Damien Gayet

Jean-Yves Welschinger

What is the total Betti number of a random real hypersurface?

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve Hörmander's theory of peak sections as well as the formula of Poincaré-Martinelli.

Mathematics subject classification 2010: 14P25, 32U40, 60F10 paragraph, we first establish the real analogue of Theorem 3, see Theorem 6, and then deduce Theorem 1 from it, that is upper bounds for the expected total Betti numbers of the real locus of random real hypersurfaces, see Theorems 4 and 5.

Aknowledgements.

Introduction

The topology of real projective manifolds is under study since the nineteenth century, when Axel Harnack and Felix Klein discovered that the number of connected components of the real locus of a smooth real projective curve is bounded from above by the sum of its genus and the number of connected components of its complex domain, see [START_REF] Harnack | Ueber die Vieltheiligkeit der ebenen algebraischen Curven[END_REF], [START_REF] Klein | Ueber den Verlauf der Abel'schen Integrale bei den Curven vierten Grades[END_REF], while David Hilbert has devoted his sixteenth problem to such a study. Recall that by definition, a real projective manifold X is the vanishing locus in some complex projective space of a collection of homogeneous polynomials with real coefficients. It inherits an antiholomorphic involution c X from the ambient complex conjugation. The real locus RX is the set of real solutions of the polynomial equations, that is the fixed point set of c X . René Thom [START_REF] Thom | Sur l'homologie des variétés algébriques réelles, Differential and Combinatorial Topology[END_REF] later observed as a consequence of Smith's theory in equivariant homology, that the total Betti number of the real locus of a smooth real projective manifold is actually always bounded from above by the total Betti number of its complex locus, extending Harnack-Klein's inequality, see Theorem 2. On the other hand, John Nash proved that every closed smooth manifold can be realized as a component of the real locus of a smooth real projective manifold.

Real projective manifolds achieving the upper bound given by Harnack-Klein or Smith-Thom's inequalities are called maximal. Real maximal curves in smooth real projective surfaces appear to be exponentially rare in their linear system as their degree grows, see [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF]. What is then the expected topology of real hypersurfaces in a given smooth real projective manifold X? We tackle here this question, measuring the topology of hypersurfaces by the total Betti numbers of their real loci. The answer to this question indeed turns out to be only known for the real projective line thanks to Mark Kac [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF], Michael Shub and Stephen Smale [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF] or Alan Edelman and Eric Kostlan [START_REF] Edelman | How many zeros of a random polynomial are real?[END_REF]. From these works follows that the expected number of real roots of a random real polynomial in one variable and degree d is √ d. We establish here general upper bounds for the expected total Betti numbers of real hypersurfaces in real projective manifolds. More precisely, let X be a smooth real projective manifold of positive dimension n equipped with a real ample line bundle L. The growth of the total Betti number of complex loci of hypersurfaces linearly equivalent to L d is polynomial in d of degree n, see Lemma 3. We prove the following, see Theorem 4 and 5.

Theorem 1 Let (X, c X ) be a smooth real projective manifold of dimension n greater than one equipped with a Hermitian real line bundle (L, c L ) of positive curvature. Then, the expected total Betti number of real loci of hypersurfaces linearly equivalent to L d is a o(d n ). If n = 2 or if X is a product of smooth real projective curves, then it is even a O(d

n 2 (log d) n ).
The probability measure that we consider on the complete linear system of real divisors associated to L d is the Fubiny-Study measure arising from the L 2 -scalar product induced by the Hermitian metric of positive curvature fixed on L, see §3.1. When X is one-dimensional, upper bounds as the ones given by Theorem 1 can already be deduced from our work [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF]. In order to prove Theorem 1, we first fix a real Lefschetz pencil on X, which restricts to a Lefschetz pencil on every generic hypersurface of X. The number of critical points of such a restriction has the same asymptotic as the total Betti number of the hypersurface, see §1.2. We then prove that these critical points get uniformly distributed in X when the degree increases and more precisely that the expected normalized counting measure supported by these critical points converges to the volume form induced by the curvature of the Hermitian bundle L, see Theorem 6. The latter weak convergence proved in Theorem 6 is established outside of the critical locus of the original pencil when n > 2 and away from the real locus of X. Note that we first prove this equirepartition result over the complex numbers, see Theorem 3. In order to deduce Theorem 1 from this equirepartition result, we observe that the total Betti number of real loci of hypersurfaces is bounded from above by the number of critical points of the restricted Lefschetz pencil in a neighborhood of the real locus, which we choose of size log d √ d thanks to the theory of Hörmander's peak sections, see §2. Theorems 3 and 6 on equirepartition of critical points are independent of Theorem 1 which motivated this work. Note also that the expected Euler characteristic of the real locus of such random hypersurfaces has been computed in [START_REF] Ibragimov | On random real algebraic surfaces[END_REF] and [START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF]. Finally, while we were writing this paper in june 2011, Peter Sarnak informed us that he is able to prove together with Igor Wigman in a work in progress that the expected number of connected components of real curves of degree d in RP 2 is even a O(d).

Our paper is organized as follows. In the first paragraph, we recall few results about total Betti numbers of real projective manifolds, critical points of Lefschetz pencils and their asymptotics. The second paragraph is devoted to Theorem 3 about equirepartition of critical points of Lefschetz pencils restricted to complex random hypersurfaces. The theory of peak sections of Hörmander and Poincaré-Martinelli's formula play a crucial rôle in the proof, see §2.2.2 and §2. [START_REF]Critical points and supersymmetric vacua. II: Asymptotics and extremal metrics[END_REF]

. Finally in the third

This first paragraph is devoted to Lefschetz pencils, total Betti numbers and their asymptotics.

Real Lefschetz pencils and Betti numbers

Let X be a smooth complex projective manifold of positive dimension n.

Definition 1 A Lefschetz pencil on X is a rational map p : X CP 1 having only non degenerated critical points and defined by two sections of a holomorphic line bundle with smooth and transverse vanishing loci.

We denote by B the base locus of a Lefschetz pencil p given by Definion 1, that is the codimension two submanifold of X where p is not defined. A Lefschetz pencil without base locus is called a Lefschetz fibration. Blowing up once the base locus of a Lefschetz pencil turns it into a Lefschetz fibration. When the dimension n of X equals one, the base locus is always empty and a Lefschetz fibration is nothing but a branched cover with simple ramifications. Hence, the following Proposition 1 extends to Lefschetz fibrations the classical Riemann-Hurwitz formula.

Proposition 1 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz fibration p : X → CP 1 and let F be a regular fiber of p. Then, the Euler characteristics of X and F satisfy the relation

χ(X) = 2χ(F ) + (-1) n #Crit(p),
where Crit(p) denotes the set of critical points of p.

