ ≥ 0.

Theorem 1 . 1 . 15 .

 1115 Let X be equipped with a Hermitian metric h. Deonte by ω the associated form of h. Let Y be an oriented real submanifold of class C 1 and real dimension 2p in X. Denote by dV Y the Riemannian volume form on Y with respect to the induced metric h |Y . There exists a continuous function α in Y such that αdV Y = 1 2 p p! ω p |Y .

Theorem 1 . 1 . 16 .

 1116 Let Y be an analytic subset of pure dimension n -p of X. Denote by reg(Y ) be the regular part of Y . Define the (p, p)-current [Y ] by

Proposition 1 . 2 . 4 .

 124 One counterexample was shown by Fornaess [42, Example 2.9.4]. Proposition 1.2.3. Let Ω be an open subset of C n . Then we have P sh(Ω) ⊂ Sh(Ω) ⊂ L 1 loc (Ω). P.s.h. functions satisfy the maximum principal in bounded domains. Let Ω be a bounded connected open subset of C n . Let u ∈ P sh(Ω). Then either u is constant or, for each z ∈ Ω,

Positive closed ( 1 , 1 )

 11 -currents on P n can be associated to p.s.h. functions on C n+1 with certain homogeneity properties. See[START_REF] Fornaess | Complex dynamics in higher dimensions II[END_REF] Section 4] and[53, A.5]. Let L 1 be the set of positive closed (1, 1)-currents on P n . Consider a p.s.h. function v on C n+1 satisfying the conditionv(λz) = c log |λ| + v(z)for some constant c ≥ 0 and all z ∈ C n+1 . Let [v] be the class of functions which is equal to v up to a constant. Denote by L 2 the set of the classes of p.s.h. functions with the above condition. Proposition 1.2.41. The two sets L 1 and L 2 are isomorphic. If T ∈ L 1 has the associated p.s.h. function v such that v(λz) = c log |λ| + v(z) for some constant c ≥ 0, then the mass of T is c.

  Now we introduce the notion of moderate measures which is essential in the thesis. First we recall a classical result about an uniform estimate for certain compact family of p.s.h. functions in the unit ball of C n [36, Theorem 4.4.5].Theorem 1.2.42.There is a constant c > 0 such that for any p.s.h. function u in the unit ball of C n with u(0) > -1 and u(z) ≤ 0, we have |z|<1/2 exp(-u(z))dλ ≤ c, where dλ denotes the Lebesgue measure on the unit ball.

Definition 1 . 3 . 3 .

 133 The first Chern class of a complex line bundle L is defined to c 1 (L) := δ({g αβ }), where {g αβ } is the Cěch cohomology class of the 1-cocycle associated to L.

Definition 1 . 3 . 6 .

 136 The above unique Hermitian connection D is called Chern connection and the curvature Θ(D) of D is called Chern curvature.

  -Poincaré formula that dd c log |f α | = div(f α ). Note that L is a Hermitian holomorphic line bundle with the Hermitian metric h. There exists a local holomorphic frame e L of L over U α such that |e L | 2 h = h(e L , e L ) = e -2ϕ , where ϕ is a real function on U α . It is possible since we can shrink U α properly. Then |s| 2 h = |f α | 2 e -2ϕ . There is a characterization of Chern curvature by the Hermitian metric. Proposition 1.3.11. We have locally on U α i 2π Θ(L) = dd c ϕ.

  is called local weight. A singular Hermitian holomorphic line bundle is a holomorphic line bundle with a singular Hermitian metric. If ϕ is smooth, the L is a Hermitian line bundle. It is easy to see that dd c ϕ is independent of the choice of open coverings. We can define the curvature current by Θ(L) = -2πidd c ϕ. The current c L,h =: i 2π Θ(L) = dd c ϕ represents the first Chern class of L. The Lelong-Poincaré formula for meromorphic sections of singular Hermitian holomorphic line bundles also holds. Given an arbitrary divisor D on X, we have c 1 (O(D)) = {[D]}. Indeed, let D = a j D j , the line bundle O(D) can be endowed with a singular Hermitian metric h such that |f | h = |f |, where f is a meromorphic function with div(f )+ D ≥ 0. Let g j be the defining holomorphic function of D j on U α . Then there is a trivialization θ α (f ) = f g a j j of O(D) on U α . The singular Hermitian metric has local weight ϕ = a j log |g j |, The Lelong-Poincaré formula yields the equation c 1 (O(D)) = {[D]}. Definition 1.3.14. A Hermitian holomorphic line bundle L is called positive if there exists a smooth Hermitian metric on L with the Chern curvature Θ(L) such that iΘ(L) is a positive (1, 1)-form. A divisor D is positive if the line bundle O(D) is positive. The following result shows that the positivity of a line bundle is a topological property. Proposition 1.3.15. Let φ be a real closed (1, 1)-form whose cohomology class is c 1 (L). Then there exists a smooth Hermitian metric on L such that i 2π Θ(L) = φ. Hence L is positive if and only if c 1 (L) can be represented by a positive closed (1, 1)-form. Example 1.3.16. The hyperplane bundle O(1) on P n is a positive line bundle. The dual of O(1) is the tautological line bundle O(-1) whose fiber at a point z = [z 0 , ..., z n ] ∈ P n is the complex line {λz : λ ∈ C} ⊂ C n+1 . There is a natural Hermitian metric h on O(-1) such that |z| 2 h = |z i | 2 . Then the curvature form Θ of O(1) satisfy i 2π Θ = dd c log z 2 .