Proof. Denote by ∞ = p(F ) ∈ CP 1 and by F 0 the fiber of p associated to a regular value 0

∈ CP 1 \ {∞}. Let U 0 (resp. U ∞ ) be a neighborhood of 0 (resp. ∞) in CP 1 , without any critical value of p. Since U ∞ (resp. U 0 ) retracts on F (resp. F 0 ), we know that χ(p -1 (U 0 )) = χ(p -1 (U ∞ )) = χ(F ), whereas from additivity of the Euler characteristic, χ(X) = χ(X) -2χ(F ), where X denotes the complement X \ p -1 (U 0 ∪ U ∞ ).
Without loss of generality, we may assume that in an affine chart C = CP 1 \ {∞}, 0 corresponds to the origin, U 0 to a ball centered at the origin and U ∞ to a ball centered at ∞. The manifold X comes then equipped with a function

f : x ∈ X → |p(x)| 2 ∈ R * + ⊂ C, taking values in a compact interval [a, b] of R * + .
This function f is Morse and has the same critical points as p, all being of index n. Indeed, the differential of f writes df = p∂p + p∂p and vanishes at x ∈ X if and only if ∂p |x vanishes. Moreover, its second differential is the composition of the differential of the norm |.| 2 with the second differential of p. The multiplication by i exchanges stable and unstable spaces of these critical points which are non degenerated. Hence, X is equipped with a Morse function f : X → [a, b] having #Crit(p) critical points, all of index n. By the Morse Lemma (see [START_REF] Milnor | Morse theory[END_REF]), the topology of f -1 ([a, a + ǫ]) changes, as ǫ grows, only at the critical points, where a handle D n × D n of index n is glued on a submanifold diffeomorphic to D n × S n-1 . From this Morse theory we deduce that

χ(X) = #Crit(p)(1 -χ(S n-1 )) = (-1) n #Crit(p)
and the result. 2

Recall that a complex projective manifold X ⊂ CP N is said to be real when it is defined over the reals, as the vanishing locus of a system of polynomial equations with real coefficients. It inherits then an antiholomorphic involution c X : X → X, which is the restriction of the complex conjugation conj : (z 0 :

• • • : z n ) ∈ CP n → (z 0 : • • • : z n ) ∈ CP n . Its fixed point set RX ⊂ RP n is called the real locus of X.
When X is smooth, the latter is either empty or half-dimensional.

Definition 2 Let (X, c X ) be a smooth real projective manifold of positive dimension n. A Lefschetz pencil p : X CP 1 is said to be real iff it satisfies p • c X = conj • p.

Such a real Lefschetz pencil given by Definition 2 is then defined by two real sections σ 0 , σ 1 of a holomorphic real line bundle π :

(N, c N ) → (X, c X ), where π • c N = c X • π. Now, if M is a smooth manifold of positive dimension n, we denote by b * (M; Z/2Z) = n i=0 dim H i (M; Z/2Z)
its total Betti number with Z/2Z-coefficients.

Lemma 1 Let M be a smooth manifold equipped with a smooth fibration p : M → RP 1 and F be a regular fiber of p. Then, the total Betti numbers of M and F satisfy

b * (M; Z/2Z) ≤ 4b * (F ; Z/2Z) + #Crit(p).
This relation also holds when M is the real locus of a smooth real projective manifold and p the restriction of a real Lefschetz pencil.

Proof. Denote by ∞ = p(F ) ∈ RP 1 and by F 0 the fiber of p associated to a regular value 0 ∈ RP 1 \ {∞}. Let I 0 (resp. I ∞ ) be a neighborhood of 0 (resp. ∞) in RP 1 , so that I 0 and I ∞ cover RP 1 , such that I ∞ contains only regular values of p. Now, set U 0 = p -1 (I 0 ) and

U ∞ = p -1 (I ∞ ), so that U 0 ∪ U ∞ = M. From the Mayer-Victoris formula follows that b * (M) ≤ b * (U 0 ) + b * (U ∞ ) + b * (U 0 ∩ U ∞ ) ≤ b * (U 0 ) + 3b * (F ),
* (R X; Z/2Z) ≤ 4b * (R F , Z/2Z) + #Crit( p |R X ),
where F denotes the fiber of p associated to F . Moreover the morphism H * (R X; Z/2Z) → H * (RX; Z/2Z) is onto, since every element of H * (RX; Z/2Z) has a representative transverse to RB and a proper transform in R X.

It follows that b * (RX) ≤ b * (R X) whereas the projection R F → RF is a diffeomorphism. 2
Recall finally the following Theorem 2 proved by R. Thom in [START_REF] Thom | Sur l'homologie des variétés algébriques réelles, Differential and Combinatorial Topology[END_REF], as a consequence of Smith's exact sequence in equivariant homology.

Theorem 2 Let (X, c X ) be a smooth real projective manifold with real locus RX. Then, the total Betti numbers of X and RX satisfy b * (RX; Z/2Z) ≤ b * (X; Z/2Z).

The manifolds for which equality holds in Theorem 2 are called maximal. For instance, real projective spaces are maximal. When X is one-dimensional and irreducible, Smith-Thom's inequality given by Theorem 2 reduces to the Harnack-Klein's inequality, up to which the number of connected components of RX is bounded from above by g(X) + 1, where g(X) denotes the genus of the curve X, see [START_REF] Wilson | Hilbert's sixteenth problem[END_REF] and references therein. Real maximal curves in real projective surfaces turn out to become exponentially rare in their linear system as their degree grows, see [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF].

Asymptotics

Given a holomorphic line bundle L over a smooth complex projective manifold X, we denote, for every non trivial section σ of L, by C σ its vanishing locus.

Lemma 2 Let L be a holomorphic line bundle over a smooth complex projective manifold X of positive dimension n. For every section σ of L which vanishes transversally, the Chern classes of its vanishing locus C σ write:

∀j ∈ {1, • • • , n -1}, c j (C σ ) = j k=0 (-1) k c 1 (L) k ∧ c j-k (X) |Cσ ∈ H 2j (C σ ; Z).
In particular, if (σ d ) d>0 is a sequence of sections of L d given by Lemma 2, the Euler characteristic of C σ d is a polynomial of degree n in d with leading coefficient (-1) n-1 X c 1 (L) n . Proof. The adjunction formula for X, C σ and L writes c(X

) |Cσ = c(C σ )∧c(L) |Cσ , since the restriction of L to C σ is isomorphic to the normal bundle of C σ in X. As a consequence, c 1 (X) |Cσ = c 1 (C σ ) + c 1 (L) |Cσ and for every j ∈ {2, • • • , n -1}, c j (X) |Cσ = c j (C σ ) + c j-1 (C σ ) ∧ c 1 (L) |Cσ .
Summing up, we get the result. 2 Lemma 3 Let L be an ample line bundle over a smooth complex projective manifold of positive dimension n. Let (σ d ) d>0 be a sequence of sections of L d vanishing transversally. Then,

b * (C σ d ; Z/2Z) = (-1) n-1 χ(C σ d ) + O(1) = X c 1 (L) n d n + O(d n-1 ).
Proof. When d is large enough, L d is very ample and we choose an embedding of X in CP N , N > 0, such that L d coincides with the restriction of O CP N (1) to X. Then, C σ d writes X ∩ H where H is a hyperplane of CP N . By Lefschetz's theorem of hyperplane sections, for 0

≤ i ≤ n -1, dim H i (C σ d ; Z/2Z) = dim H i (X; Z/2Z) and then by Poincaré duality, dim H 2n-2-i (C σ d ; Z/2Z) = dim H 2n-i (X; Z/2Z). Hence b * (C σ d ; Z/2Z) = dim H n-1 (C σ d ; Z/2Z) + O(1) = (-1) n-1 χ(C σ d ) + O(1).
The result now follows from Lemma 2. 2

Proposition 2 Let X be a smooth complex projective manifold of dimension n greater than one equipped with a Lefschetz pencil p : X CP 1 . Let L → X be a holomorphic line bundle and σ d be a section of L d which vanishes transversally, where d > 0. Assume that the restriction of p to C σ d is Lefschetz. Then, the number of critical points of the restriction p |Cσ d equals

( X c 1 (L) n )d n + O(d n-1
(-1) n-1 #Crit(p |Cσ d ) = χ( C σ d ) -2χ( F ∩ C σ d ).
From additivity of the Euler characteristic, we know that χ( C

σ d ) = χ(C σ d ) + χ(B ∩ C σ d ). The exceptionnal divisor of C σ d over B ∩ C σ d is indeed a ruled surface over B ∩ C σ d of Euler characteristic 2χ(B ∩ C σ d ). Likewise, χ( F ∩ C σ d ) = χ(F ∩ C σ d ), since the projection F ∩ C σ → F ∩ C σ is a diffeomorphism.
The result now follows from Lemma 2, which provides the equivalents

χ(C σ d ) ∼ d→∞ (-1) n-1 X c 1 (L) n d n , χ(B ∩ C σ d ) ∼ d→∞ (-1) n-3 B c 1 (L) n-2 |B d n-2 and χ(F ∩ C σ d ) ∼ d→∞ (-1) n-2 F c 1 (L) n-1 |F d n-1 .

Random divisors and distribution of critical points

Let X be a smooth complex projective manifold equipped with a Lefschetz pencil. The restriction of this pencil to a generic smooth hypersurface C of X is a Lefschetz pencil of C. The aim of this paragraph is to prove the equidistribution in average of critical points of such a restriction to a random hypersurface C of large degree, see Theorem 3. The estimations of the total Betti number of real hypersurfaces will be obtained in paragraph 3 as a consequence of a real analogue of this Theorem 3, see Theorem 6. We first formulate this equirepartition Theorem 3, then introduce the main ingredients of the proof, namely Poincaré-Martinelli's formula and Hörmander's peak sections. Finally, we prove Theorem 3. Note that this paragraph is independent of the remaining part of the paper, it does not involve any real geometry.

Notations and result

Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 with base locus B ⊂ X. Let L → X be a holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω ∈ Ω (1,1) (X; R). The latter is defined in the neighborhood of every point x ∈ X by the relation ω = 1 2iπ ∂ ∂ log h(e, e), where e is a local non vanishing holomorphic section of L defined in the neighborhood of x. The curvature form induces a Kähler metric on X and we denote by dx = ω n X ω n its associated normalized volume form. For every integer d > 0, we denote by h d the induced Hermitian metric on the bundle L d and by the induced L 2 -Hermitian product on the space H 0 (X; L d ) of global sections of L d . This product is defined by the relation

(σ, τ ) ∈ H 0 (X; L d ) × H 0 (X; L d ) → σ, τ = X h d (σ, τ )dx ∈ C.
Denote by N d the dimension of H 0 (X; L d ) and by µ its Gaussian measure, defined by the relation For every section σ ∈ H 0 (X; L d ) \ ∆ d , denote by R σ the set of critical points of the restriction p |Cσ of p to C σ , so that by Proposition 2, the cardinal #R σ of this set is equivalent to ( X ω n )d n as d grows to infinity. For every x ∈ X, we finally denote by δ x the Dirac measure χ ∈ C 0 (X, R) → χ(x) ∈ R.

∀A ⊂ H 0 (X; L d ), µ(A) = 1 π N d A e -||σ||
Definition 3 For every σ ∈ H 0 (X; L d ) \ ∆ d , the measure ν σ = 1 #R d x∈Rσ δ
x is called the probability measure of X carried by the critical points of p |Cσ .

Our goal in this paragraph is to prove the following Theorem 3 which asymptotically computes the expected probability measure given by Definition 3.

Theorem 3 Let X be a smooth complex projective manifold of dimension n greater than one equipped with a Lefschetz pencil p : X CP 1 with critical locus Crit(p). Let L → X be a holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω. Then, for every function χ :

X → R of class C 2 such that, when n > 2, the support of ∂ ∂χ is disjoint from Crit(p), we have lim d→∞ E( ν σ , χ ) = X χdx, where E( ν σ , χ ) = H 0 (X;L d )\ ∆ d ν σ , χ dµ(σ).
Note that similar results as Theorem 3, on equirepartition of critical points of sections, have been obtained in [START_REF] Michael | Critical points and supersymmetric vacua. I[END_REF], [START_REF]Critical points and supersymmetric vacua. II: Asymptotics and extremal metrics[END_REF] by M. Douglas, B. Shiffman and S. Zelditch.

Note also that the equirepartition Theorem 3, as well as Theorem 6, is local in nature and does not depend that much on a Lefschetz pencil. Any local holomorphic Morse function could be used instead of a Lefschetz pencil, leading to the same proof and conclusions.

Poincaré-Martinelli's formula and adapted atlas

Adapted atlas and associated relative trivializations

Definition 4 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 . An atlas U of X is said to be adapted to p iff for every open set U ∈ U, the restriction of p to U is conjugated to one of the following three models in the neighborhood of the origin in In the model (r), the vertical tangent bundle ker(dp) is trivialized by the vector fields

C n : (r) (z 1 , • • • , z n ) ∈ C n → z n ∈ C (b) (z 1 , • • • , z n ) ∈ C n \ C n-2 → [z n-1 : z n ] ∈ CP 1 (c) (z 1 , • • • , z n ) ∈ C n → z 2 1 + • • • + z 2 n ∈ C
∂ ∂z 1 , • • • , ∂ ∂z n-1 of C n .
In the model (b), it is trivialized outside of the base locus by the vector fields z n-1

∂ ∂z n-1 + z n ∂ ∂zn , ∂ ∂z 1 , • • • , ∂ ∂z n-2 of C n .
In the model (c), when n = 2, it is trivialized outside of the critical point by the vector field z 1

∂ ∂z 2 -z 2 ∂ ∂z 1 of C 2 .
Definition 5 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 and an adapted atlas U. A relative trivialization associated to U is the data, for every open set U ∈ U, of n-1 vector fields on U corresponding to the vector fields

∂ ∂z 1 , • • • , ∂ ∂z n-1 in the model (r), to z n-1 ∂ ∂z n-1 + z n ∂ ∂zn , ∂ ∂z 1 , • • • , ∂ ∂z n-2 in the model (b) and to z 1 ∂ ∂z 2 -z 2 ∂ ∂z 1 in the model (c) when n = 2.
Note that in the model (c) given by Definition 4 the vertical tangent bundle ker(dp) restricted to C n \ {0} is isomorphic to the pullback of the cotangent bundle of CP n-1 by the projection π : C n \ {0} → CP n-1 . Indeed, the fibers of this vertical tangent bundle are the kernels of the 1-form α = n i=1 z i dz i , so that the restriction map induces an isomorphism (C n ) * / < α > ∼ = (ker dp) * . But the canonical identification between C n and (C n ) * gives an isomorphism between the bundle (C n ) * / < α > and π * (T P n-1 ) over C n \ {0}. By duality, we get the isomorphism ker(dp) ∼ = π * T * CP n-1 . When n > 2, we no more see trivialisations of this bundle over C n \ {0} and thus restrict ourselves to n = 2 for the model (c) in Definition 6.

Definition 6 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 and with a holomorphic line bundle L → X. An atlas U is said to be adapted to (p, L) if it is adapted to p in the sense of Definition 4 and if for every open set U ∈ U, the restriction of L to U is trivializable. A relative trivialization associated to U is a relative trivialization in the sense of Definition 5 together with a trivialization e of L |U , for every open set U of U.

2.2.2

Poincaré-Martinelli's formula Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 of critical locus Crit(p). Let L → X be an ample holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω ∈ Ω (1,1) (X; R). Let U be an atlas of X adapted to (p, L) and (v 1 , • • • , v n-1 , e) be an associated relative trivialization given by Definition 6. Let U be an element of U. For every section σ ∈ H 0 (X; L d ), we denote by f σ,U : U → C the holomorphic function defined by the relation

σ |U = f σ,U e d |U . When σ / ∈ ∆ d , the set R σ ∩ U
coincides by definition with the transverse intersection of the hypersurfaces

{f σ,U = 0}, {∂f σ,U (v 1 ) = 0}, • • • , {∂f σ,U (v n-1 ) = 0}.
For every function χ : X → R with compact support in U, Poincaré-Martinelli's formula (see [START_REF] Griffiths | Nevanlinna theory and holomorphic mappings between algebraic varieties[END_REF]) then writes here:

ν σ , χ = ( i 2π ) n 1 #R σ X λ U ∂ ∂χ ∧ (∂ ∂λ U ) n-1 , (PM)
where

λ U = log d 2 |f σ,U | 2 + n-1 i=1 |∂f σ,U (v i )| 2 .
Note that in this definition of λ U , we have chosen for convenience to use df σ,U instead of f σ,U as the first function. This formula of Poincaré-Martinelli computes the integral of χ for the measure ν σ introduced in Definition 3; its left hand side does not involve any trivialization of L over U, contrary to the right hand side. It makes it possible to estimate the expectation of the random variable ν σ , χ . However, it appears to be useful for this purpose to choose an appropriate trivialization of L in the neighborhood of every point x ∈ X, whose norm reaches a local maximum at x where it equals one. We are going to make such a choice instead of the trivialization e defined on the whole U, as discussed in the following Proposition 3 and §2.3.