Definition 1 . 3 . 21 .

 1321 The Kodaira-Iitaka dimension of L is κ(L) := max{ p : p ≥ 1}. Note that κ(L) ≤ dim X. Definition 1.3.22. A big line bundle is a holomorphic line bundle L such that κ(L) = dim X.There are several criterions for big line bundles[45, 2.2.1, 2.3.3].

Example 2 . 1 . 4 .

 214 The examples are constructed based on Grassmannian manifolds.

Example 2 . 3 . 7 .

 237 Let π : C k+1 \ {0} → P k be the natural map. Consider the map f : P k → P k with f [z 0 , ..., z k ] = [z k 0 , ..., z k k ]. From [53, Example 1.6.4], its Green function is s(z) = max(log |z 0 |, ..., log |z k |). Moreover, s is a Hölder continuous function with any exponent 0 < ρ < 1. We obtain a well-defined function v := max(log |z 0 | |z| , ..., log |z k | |z| ) (2.17) on P k . Since π (dd c v + ω F S ) = dd c s ≥ 0, then v is ω F S -p.s.h. and Hölder continuous with any exponent 0 < ρ < 1. Denote by d F S the distance induced by Fubini-Study metric. Let d k = sup z,w∈P k z =w |v(z)-v(w)|

Proof.

  The base loci A 1,p , ..., A m,p are in general position for all p sufficiently large, by Proposition 3.1.1. Then (i) follows by Proposition 3.2.2. We fix such p and consider the current [Z(t k ) = 0]. Recall that e k is a local holomorphic frame for L k and S j k,p = s k,p j e ⊗p k , where s k,p j is a holomorphic function on a contractible Stein open subset in X. By Poincaré-Lelong formula, we have locally on U

Remark 3 . 3 . 6 .

 336 Lemma 3.3.4 implies that δ p ≤ Cp m-1 for some constant C > 0 which depends on(L k , h k ), 1 ≤ k ≤ m.Recall that γ k,p = Φ k,p ω F S is the Fubini-Study current. Proposition 3.3.7. [14, Lemma 4.5] Φ p (ω d 0,p p ) = γ 1,p ∧ ... ∧ γ m,p for all p sufficiently large.

  exp

(3. 21 )

 21 The following short exact sequence 0 → L p ⊗ I(h p ) ⊗ I(pψ) → L p ⊗ I(h p ) → L p ⊗ I(h p ) ⊗ O X /I(pψ) → 0 with(3.21) implies thatH 0 (X, L p ⊗ I(h p )) → H 0 (X, L p ⊗ I(h p ) ⊗ O X /I(pψ)) → 0 for all p ≥ p 0 .Here O X is the sheaf of holomorphic functions on X. Note that I(pψ) x = O X,x for every point x = x 0 . That is to say, O X,x /I(pψ) x = 0. Since h is continuous at x 0 , then I(h p ) x 0 = O X,x 0 . So we haveH 0 (X, L p ⊗ I(h p ) ⊗ O X /I(pψ)) = L p x 0 ⊗ I(h p ) x 0 ⊗ O X,x 0 /I(pψ) x 0 = L p x 0 ⊗ O X,x 0 /I(pψ) x 0 . Hence H 0 (X, L p ⊗ I(h p )) → L p x 0 ⊗ O X,x 0 /I(pψ) x 0 → 0 (3.22)for all p ≥ p 0 . Let M X,x 0 be the maximal ideal of O X,x 0 . Consider a germ f ∈ I(pψ) x 0 in a local coordinate z = (z 1 , ..., z n ) with x 0 = 0. Let f = a J z J be its Taylor expansion. Note that only the terms |a J | 2 |z| 2J contribute to the following integralB(0,a) |f | 2 |z| -2pη dV,where B(0, a) is a neighborhood of the origin and dV denotes the Lebesgue measure. Hence for each multi-index J, we obtainB(0,a) |z| 2J |z| -2pη dV < ∞.It is equivalent toa 0 r ( |J| -pη + n -1)dr < ∞ ⇐⇒ |J| ≥ [pη] -n + 1.ThenI(pψ) x 0 ⊂ M [pη]-n+1 X,x 0 , dim O X,x 0 /M k+1 X,x 0 = oflower bound of the dimensions follows from (3.22) and (3.23). The proof is completed.3.4.2 Proof of Theorem 0.3.1