Proposition 3 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz pencil p : X CP 1 . Let L be an ample holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω. Let U be an element of an atlas adapted to (p, L) and (v 1 , • • • , v n-1 , e) be an associated relative trivialization. Finally, let (g x ) x∈U be a family of germs of holomorphic functions such that g x is defined in a neighborhood of x and ℜg x (x) = -log h d (e d , e d ) |x . Then, for every function χ : X → R of class C 2 with support in U, disjoint from the critical set of p when n > 2, and for every σ ∈ H 0 (X; L d ) \ ∆ d , we have:

ν σ , χ = 1 #R σ d n X χω n + 1 #R σ n-1 k=0 ( i 2π ) n-k d k X ∂ ∂χ ∧ ω k ∧ λ x (∂ ∂λ x ) n-1-k ,
where

λ x = log d 2 |f σ,x | 2 + n-1 i=1 |∂f σ,x (v i ) + f σ,x ∂g x (v i )| 2
and σ = f σ,x exp(g x )e d in the neighborhood of every point x ∈ U.

The condition on (g x ) x∈U in Proposition 3 ensures that exp(g x )e d is a holomorphic trivialization of norm one at x, so that h d (σ, σ) |x coincides with |f σ,x (x)| 2 . The point x in λ x is a parameter and not a variable, so that λ x reads in the neighborhood of x as a function

z → log d 2 |f σ,x | 2 (z) + n-1 i=1 |∂f σ,x (v i ) |z + f σ,x (z)∂g x (v i ) |z | 2
and ∂ ∂λ x in the formula given by Proposition 3 stands for its second derivative computed at the point x. Note that if U is a locally finite atlas adapted to (p, L), and if (ρ U ) U ∈U is an associated partition of unity, then for every function χ : X → R of class C 2 , with support disjoint from the critical locus of p when n > 2, and for every open set U ∈ U, the function χ U = ρ U χ satisfies the hypotheses of Proposition 3, while χ = U ∈U χ U and ν σ , χ = U ∈U ν σ , χ U .

Proof. Let σ ∈ H 0 (X; L d ) \ ∆ d and x ∈ U \ Crit(p)
. By definition, f σ,U = f σ,x exp(g x ) and for every 1

≤ i ≤ n -1, ∂f σ,U (v i ) = (∂f σ,x (v i ) + f σ,x ∂g x (v i )) exp(g x ).
As a consequence, λ U = ℜg x +λ x , so that at the point x, λ U (x) = -log h d (e d , e d )(x)+ λ x (x). Since ∂ ∂ℜg x vanishes, the equality ∂ ∂λ U = ∂ ∂λ x holds in a neighborhood of x, Hence, formula (PM) rewrites

ν σ , χ = i n (2π) n #R σ X ∂ ∂χ -log h d (e d , e d ) + λ x 0 (∂ ∂λ U ) n-1 = i n-1 d (2π) n-1 #R σ X ∂ ∂χ ∧ ω ∧ λ U (∂ ∂λ U ) n-2 + i n (2π) n #R σ X ∂ ∂χ ∧ λ x 0 (∂ ∂λ x ) n-1
The first part of the latter right hand side follows from the relation

∂ ∂(λ U (∂ ∂λ U ) n-2 ) = (∂ ∂λ U ) n-1 , the curvature equation ω = i 2πd ∂ ∂(-log h d (e, e)
) and Stokes's theorem. The second part of this right hand side comes from ∂ ∂λ U = ∂ ∂λ x . Applying this procedure (n -1) times, we deduce by induction and Stokes's theorem the relation

ν σ , χ = 1 #R σ d n X χω n + 1 #R σ n-1 k=0 ( i 2π ) n-k d k X ∂ ∂χ ∧ ω k ∧ λ x (∂ ∂λ x ) n-1-k .
2

Corollary 1 Under the hypotheses of Proposition 3,

E( ν σ , χ ) = 1 #R σ d n X χω n + 1 #R σ n-1 k=0 ( i 2π ) n-k d k X ∂ ∂χ ∧ ω k ∧ H 0 (X;L d )\ ∆ d λ x (∂ ∂λ x ) n-1-k dµ(σ).
Proof. The result follows by integration over H 0 (X; L d ) \ ∆ d of the relation given by Proposition 3. 2

Hörmander's peak sections

Let L be a holomorphic line bundle over a smooth complex projective manifold, equipped with a Hermitian metric h of positive curvature ω. Let x be a point of X. There exists, in the neighborhood of x, a holomorphic trivialization e of L such that the associated potential φ = -log h(e, e) reaches a local minimum at x with Hessian of type [START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF][START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF]. The latter coincides, by definition, with ω(., i.). The Hörmander L 2estimates makes it possible, for all d > 0 and maybe after modifying a bit e d in L 2 -norm, to extend e d to a global section σ of L d . The latter is called peak section of Hörmander, see Definition 7. Moreover, G. Tian (Lemma 1.2 in [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF]) showed that this procedure can be applied to produce global sections whose Taylor expansion at x can be controlled at every order, as long as d is large enough. We recall this result in the following Lemma 4 where we denote, for every r > 0, by B(x, r) the ball centered at x of radius r in X.

Lemma 4 [See [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], Lemma 1.2] Let (L, h) be a holomorphic Hermitian line bundle of positive curvature ω over a smooth complex projective manifold X.

Let x ∈ X, (p 1 , • • • , p n ) ∈ N n and p ′ > p 1 + • • • + p n .
There exists d 0 ∈ N such that for every d > d 0 , the bundle L d has a global holomorphic section σ satisfying X h d (σ, σ)dx = 1 and

X\B(x, log d √ d ) h d (σ, σ)dx = O( 1 d 2p ′ ). Moreover, if z = (z 1 , • • • , z n ) are local coordinates in the neighborhood of x, we can assume that in a neighborhood of x, σ(z) = λ(z p 1 1 • • • z pn n + O(|z| 2p ′ ))e d (1 + O( 1 d 2p ′ )),
where

λ -2 = B(x, log d √ d ) |z p 1 1 • • • z pn n | 2 h d (e d , e d )dx
and e is a trivialization of L in the neighborhood of x whose potential φ = -log h(e, e) reaches a local minimum at x with Hessian ω(., i.).

Definition 7

We call Hörmander's peak section of the ample line bundle L d over the smooth complex projective manifold X any section given by Lemma 4 with p 1 = • • • = p n = 0 and p ′ > 1, where d is large enough.

Note that such a peak section σ 0 given by Definition 7 has its norm concentrated in the neighborhood of the point x given by Lemma 4, so that it is close to the zero section outside of a log d √ d -ball. Moreover, the derivatives and second derivatives of σ 0 at x vanish and the value of λ at x is equivalent to ( X c 1 (L) n )d n as d grows to infinity, see Lemma 2.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF].

Note also that if the coordinates (z 1 , • • • , z n ) in Lemma 4 are orthonormal at the point x, then two sections given by this lemma for different values of (p 1 , • • • , p n ) are asymptotically orthogonal, see Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF].