C 3 1 pp φ C 2 .

 12 3.1, see [14, Proposition 3.1]. Proposition 3.4.2. In the setting of Theorem 0.3.1, we have 1 p m γ 1,p ∧ ... ∧ γ m,p → c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) when p tends to ∞. Proof. we consider the p.s.h. functions u k , u k,p on the contractible Stein open subset U defined in Section 3.1. Recall that ddc u k = c 1 (L k , h k ), dd c u k,p = 1 p γ k,p on U . We know that 1 p log P k,p → 0 in L 1 (X, ω n ) by Theorem 5.1 in [10]. So we have that u k,p → u k in L 1 loc (U ), ∀1 ≤ k ≤ m. By (3.3), the inequality u k,p ≥ u k -Cp holds on U for some constant C > 0 and all p sufficiently large. By Theorem 1.2.28, we havedd c u 1,p ∧ ... ∧ dd c u m,p → dd c u 1 ∧ ... ∧ dd c u min the weak sense of currents as p → ∞. This completes the proof.We also need the following convergence property. Proposition 3.4.3. In the setting of Theorem 0.3.1, there exists a positive constant C depending only on X, (L 1 , h 1 ), ..., (L m , h m ) such that1 p m (Φ p (σ p ) -Φ p (ω d 0,p p )), φ ≤ C log p p φ C 2for any (n -m, n -m)-form of class C 2 on X and p sufficiently large. In particular,1 p m (Φ p (σ p ) -Φ p (ω d 0,pp)) converges weakly to 0 as p → ∞.Proof. By Theorem 3.4.1, there exist a positive constant C 1 and p 0 ∈ N such that for all p ≥ p 0 , 1 ≤ k ≤ m, we havep n /C 1 ≤ d k,p ≤ C 1 p n .Then by Lemma 3.3.10, r(X p , ω p ) ≤ mC 2 1 . Moreover, d 1,p , ..., d m,p satisfy the conditions in (3.20) for p sufficiently large. Hence it follows from Proposition 3.3.12 that there exists a positive constant C 2 , S(X p , ω p , σ p ) ≤ C 2 log p. Thanks to Proposition 3.3.5 and Remark 3.3.6, we can deduce that δ p d -1 p ≤ for some positive constant C 3 . Note that the constants C 1 , C 2 , C 3 all depend only on X, (L 1 , h 1 ), ..., (L m , h m ). Then 2S(X p , ω p , σ p )δ p d -1 p ≤ C log p p for some positive constant C depending only on X, (L 1 , h 1 ), ..., (L m , h m ) Hence the proof is completed by applying Theorem 3.3.2. The basic proof of the main theorem will end with the following theorem, which extends [18, Corollary 3.9] and [14, Theorem 4.2]. Theorem 3.4.4. In the setting of Theorem 0.3.1, there exist a positive constant ξ which depends only on m and a positive constant C which depends only on X, (L 1 , h 1 ), ..., (L m , h m ) with the following property: Given any sequence of positive numbers {λ p } ∞ p=1 with the following conditions lim inf p→∞ λ p log p > (1 + ξn)C, lim p→∞ λ p p n = 0,there exist subsets E p ⊂ X p such that for all p sufficiently large, (i)σ p (E p ) ≤ Cp ξn exp(-λ p C ),(ii) for any point S p ∈ X p \ E p and any(n -m, n -m)-form φ of class C 2 , 1 p m [S p = 0] -Φ p (σ p ), φ ≤ Cλ pProof. To simplify the notations, letR p : = R(X p , ω p , σ p ), ∆ p (t) : = ∆(X p , ω p , σ p , t), E p ( ) : = φ C 2 ≤1 {S p ∈ X p : | [S p = 0] -Φ p (σ p ), φ | ≥ d p },where t ≥ 0, > 0. By Theorem 0.2.1 with its proof, there exists a constant c > 1 which depends only on X, L 1 , ..., L m , ρ such that each componentd k,p j=1 π k,p (dd c u k,p j + ω F S )of σ p is a probability moderate measure satisfying Proposition 3.3.9. Theorem 3.4.1 implies that d 1,p , ..., d m,p satisfy the conditions in (3.20) for p sufficiently large. Hence σ p satisfy Proposition 3.3.12. Let C be a positive constant depending only on X, (L 1 , h 1 ), ..., (L m , h m ) such that for all p ≥ p 0 , 1 ≤ k ≤ m, we have p n / C ≤ d k,p ≤ Cp n , Here p 0 is a positive integer large enough. Then we have for p ≥ p 0 and 0 ≤ t ≤ p n / C, R p ≤ mβ 6 C2 (1 + log(m Cp n )) ≤ C 1 log p, ∆ p (t) ≤ β 6 (m Cp n ) ξ exp( -βt m C2 ) ≤ C 1 p ξn exp(-t C 1