Evaluation of the two-jets of sections

Again, let (L, h) be a holomorphic Hermitian line bundle of positive curvature ω over a smooth n-dimensional complex projective manifold X. Let x be a point of X and d > 0. We denote by H x the kernel of the evaluation map σ ∈ H 0 (X;

L d ) → σ(x) ∈ L d
x , where L d x denotes the fiber of L d over the point x. Likewise, we denote by H 2x the kernel of the map σ ∈ H

x → ∇σ |x ∈ T * x X ⊗ L d x .
This map does not depend on a chosen connection ∇ on L. Denote by H 3x the kernel of the map

σ ∈ H 2x → ∇ 2 σ |x ∈ Sym 2 (T * x X) ⊗ L d x .
We deduce from these the jet maps:

eval x : σ ∈ H 0 (X; L d )/H x → σ(x) ∈ L d x , eval 2x : σ ∈ H x /H 2x → ∇σ |x ∈ T * x X ⊗ L d x , and
eval 3x : σ ∈ H 2x /H 3x → ∇ 2 σ |x ∈ Sym 2 (T * x X) ⊗ L d x .
When d is large enough, these maps are isomorphisms between finite dimensional normed vector spaces. We estimate the norm of these isomorphisms in the following Proposition 4, closely following [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF].

Proposition 4 Let L be a holomorphic Hermitian line bundle of positive curvature over a smooth n-dimensional complex projective manifold X. Let x be a point of X.

Then the maps d -n 2 eval x , d -n+1 2 eval 2x and d -n+2 2 eval 3x as well as their inverse have norms and determinants bounded from above independently of d as long as d is large enough.

Note that Proposition 4 provides an asymptotic result while the condition that d be large ensures that the three maps are invertible.

Proof. Let σ 0 be a peak section of Hörmander given by Definition 7. By Lemma 2.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], d -n h d (σ 0 , σ 0 ) |x converges to a positive constant as d grows to infinity. Let σ Hx 0 be the orthogonal projection of σ 0 onto H x . The Taylor expansion of σ Hx 0 does not contain any constant term, so that by Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF] (see also Lemma 3.2 in [START_REF] Ruan | Canonical coordinates and Bergmann [Bergman] metrics[END_REF]), the Hermitian product σ 0 ,

σ ⊥ 0 = σ 0 -σ Hx 0 ||σ 0 -σ Hx 0 || . As a consequence, d -n h d (σ ⊥ 0 , σ ⊥ 0 )
|x converges to a positive constant as d grows to infinity. Hence, d -n 2 eval x as well as its inverse, has norm and determinant bounded when d is large enough. The two remaining assertions of Proposition 4 follow along the same lines. For i ∈ {1, • • • n}, let σ i be a section given by Lemma 4, with p ′ = 2, p i = 1 and p j = 0 if j = i, i ∈ {1, • • • n}, and where the local coordinates

(z 1 , • • • , z n ) are orthonormal at the point x. By Lemma 2.1 of [18], for i ∈ {1, • • • n}, d -(n+1) h d (∇σ i , ∇σ i ) |x converges
to a positive constant as d grows to infinity. The sections σ i , i ∈ {1, • • • n}, belong by construction to H x and we set as before

σ ⊥ i = σ i -σ H 2x i ||σ i -σ H 2x i || ,
where σ H 2x i denotes the orthogonal projection of σ i onto H 2x . We deduce as before from Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF] that d -(n+1) h d (∇σ ⊥ i , ∇σ ⊥ i ) |x converges to a positive constant when d grows to infinity and that h d (∇σ ⊥ i , ∇σ ⊥ j ) |x = 0 if i = j. The norms of the sections σ ⊥ i , i ∈ {1, • • • n}, all equal one but these sections are not a priori orthogonal.

However, by Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], the products ∇σ ⊥ i , ∇σ ⊥ i are O( 1 d ) if j = i, so that asymptotically, the basis is orthonormal. We deduce that d -n+1

2 eval 2x and its inverse are of norms and determinants bounded as d is large enough. The last case follows along the same lines.

||v 1 || 2 d k H 0 (X;L d )\ ∆ d ||λ x (∂ ∂λ x ) k ||dµ(σ)
is uniformly bounded by a O((log d) 2 ) on every compact subset of X \ Crit(p) when n > 2 and on the whole X when n ≤ 2.

The norms || || appearing in the statement of Proposition 5 are induced by the Kähler metric of X on elements and 2k-linear forms of T x X, where x ∈ X. It follows from Definition 4 that ||v 1 || may vanish in the models (b) and (c). Before proving Proposition 5, for which we will spend the whole paragraph, let us first deduce a proof of Theorem 3.

Proof of Theorem 3. Let χ : X → R be a function of class C 2 such that the support K of ∂ ∂χ be disjoint from Crit(p) when n > 2. Choose a finite atlas U adapted to (p, L) given by Definition 6, such that when n > 2, K be covered in X \ Crit(p) by elements of U. Let U be such an element of U and (v 1 , • • • , v n , e) be an associated relative trivialization given by Definition 6. Without loss of generality, we can assume that χ has support in U. By Proposition 3 and with the notations introduced there, when d is large enough, the expectation E( ν σ , χ ) equals

1 #R σ d n X χω n + 1 #R σ n-1 k=0 ( i 2π ) n-k d k X ∂ ∂χ∧ω k ∧ H 0 (X;L d )\ ∆ d λ x (∂ ∂λ x ) n-1-k dµ(σ).
By Proposition 2, #R σ is equivalent to ( X ω n )d n as d grows to infinity, so that the first term converges to X χdx. By Proposition 5, the last integral over H 0 (X;

L d )\ ∆ d is a O( (log d) 2 d||v 1 || 2
), since we integrate on the support of ∂ ∂χ which is disjoint of Crit(p) when n > 2. The result now follows from the fact that the function ||v 1 || -2 is integrable over X. Recall that X is equipped with an atlas U adapted to (p, L) and with an associated relative trivialization. The compact K given by Proposition 5 is covered by a finite number of elements of U. Moreover, we can assume that these elements are all disjoint from the critical set Crit(p) when n > 2. Let U be such an element ; it is either of type (r) given by Definition 4, or of type (b) or (c). Let us prove now Proposition 5 in the case U be of type (r) and postpone the remaining cases to §2.4.2.

Let x be a point of K ∩ U. For every σ ∈ H 0 (X;

L d ) \ ∆ d , define h 0 = df σ,x and for i ∈ {1, • • • , n -1}, h i = ∂f σ,x (v i ) + df σ,x ∂g x (v i ), so that ∂h i = ∂(∂f σ,x (v i )) + d∂g x (v i )∂f σ,x + df σ,x ∂(∂g x (v i )).
Recall here that f σ,x was introduced in Proposition 3 and defined by the relation σ = f σ,x exp(g x )e d , where the local section exp(g x )e d has norm one at x. It is enough to bound by O((log d) 2 ) the integral

1 d k H 0 (X;L d )\ ∆ d || log( n-1 i=0 |h i | 2 )(∂ ∂ log( n-1 i=0 |h i | 2 )) k ||dµ(σ),
since ||v 1 || 2 is bounded from below and above by positive constants in the model (r). Recall that

∂ ∂( n-1 i=0 |h i | 2 ) = n-1 i=0 ∂h i ∧ ∂h i n-1 i=0 |h i | 2 + n-1 i=0 h i ∂h i ∧ n-1 j=0 h j ∂h j ( n-1 i=0 |h i | 2 ) 2
.

From this we deduce, for k ∈ {1, • • • , n -1}, the upper bound

||(∂ ∂ log( n-1 i=0 |h i | 2 )) k || ≤ 2 k ( n-1 i=0 |h i | 2 ) k |I|=k,|J|=k ||∂h I ∧ ∂h J ||,
where I and J are ordered sets of k elements in {1, • • • , n} and ∂h

I = ∂h i 1 ∧ • • • ∧ ∂h i k if I = (i 1 , • • • , i k ).
Our integral gets then bounded from above by

2 k d k |I|=k,|J|=k H 0 (X;L d )\ ∆ d | log( n-1 i=0 |h i | 2 )| ( n-1 i=0 |h i | 2 ) k ||∂h I ∧ ∂h J ||dµ(σ).
Denote by H ⊥ x the orthogonal complement of H x in H 0 (X; L d ), see §2.3.1. Likewise, with a slight abuse of notation, denote by H x /H 2x (resp. H 2x /H 3x ) the orthogonal complement of H 2x (resp. H 3x ) in H x (resp. H 2x ). The space H 0 (X; L d ) then writes as a product H 0 (X;

L d ) = H ⊥ x × (H x /H 2x ) × (H 2x /H 3x
) × H 3x while its Gaussian measure µ is a product measure. The terms in our integral only involve jets at the second order of sections and hence are constant on H 3x . Using Fubini's theorem, it becomes thus enough to bound the integral over the space H ⊥

x × (H x /H 2x ) × (H 2x /H 3x ), whose dimension no more depends on d.

Moreover, the subspace

V ⊥ ⊂ T * x X ⊗ L d x of forms that vanish on the n -1 vectors v 1 (x), • • • , v n-1 (x)
given by the relative trivialization is one-dimensional. It induces an orthogonal decomposition T *

x X ⊗ L d x = V ⊕ V ⊥ , where V is of dimension n -1. The inverse image of V ⊥ in H x /H 2x by the evaluation map eval 2x is the line D of H x /H 2x containing the sections σ of H x whose derivatives at x vanish against v 1 (x), • • • , v n-1 (x). We denote by H x the orthogonal complement of D in H x /H 2x and by H 2x the direct sum D ⊕ (H 2x /H 3x ).

We then write σ = (σ between positive constants as d grows to infinity. For i ∈ {1, • • • , n -1}, h i linearly depends on σ 0 and σ 1 ; we write h i (σ 0 , σ 1 ) this linear expression. The derivatives ∂h I and ∂h J depend on σ 2 . The expression ∂h I ∧ ∂h J expands as a sum of 3 k terms, some of which vanish since the forms ∂f σ,x and ∂f σ,x can only appear once in the expression. Denote by h I J one of these 3 k terms. It is a monomial of degree 2k in σ 0 , σ 1 , σ 2 and we denote by l 0 the degree of σ 0 , by l 1 the degree of σ 1 and by l 2 = 2k -l 0 -l 1 the degree of σ 2 in this monomial. Now, it is enough to bound from above by a O((log d) 2 ) the following integral, where I, J ⊂ {1, • • • , n} k are given:

0 , σ 1 , σ 2 ) ∈ H ⊥ x × H x × H
1 d k H ⊥ x × Hx× H 2x | log n-1 i=0 |h i (x)| 2 | ( n-1 i=0 |h i (x)| 2 ) k ||h I J (σ 0 , σ 1 , σ 1 )||dµ(σ) ≤ 1 c 2k d d k H ⊥ x × Hx× H 2x | log(c 2 d ||σ 0 || 2 ) + log(1 + n-1 i=1 1 c 2 d |h i ( σ 0 ||σ 0 || , σ 1 ||σ 0 || )| 2 )| (1 + n-1 i=1 1 c 2 d |h i ( σ 0 ||σ 0 || , σ 1 ||σ 0 || )| 2 ) k ||σ 0 || 2k ... ... ||h I J ( σ 0 ||σ 0 || , σ 1 , σ 2 )||||σ 0 || l 0 e -||σ 0 || 2 -||σ 1 || 2 dσ 0 dσ 1 dµ(σ 2 ).
We replace σ 0 ||σ 0 || by 1 in this integral, without loss of generality, since it remains bounded. Define

α 1 = σ 1 /||σ 0 ||, so that dα 1 = 1 ||σ 0 || 2(n-1) dσ 1 . The integral rewrites 1 c 2k d d k H ⊥ x × Hx× H 2x | log(c 2 d ||σ 0 || 2 ) + log(1 + n-1 i=1 1 c 2 d |h i (1, α 1 )| 2 )| (1 + n-1 i=1 1 c 2 d |h i (1, α 1 )| 2 ) k ... ... ||h I J (1, α 1 , σ 2 )||||σ 0 || 2(n-1)-l 2 e -||σ 0 || 2 (1+||α 1 || 2 ) dσ 0 dα 1 dµ(σ 2 ). Now set β 0 = σ 0 1 + ||α 1 || 2 and β 1 = α 1 √ d , so that dβ 0 dβ 1 = 1+||α 1 || 2 d n-1 dσ 0 dα 1 . The integral becomes 1 c 2k d d 1+ l 0 2 H ⊥ x × Hx× H 2x log(c 2 d ||β 0 || 2 )-log(1+d||β 1 || 2 )+log(1+ n-1 i=1 1 c 2 d |h i (1, √ dβ 1 )| 2 ) (1+ n-1 i=1 1 c 2 d |h i (1, √ dβ 1 )| 2 ) k ( 1 d +||β 1 || 2 ) n-l 2 2 ... ... ||h I J (1, β 1 , σ 2 )||||β 0 || 2(n-1)-l 2 e -||β 0 || 2 dβ 0 dβ 1 dµ(σ 2 ).
The only terms depending on β 0 in this integral are log ||β 0 || 2 , ||β 0 || 2(n-1)-l 2 and e -||β 0 || 2 . They can be extracted from it and integrated over H ⊥ x thanks to Fubini's theorem. The latter integral over H ⊥

x turns out to be bounded independently of d. As a consequence, it becomes enough to bound from above the following 1 . Let us denote by g i (β 1 ) these expressions. Likewise, the monomial h I J is the product of three monomials of degrees l 0 , l 1 and l 2 in σ 0 , σ 1 and σ 2 respectively. By Proposition 5, the coefficients of these monomials are O(d 1 equals 0 if neither ∂f σ,x nor ∂f σ,x appears in the monomial h I J , equals 1 if one of these two forms appears and 2 if they both appear. As a consequence, ||h I J (1, β 1 , σ 2 )|| is bounded from above, up to a constant, by

c 2k d d 1+ l 0 2 Hx× H 2x log(c 2 d /d)-log( 1 d +||β 1 || 2 )+log(1+ n-1 i=1 1 c 2 d |h i (1, √ dβ 1 )| 2 ) (1+ n-1 i=1 1 c 2 d |h i (1, √ dβ 1 )| 2 ) k ( 1 d +||β 1 || 2 ) n-l 2 2 ... ... ||h I J (1, β 1 , σ 2 )||dβ 1 dµ(σ 2 ).

Note that by Proposition 4 and by definition of the functions h

i , 1 ≤ i ≤ n -1, the expressions 1 c d h i (1, √ dβ
d (n+2)k+l ′ 1 -l 1 /2 ||β 1 || l 1 ||σ 2 || l 2
. Now, only the term in ||σ 2 || l 2 depends on σ 2 in our integral. Again using Fubini's theorem, we may first integrate over H 2x equipped with the Gaussian measure dµ(σ 2 ) to get an integral bounded independently of d. The upshot is that we just need to bound the integral

I = 1 d 1+ l 0 +l 1 2 -l ′ 1 Hx | log(c 2 d /d) -log( 1 d + ||β 1 || 2 ) + log(1 + n-1 i=1 |g i (β 1 )| 2 )| (1 + n-1 i=1 |g i (β 1 )| 2 ) k ( 1 d + ||β 1 || 2 ) n-l 2 2 ||β 1 || l 1 dβ 1 .
There is a compact subset Q of H x independent of d and a constant C > 0 independent of x and d such that

∀β 1 ∈ Q, 1 ≤ 1 + n-1 i=1 |g i (β 1 )| 2 ≤ C and ∀β 1 ∈ H x \ Q, 1 + n-1 i=1 |g i (β 1 )| 2 ≥ 1 C ||β 1 || 2 ,
since by Definition 4 the vector fields v i remain uniformly linearly independent on the whole U.