  := p d p δ -1 p -3R p .It follows from Proposition 3.3.5 and Remark 3.3.6 that for p ≥ p 0 ,η p ≥ C 2 λ p -3C 1 log p.

Proposition 3 . 4 . 5 . 3 ∞ p=1 1 pC 2 . 3 . 4 . 3 5

 3453123435 In the setting of Theorem 0.3.1, for almost all S = {S p } ∞ p=1 ∈ P X with respect to σ, we have1 p m [S p = 0] -1 p m Φ p (σ p ) → 0in the weak sense of currents as p → ∞ on X.Proof. Note that∞ p=1 σ p (E p ) ≤ C η < ∞for some positive constant C 3 and η > 1. The proof follows from the same argument in Remark 2.3.8.End of the proof of Theorem 0.3.1. By Theorem 0.2.1 with its proof, there exists a constant c > 1 which depends only on X, L 1 , ..., L m , ρ such that each componentd k,pj=1 π k,p (dd c u k,p j + ω F S ) of σ p is a probability moderate measure satisfying Proposition 3.3.9. Hence σ p satisfy Proposition 3.3.12.Note that c is independent of the choices of the metrics h 1 , ..., h m . It follows from Proposition 3.3.7 and Proposition 3.4.3 that 1 p m (Φ p (σ p ) -γ 1,p ∧ ... ∧ γ m,p ) → 0 in the weak sense of currents as p → ∞. Then Proposition 3.4.5 implies that for allmost all S = {S p } ∞ p=1 ∈ P X with respect to σ 1 p m ([S p = 0] -γ 1,p ∧ ... ∧ γ m,p ) → 0 in the weak sense of currents as p → ∞. The proof is finally completed by application of Proposition 3.4.2. Theorem 0.3.3 follows from Proposition 3.4.3 and Theorem 3.4.4 with the following theorem [14, Proposition 5.1].Theorem 3.4.6. In the setting of Theorem 0.3.3, there exists a positive constant C which depends only on X, (L 1 , h 1 ), ..., (L m , h m ) such that for all p sufficiently large and any (n -m, n -m)-form φ of class C 2 , we have1 p m γ 1,p ∧ ... ∧ γ m,p -c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ), φ ≤ C log p p φ Proof of Theorem 0.3.To prove Theorem 0.3.5, we need the following result[START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] Theorem 15.1.6].Theorem 3.4.7. Let P A be the set of all functions of the form p -1 log |f (z)| where p is a positive integer and f an entire function ≡ 0 in C n . Then the closure of P A in L 1 loc (C n ) consists of all plurisubharmonic functions. Let ω 0 be the Fubini-Study form with mass 1 in P n . Given any positive closed current T of bidegree (1, 1) with mass 1 in P n . Then by Proposition 1.2.19, there exists a q.p.s.h. function ϕ such that T -ω 0 = dd c ϕ.

  Nontrivial examples of moderate measures . . . . . . . . 2.3.4 Proof of Theorem 0.2.2 . . . . . . . . . . . . . . . . . . . 3 Equidistribution on big line bundles for moderate measures. 3.1 Intersection of Fubini-Study currents . . . . . . . . . . . . . . . 3.2 Bertini theorem associated to moderate measures . . . . . . . . 3.3 Estimate on multi-projective spaces . . . . . . . . . . . . . . . . 3.3.1 Meromorphic transforms for several line bundles . . . . . 3.3.2 Intermediate degrees . . . . . . . . . . . . . . . . . . . . 3.3.3 Main result and its proof . . . . . . . . . . . . . . . . . . 3.4 Proof of main theorems . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Lower bound of dimensions . . . . . . . . . . . . . . . . 3.4.2 Proof of Theorem 0.3.1 . . . . . . . . . . . . . . . . . . . 3.4.3 Proof of Theorem 0.3.5 . . . . . . . . . . . . . . . . . . . 3.5 Further problems . . . . . . . . . . . . . . . . . . . . . . . . . .