Bounding from above the term

||β 1 || l 1 by ( 1 d + ||β 1 || 2 ) l 1 /2
, we finally just have to estimate from above the integrals

I 1 = log d d 1+ l 0 +l 1 2 -l ′ 1 Q 1 ( 1 d + ||β 1 || 2 ) n-l 1 +l 2 2 dβ 1
and

I 2 = 1 d 1+ l 0 +l 1 2 -l ′ 1 Hx\Q log d + log(||β 1 || 2 )dβ 1 ||β 1 || 2k ( 1 d + ||β 1 || 2 ) n-l 1 +l 2 2 , since log(1/d + ||β 1 || 2 ) over Q and log(c d /d) are O(log d). Note that l ′ 1 ≤ max(2, l 1
), so that the exponent 1 + l 0 +l 1 2 -l ′ 1 is never negative and vanishes if and only if l 0 = 0, l 1 = l ′ 1 = 2 and thus l 2 = 2k -2. There exists R > 0 such that

I 1 ≤ V ol(S 2n-3 ) log d d 1+ l 0 +l 1 2 -l ′ 1 R 1 d du u 2-l 1 +l 2 2 du = O((log d) 2 ), where u = 1 d + ||β 1 || 2 .
Likewise, there exists T > 0 such that

Hx\Q log d + log ||β 1 || 2 dβ 1 ||β 1 || 2k ( 1 d + ||β 1 || 2 ) n-l 1 +l 2 2 ≤ V ol(S 2n-3 ) ∞ T log d + log u u 2+l 0 /2 du.
This last integral is a O(log d), which implies Proposition 5 when U is of type (r).

Proof of Proposition 5 along the base and critical loci of p

The compact K given by Proposition 5 is covered by a finite number of elements of the atlas U adapted to (p, L). Moreover, we may assume that such elements are all disjoint from the critical locus Crit(p) when n > 2. Proposition 5 was proved in §2.4.1 for elements U of type (r) of U. Let us assume now that U is such an element of type (b) given by Definition 4 and that x ∈ K ∩ U. The case where U is of type (c) when n = 2 just follows along the same lines. The main part of the proof is similar to the one given in §2.4.1 and we have to bound from above the integral

I = 1 d 1+ l 0 +l 1 2 -l ′ 1 Hx | log(c 2 d /d) -log( 1 d + ||β 1 || 2 ) + log(1 + n-1 i=1 |g i (β 1 )| 2 )| (1 + n-1 i=1 |g i (β 1 )| 2 ) k ( 1 d + ||β 1 || 2 ) n-l 2 2 ||β 1 || l 1 dβ 1 .
However, the norm ||v 1 || of the vector v 1 given by Definition 5 converges now to 0 when x approaches the base locus of p. Denote by H ′′ x the hyperplane of H x consisting of the sections whose 1-jet at x vanish against v 1 , that is the sections whose image under eval 2x vanish against v 1 , see §2.3.1. Denote then by H ′

x the line orthogonal to H ′′

x in H x and by

β 1 = (β ′ 1 , β ′′ 1 ) the coordinates on H x = H ′ x × H ′′ x . This time there exists a compact subset Q = Q ′ × Q ′′ of H x , independent of d and of x ∈ K ∩ U, as well as a constant C > 0 such that ∀β 1 ∈ Q, 1 ≤ 1 + n-1 i=1 |g i (β 1 )| 2 ≤ C and ∀β 1 ∈ H x \ Q, 1 + n-1 i=1 |g i (β 1 )| 2 ≥ 1 C (1 + ||v 1 (x)|| 2 ||β ′ 1 || 2 + ||β ′′ 1 || 2 ).
The integral I over the compact Q is bounded from above by a O((log d) 2 ), see §2.4.1.

Only the second integral differs. In order to estimate the latter, let us bound from above

||β 1 || l 1 by ( 1 d + ||β 1 || 2 ) l 1 /2 .
We have to bound the integral 1

d 1+ l 0 +l 1 2 -l ′ 1 Hx\Q log d + log ||β 1 || 2 dβ 1 (1 + ||v 1 || 2 ||β ′ 1 || 2 + ||β ′′ 1 || 2 ) k ( 1 d + ||β 1 || 2 ) n-l 1 +l 2 2 .
When n -l 1 +l 2 2 > 1, let us bound from above this integral by 1

d 1+ l 0 +l 1 2 -l ′ 1 H ′′ x \Q ′′ dβ ′′ 1 ||β ′′ 1 || 2k H ′ x \Q ′ log d + log ||β 1 || 2 dβ ′ 1 (||β 1 || 2 ) n-l 1 +l 2 2 .
There exists R > 0 such that

1 π H ′ x \Q ′ log d + log ||β 1 || 2 dβ ′ 1 (||β 1 || 2 ) n-l 1 +l 2 2 ≤ ∞ R log d + log(||β ′′ 1 || 2 + u)du (||β ′′ 1 || 2 + u) n-l 1 +l 2 2 = log d + log(||β ′′ 1 || 2 + u) ( l 1 +l 2 2 -n + 1)(||β ′′ 1 || 2 + u) n-l 1 +l 2 2 -1 ∞ R + ∞ R du (n -l 1 +l 2 2 -1)(||β ′′ 1 || 2 + u) n-l 1 +l 2 2 = log d + log(||β ′′ 1 || 2 + R) (n -l 1 +l 2 2 -1)(||β ′′ 1 || 2 + R) n-l 1 +l 2 2 -1 + 1 (n -l 1 +l 2 2 -1) 2 (||β ′′ 1 || 2 + R) n-l 1 +l 2 2 -1
Hence, our integral gets bounded from above, up to a constant, by the integral

H ′′ x \Q ′′ log d + log(||β ′′ 1 || 2 + R) ||β ′′ 1 || 2(n-1)+l 0 dβ ′′ 1 , which is itself a O(log d) since H ′′ x is of dimension n -2. When n -l 1 +l 2 2
= 1, which implies that l 0 = 0 and k = n -1, we observe that

(1 + ||v 1 || 2 ||β ′ 1 || 2 + ||β ′′ 1 || 2 ) k = (1 + ||v 1 || 2 ||β ′ 1 || 2 + ||β ′′ 1 || 2 )(1 + ||v 1 || 2 ||β ′ 1 || 2 + ||β ′′ 1 || 2 ) k-1 ≥ ||v 1 || 2 (1 + ||β 1 || 2 )(1 + ||β ′′ 1 || 2 ) k-1 ≥ ||v 1 || 2 ||β 1 || 2 ||β ′′ 1 || 2(k-1)
as long as ||v 1 || ≤ 1, which can be assumed. Our integral gets then bounded by 1

d 1+ l 0 +l 1 2 -l ′ 1 ||v 1 || 2 H ′′ x \Q ′′ dβ ′′ 1 ||β ′′ 1 || 2(k-1) H ′ x \Q ′ log d + log ||β 1 || 2 dβ ′ 1 ||β 1 || 4
which is similar to the previous one. The latter is then bounded from above by a O( log d ||v 1 || 2 ), implying the result.

3 Total Betti numbers of random real hypersurfaces 3.1 Statement of the results

Expectation of the total Betti number of real hypersurfaces

Let (X, c X ) be a smooth real projective manifold of positive dimension n, meaning that X is a smooth n-dimensional complex projective manifold equipped with an antiholomorphic involution c X . Let π : (L, c L ) → (X, c X ) be a real holomorphic ample line bundle, so that the antiholomorphic involutions satisfy π For every section σ ∈ RH 0 (X; L d ) \ R∆ d , C σ = σ -1 (0) is a smooth real hypersurface of X. By Smith-Thom's inequality, see Theorem 2, the total Betti number b * (RC σ ; Z/2Z) = n-1 i=0 dim H i (RC σ , Z/2Z) of its real locus is bounded from above by the total Betti number b * (C σ ; Z/2Z) = 2n-2 i=0 dim H i (C σ ; Z/2Z) of its complex locus, which from Lemma 3 is equivalent to ( X c 1 (L) n )d n as d grows infinity. What is the expectation of this real total Betti number? If we are not able to answer to this question, we will estimate this number from above, see Theorems 4 and 5. Note that in dimension one, such an upper bound can be deduced from our recent work [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF].