	2.3.3 Bibliographie. Chapter 0
	Thèse préparée au
	Département de Mathématiques d'Orsay
	Laboratoire de Mathématiques (UMR 8628), Bât. 425
	Université Paris-Sud 11
	91 405 Orsay CEDEX

  3.4. The primary difference between Theorem 0.3.3 and [14, Theorem 1.4] is that the measures in Theorem 0.3.3 are only moderate. The optimal estimate of the convergence speed in Theorem 0.3.

3 (ii) is of order O(log p/p). In this case, the measures σ p (E p ) are polynomially small. So Theorem 0.3.3 generalizes Theorem 0.2.2 and [18, Theorem 1.2].

  Dinh-Sibony[START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] ont généralisé le résultat dans le cas de mesures PLB et obtenu une bonne estimation de la vitesse de convergence sur une variété projective doté d'une métrique positivement courbe lisse. Voir Section 2.1 et Section 2.3 pour les définitions et notations en détail.

Les ingrédients clés pour prouver le théorème sont le théorème de Tian

[START_REF]The Bergman kernel and a theorem of Tian, Analysis and Geometry in Several Complex Variables[END_REF][START_REF] Zelditch | Szegö kernels and a theorem of Tian[END_REF]

(autres résultats sur les comportements asymptotiques des noyaux Bergman) et les estimations des variances des courants [s p = 0]. Quand X = P n , ω = ω F S et L = O(1) le dual de fibré en droites tautologique, le théorème est d'étudier la distribution des zéros de polynômes homogènes aléatoire ou les polynômes complexes correspondants. La distribution du SU (2) polynômes est un cas particulier, lorsque n = 1. Si on considère les polynômes aléatoires complexes d'une variable et le produit scalaire défini sur le cercle unitaire de C avec la mesure de Haar, on obtient alors le résultat classique de Hammersley

[START_REF] Hammersley | The zeros of a random polynomial[END_REF]

.

  soit m fibrés en droites gros singulières hermitiennes holomorphes sur X. La métrique h k est continue en dehors d'un sous-ensemble analytique propre A k ⊂ X, c 1 (L k , h k ) ≥ ω sur X pour une constante > 0,et A 1 , ..., A m sont en position générale. Soit 0 < ρ < 1.

	Alors il existe une
	constante c > 1 qui ne dépend que X, L k , ρ avec la propriété suivante: Si σ est la mesure de probabilité sur P X généré par une famille de (1/c p n , ρ)-fonctions {u j k,p } sur {PH 0 (2) (X, L p k )} défini par (2), alors pour presque partout
	{S p } p≥1 ∈ P X par rapport à σ, nous avons dans le sens faible des courants
	comme

  Théorème. Nous gardons les notations et les hypothèses du Théor me 0.3.1. Supposons, en outre, que h k est Hölder avec singularités le long A k , 1 ≤ k ≤ m. Alors il existe une constante ξ positive qui ne dépend que de m, et une autre constante positive C qui dépend X, (L 1 , h 1 ), ..., (L m , h m ) avec la propriété suivante: Compte tenu de toute séquence des nombres positifs {λ p } ∞

	il existe des sous-ensembles E p ⊂ X p tel que pour tout p suffisamment grand,
	(i)
				. Ce théorème
	est aussi une généralisation du théorème 0.2.2.
				p=1 avec les
	conditions suivantes
	lim inf p→∞	λ p log p	> (1 + ξn)C and lim p→∞	λ p p n = 0,

  or the proof of [45, Theorem 2.2.20]. The Siegel's lemma gives an upper bound of d p [45, Lemma 2.2.6].

	Proposition 1.3.24.

  Now we give a local analytic description of the above map. Let U ⊂ X be a contractible Stein open subset, e k a local holomorphic frame of L k on U . Then there exists a holomorphic function s k,p It is easy to see that this definition is independent of the choice of basis.Recall that ω F S is the normalized Fubini-Study form onP d k,p . The Fubini-Study current γ k,p of H 0 (2) (X, L p k ) is γ k,p = Φ k,p (ω F S ).Lemma 2.4.1 says that the Fubini-Study current is a L 1 -form, which is smooth outside the the indeterminacy set of Φ k,p . We have in the local Stein open subset U , Let u k be the local weight of the Hermitian metric h k on U . Then the following p.s.h. function verifies

	It is called the Kodaira map defined by the basis {S j k,p }	d k,p j=0 . Denote by P k,p
	the Bergman kernel function defined by
	d k,p
	P k,p (x) =	|S j k,p (x)| 2 h k,p , |S j k,p (x)| 2 h k,p = h k,p (S j k,p (x), S j k,p (x)).
	j=0
			γ k,p U =	1 2	dd c log	d k,p j=0	|s k,p j | 2 .
	u k,p :=	1 2p	log	d k,p j=0	|s k,p j | 2 = u k +	1 2p	log P k,p .
								3. Consider the Kodaira
	map
				Φ k,p : X → P(H 0 (2) (X, L p k ) ).
	Here H 0 (2) (X, L p k ) is the dual space of H 0 (2) (X, L p k ). By a similar argument in
	Section 1.3.3, choose {S j k,p } d k,p j=0 as an orthonormal basis of H 0 (2) (X, L p k ). By an
	identification via the basis, it boils down to a meromorphic map
						Φ k,p : X → P d k,p .
							j	on U such that S j k,p = s k,p j e ⊗p k . Then
	the map is expressed locally as
		Φ k,p (x) = [s k,p 0 (x) : ... : s k,p d k,p (x)], ∀x ∈ U	(3.1)

  . Then the base locus of V k,p is A k,p . Let Z(t k ) be an analytic hypersurface for any t k = [t k,0 : ... : t k,d k,p ] ∈ P d k,p , defined in Proposition 3.2.2 (ii). Let σ p be the product measure of probability moderate measures on P d 1,p × ... × P dm,p in Theorem 0.3.1. Arguing as in the proof of [14, Proposition 3.3], we obtain the following Proposition 3.2.3. In the above setting, (i) The analytic subsets Z(t 1 ), ..., Z(t m ) are in general position, for almost every (t 1 , ..., t m ) ∈ P d 1,p × ... × P dm,p with respect to σ p , when p is sufficiently large.

	3.1, let V k,p = H 0 (2) (X, L p k ) with orthonormal
	basis {S j k,p } d k,p

j=0

  1 ×...×P m , ω M P ) is defined after Proposition 1.2.34. We have the following lemma[START_REF] Coman | Hölder singular metrics on big line bundles and equidistribution[END_REF] Lemma 4.6].

	Lemma 3.3.10. Under the above hypotheses,

r(P 1 × ... × P m , ω M P ) ≤ r( 1 , ... m ) := max 1≤k≤m k . Proof. Consider a positive closed current T of bidegree (1, 1) with mass 1 on P 1 × ... × P m . It has a nontrivial cohomology class. By Künneth formula,

  3.9, ∀1 ≤ k ≤ m. Set σ := σ 1 × ... × σ m . Suppose that 1 , .., m are chosen sufficiently large such that Then there exist positive constants β 6 , β, ξ depending only on m such that for 0 ≤ t ≤ min( 1 , ..., m ), we haveR(P 1 × ... × P m , ω M P , σ) ≤ β 6 r( 1 , ... m )(1 + log ), S(P 1 × ... × P m , ω M P , σ) ≤ β 6 r( 1 , ... m )(1 + log ), ∆(P 1 × ... × P m , ω M P , σ, t) ≤ β 6 ξ exp(-βt/r( 1 , ... m )).Proof. When m = 2, the estimates on R and ∆ are proved in Proposition 3.3.11. When m = 3, following the notations in the proof of Proposition 3.3.11, we writeX 1 = P 1 ×P 2 , X 2 = P 3 , X = X 1 ×X 2 .The estimates on R and ∆ for X 1 (resp. X 2 ) are showed in Proposition 3.3.11 and (3.17) (resp. Proposition 3.3.9). Consequently, the results of estimates on R and ∆ for X are proved by using the analogous arguments in (3.9),(3.10),(3.13),(3.14) and (3.19) with the hypotheses (3.20). For the general case, the results can be deduced inductively by using the analogous arguments in the proof of Proposition 3.3.11. The estimate on S follows from Proposition 2.2.1 and [14, Lemma 4.6].

	(	4 ρ	) ) min( 1 ,..., m)	1, 1.	(3.20)

r( 1 , ..., m ) log min( 1 , ..., m

  1 , h 1 ), ..., (L m , h m ). p can be always chosen such that C 2 2 λ p < η p < p n / C for p sufficiently large. By applying Theorem 3.3.1 to the subset E p ⊂ X p , we obtainσ p (E p ) ≤ ∆ p (η p ) ≤ C 1 p ξn exp( -C 2 2C 1 λ p ),whereE p = E p ( p ). Now we set , c 1 (L 1 , h 1 ) ∧ ... ∧ c 1 (L m , h m ) .Then for all p sufficiently large, σ p (E p ) ≤ Cp ξn exp(

	C = max	6C 1 C 2 (1 + ξn)	,	2C 1 C 2	, C 1 -λ p C	)
	under the conditions that
		lim inf p→∞	λ p log p	> (1 + ξn)C,
				lim p→∞	λ p p n = 0.
							d p p m	λ p p	φ C 2 ≤	Cλ p p	φ C 2 .
							If there
	is a condition that	lim inf p→∞	λ p log p	>	6C 1 C 2	,

then for all p sufficiently large, η p > C 2 2 λ p . Since lim p→∞ λ p p n = 0, η By the definition of E p , it is obvious that for any S p ∈ X p \ E p and any (n -m, n -m)-form φ of class C 2 , 1 p m [S p = 0] -Φ p (σ p ), φ ≤

moderate. If u is a continuous real-valued function and uS defines a current on X (for example, if supp u ⊂ suppS), then dd c (uS) is well defined (cf. Section 1.2.2). We say that u is S-p.s.h. if dd c (uS) is a positive current. Dinh-Nguyên-Sibony [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF]Theorem 1.1] proved the following theorem. We improve their method quantitatively in this section.

Theorem 2.2.1. Let S be a locally moderate positive closed (p, p)-current on a complex manifold X. If u is a Hölder continuous S-p.s.h. function, then dd c (uS) is locally moderate.

Corollary 2.2.2. Let u be a Hölder continuous p.s.h. function on X. Then the Monge-Ampère currents (dd c u) p are locally moderate.

The following lemma is needed [START_REF] Dinh | Exponential extimates for plurisubharmonic functions and stochastic dynamics[END_REF]Lemma 2.3].

Lemma 2.2.3. Let T be a positive closed current of bidegree (k -1, k -1) and u be a T -p.s.h. function on a neighbourhood U of the unit ball B 1 in C k . Suppose that u is smooth on B 1-r \ B 1-4r for a fixed number 0 < r < 1/4. If φ is a q.p.s.h. function on U , χ is a smooth function with compact support on

Covering lemma on P k

Denote by S k the unit sphere on R k+1 , B 1 the unit ball in C k . Let π : S 2k+1 → P k be the natural projection map. More precisely, set

By the homogeneity of S 2k+1 (resp. P k ), there is a neighbourhood S 0 (resp. π(S 0 )) of any point (x 0 , y 0 , ..., x k , y k ) (resp. [z 0 , ..., z k ]) which is the image of S 0 (resp. π(S 0 )) by rotations (resp. unitary transformations). We say that S 0 (resp. π(S 0 )) is similar to S 0 (resp. π(S 0 )). Since P k is compact, there are finitely many such neighbourhoods π(S 0 ) that cover P k . Denote by M k the minimum number of such neighbourhoods π(S 0 ) that cover P k . We have the following lemma.

Let P X := Π p≥1 PH 0 (X, L p ) endowed with its measure σ = Π p≥1 σ p . Denote by δ z the Dirac measure at a point z.

Proof. By the definition of pullbacks of meromorphic transforms, we have

For any test (n -1, n -1)-form φ on X,

The proof is completed.

We specify Theorem 2.2.5 and Theorem 2.2.6 for the above case.

Theorem 2.3.4. Suppose that the sequence {R p δ p d -1 p } tends to 0 and

for all t > 0. Then for almost everywhere s = (s p ) ∈ P X with respect to σ, the sequence d -1 p (F p (δ sp ) -F p (σ p )), ψ converges to 0 uniformly on the bounded set of (n -1, n -1)-forms on X of class C 2 . Theorem 2.3.5. Suppose that the sequence

The following theorem is due to Tian, Ruan, Catlin and Zelditch, see [START_REF] Zelditch | Szegö kernels and a theorem of Tian[END_REF][START_REF]The Bergman kernel and a theorem of Tian, Analysis and Geometry in Several Complex Variables[END_REF][START_REF] Ruan | Canonical coordinates and Bergmann metrics[END_REF].

In order to prove the main theorem, we write

for any test form ψ of bidegree (n -1, n -1) on X. It is sufficient to prove that the three terms in the right side of the inequality all tend to 0 when p → ∞. The third one is right due to Theorem 2.3.6. The first one holds under the conditions that R p = o(p), p≥1 ∆(pt) < ∞, ∀t > 0 by Theorem Proposition 3.2.2. Let X be a compact complex manifold of dimension n.

(iii) σ = σ 1 × ... × σ m is the product measure on the multi-projective space P d 1 × ... × P dm , where σ k is a probability moderate measure on P d k .

Then the analytic subsets Z(t 1 ), ..., Z(t m ) are in general position for almost all (t 1 , ..., t m ) ∈ P d 1 × ... × P dm with respect to σ.

where 1 ≤ l 1 < ... < l k ≤ m, j = 0 and Ṽ0 = X, Ṽj = BsV i 1 ∩ ... ∩ BsV i j for some i 1 < ... < i j in {1, ..., m} \ {l 1 , ..., l k } for 1 ≤ j ≤ m -k. Note that the sets U k depend on the choices of l 1 , ..., l k , j and Ṽj .

It is sufficient to prove that

We only consider the case when {l 1 , ..., l k } = {1, ..., k}. Write σ k = σ 1...k for short. We first consider the case when k = 1. If j = 0, then

where the subsets D l are the irreducible components of Ṽj of dimension n -j and B is of dimension less than n -j. So {t 1 ∈ P d 1 : D l ⊂ Z(t 1 )} is a proper linear subspace of P d 1 . If not, D l ⊂ BsV 1 implies that dim Ṽj ∩ BsV 1 = n -j, which contradicts the condition that BsV 1 , ..., BsV m are in general position. We know that dim Z(t 1 ) ∩ Ṽj ≥ n -j if t 1 ∈ P d 1 \ U 1 . Since Z(t 1 ) ∩ Ṽj is an analytic subset in Ṽj , then D l ⊂ Z(t 1 ) ∩ Ṽj for some l. It follows that

Hence we have that σ 1 (P d 1 \ U 1 ) = 0 by Lemma 3.2.1.

Proof. Denote by σ 0 k,p the standard volume ω

F S on PH 0 (2) (X, L p k ). Note that ω d 0,p p is a smooth form. By the definition of Φ p , for a test (n -m, n -m)-form φ on X, we have

By the definition of direct image (Section 1.1.2), for a point S p ∈ X p ,

Then Proposition 4.2 in [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF] and the proof of Theorem 1.2 in [START_REF] Coman | Equidistribution results for singular metrics on line bundles[END_REF] imply the following

Then the proof is completed.

Main result and its proof

We recall the construction of moderate measures in the settings of Theorem 0.3.1. Consider the functions u j :

. This is a probability measure on P N . Remark 2.3.7 shows that σ N is a moderate measure for suitable c N depending only on ρ and N (e.g. c N = O(1/c N ), where the constant c > 1 depends only on ρ). We reformulate the above result and some estimates from the proofs of Theorem 0.2.1 and Theorem 0.2.2. Proposition 3.3.8. In the above setting, there exists a constant 0 < c N < 1 for the measure σ N which depends only on ρ and N such that

Hence by hypotheses,

(3.9)

Since r ≥ 1 (cf. Lemma 3.3.10), α 0 < 1, log 4 ρ > 1, inequality (3.10) holds obviously when 0 ≤ t ≤ 2 . By Fubini theorem, we obtain

We secondly estimate the measure of E . For any

Hence by the same argument,

By Fubini theorem, we obtain

So by estimates (3.11) and (3.15) for t ≥ 0,

When 0 ≤ t ≤ t 0 =: min( 1 , 2 ), (3.12) and (3.16) yield

It is obvious that the above inequality is also valid for t < 0 since ψ ≤ 0. By the definition of ∆(X, ω 12 , σ, t), we need to consider a function ϕ on X with the conditions that dd c ϕ ≥ -rω and ϕdσ = 0. Define a new function ψ := ϕ -max X ϕ. The fact that ϕdσ = 0 implies that max X ϕ ≥ 0. Then ψ ≤ ϕ. Moreover, max X ψ = 0. Then

) j t).

(3.17)

where β 4 =: β 3 exp(α 0 β 2 ) is a universal positive constant. Denote by E c p the complement of E p . Set σ p := µ pjp . So σ p (E c p ) ≤ 1 p 2 . Yau's theorem [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I[END_REF] implies that there exists a smooth real function u p with (ω F S + dd c u p ) dp = σ p . Then the theorem follows from the same argument in the proof of Remark 2.3.8. This completes the proof.