• c L = c X • π.
Let us first precise the measure of probability considered on RH 0 (X; L d ). We proceed as in §2. We equip L with a real Hermitian metric h of positive curvature ω ∈ Ω (1,1) (X, R), real meaning that c * L h = h. As in §2.1, we denote by dx = 1 X ω n ω n the associated volume form of X, by the induced L 2 -scalar product on RH 0 (X; L d ), and by µ R the associated Gaussian measure, defined by the relation

∀A ⊂ RH 0 (X; L d ), µ R (A) = 1 ( √ π) N d A e -||σ|| 2 dσ.
For every d > 0, we denote by

E R (b * (RC σ ; Z/2Z)) = RH 0 (X;L d )\R∆ d b * (RC σ ; Z/2Z)dµ R (σ)
the expected total Betti number of real hypersurfaces linearly equivalent to L d .

Theorem 4 Let (X, c X ) be a smooth real projective manifold of dimension n greater than one equipped with a Hermitian real line bundle (L, c L ) of positive curvature. Then, the expected total Betti number E R (b

* (RC σ ; Z/2Z)) is a o(d n ) and even a O(d(log d) 2 ) if n = 2.
Note that the exact value of the expectation E R (b * (RC σ ; Z/2Z)) is only known when X = CP 1 , see [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF], [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF] and [START_REF] Edelman | How many zeros of a random polynomial are real?[END_REF]. While we were writing this article in june 2011, Peter Sarnak informed us that together with Igor Wigman, he can bound this expectation by a O(d) when X = CP 2 and suspects it is equivalent to a constant times d when d grows to infinity. Such a guess was already made couple of years ago by Christophe Raffalli, based on computer experiments. It could be that this expectation is in fact equivalent to d n 2 times a constant as soon as the real locus of the manifold (X, c X ) is non empty and in particular that the bound given by Theorem 4 can be improved by a O(d √ d is a fundamental scale in Kähler geometry and Hörmander's theory of peak sections. A peak section centered at x can be symmetrized to provide a real section having two peaks near x and c X (x), see §3.2. This phenomenon plays an important rôle in the proof of Theorem 4 and seems to be intimately related to the value of the expectation E R (b * (RC σ ; Z/2Z)). Note finally that Theorem 4 contrasts with the computations made by Fedor Nazarov and Mikhail Sodin in [START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF] for spherical harmonics in dimension two, as well as with the one achieved by Maria Nastasescu for a real Fubini-Study measure, as P. Sarnak informed us. In both cases, the expectation is quadratic.

Theorem 5 Let (X, c X ) be the product of n > 1 smooth real projective curves, equipped with a real Hermitian line bundle (L, c L ) of positive curvature. Then, the expected total Betti number

E R (b * (RC σ ; Z/2Z)) is a O(d n 2 (log d) n ).

Random real divisors and distribution of critical points

Our proof of Theorem 4 is based on a real analogue of Theorem 3 that we formulate here, see Theorem 6. We use the notations introduced in §3.1.1 and equip X with a real Lefschetz pencil p : X CP 1 , see §1. For every d > 0, denote by R ∆ d = ∆ d ∩RH 0 (X; L d ) the union of R∆ d with the set of sections σ ∈ RH 0 (X; L d ) such that either C σ contains a critical point of p, or p |Cσ is not Lefschetz, see §2.1. Denote, as in §2.1, by R σ the critical locus of the restriction p |Cσ , where σ ∈ RH 0 (X;

L d ) \ R ∆ d .
Then, for every continuous function χ : X → R, denote by

E R ( ν σ , χ ) = 1 #R σ RH 0 (X;L d )\R ∆ d ( x∈Rσ χ(x))dµ R (σ)
the expectation of the probability measure ν σ carried by the critical points of p |Cσ , see Definition 3, computed with respect to the real Gaussian measure µ R and evaluated against χ.

Theorem 6 Let (X, c X ) be a smooth real projective manifold of positive dimension n equipped with a real Lefschetz pencil p : X CP 1 of critical locus Crit(p). Let (L, c L ) → (X, c X ) be a real ample line bundle equipped with a real Hermitian metric of positive curvature. Let (χ d ) d∈N * be a sequence of elements of C 2 (X, R) which converges to χ in L 1 (X, R) as d grows to infinity. Assume that

sup x∈Supp(χ d ) d(x, c X (x)) > 2 log d √ d .
When n > 2, assume moreover that the distance between Crit(p) and the supports of χ d , d > 0, are uniformly bounded from below by some positive constant. Then, the real expectation E R ( ν σ , χ ) converges to the integral X χdx as d grows to infinity. More precisely,

E R ( ν σ , χ ) = X χ d dx + O (log d) 2 d ||∂ ∂χ d || L 1 + O 1 d ||χ d || L 1 .
In this Theorem 6, Supp(χ d ) denotes the support of χ d and d(x, c X (x)) the distance between the points x and c X (x) for the Kähler metric induced by the curvature ω of L.

Real peak sections and evaluation of two-jets of sections

Let (X, c X ) be a smooth real projective manifold of positive dimension n and (L, c L ) → (X, c X ) be a real holomorphic ample line bundle equipped with a real Hermitian metric h of positive curvature.

Definition 8 A real peak section of the ample real holomorphic line bundle (L, c L ) over the real projective manifold (X, c X ) is a section which writes σ+c * σ ||σ+c * σ|| , where σ is a peak section of Hörmander given by Definition 7 and

c * σ = c L d • σ • c X .
Recall that from Lemma 4, the L 2 -norm of a peak section concentrates in a log d √ d neighborhood of a point x ∈ X. When x is real, the real peak section σ+c * σ ||σ+c * σ|| looks like a section of Hörmander given by Definition 7. When the distance between x and c X (x) is bigger than log d √ d , more or less half of the L 2 -norm of this real section concentrates in a neighborhood of x and another half in a neighborhood of c X (x). Such a real section has thus two peaks near x and c X (x). When d(x, c X (x)) < 2 log d Proof. By definition,

σ d , c * σ d = X h d (σ d , c * σ d )dx = B(x, log d √ d ) h d (σ d , c * σ d )dx + X\B(x, log d √ d ) h d (σ d , c * σ d )dx ≤ B(x, log d √ d ) h d (σ d , σ d )dx 1/2 B(x, log d √ d ) h d (c * σ d , c * σ d )dx 1/2 + X\B(x, log d √ d ) h d (σ d , σ d )dx 1/2 X\B(x, log d √ d ) h d (c * σ d , c * σ d )dx 1/2
by Cauchy-Schwarz's inequality. We deduce that

σ d , c * σ d ≤ ||σ d ||   X\B(x, log d √ d ) h d (σ d , σ d )dx 1/2 + B(x, log d √ d ) h d (c * σ d , c * σ d )dx
σ d + c * σ d , σ d + c * σ d = 2||σ d || 2 + 2ℜ σ d , c * σ d = 2||σ d || 2 + O(1/d 2 )||σ d || 2 , so that ||σ d + c * σ d || = √ 2||σ d ||(1 + O(1/d 2 )). 2 Set RH x = H x ∩ RH 0 (X; L d ), RH 2x = H 2x ∩ RH 0 (X; L d ) and RH 3x = H 3x ∩ RH 0 (X; L d ),
eval x : σ ∈ RH 0 (X; L d )/RH x → σ(x) ∈ L d x , eval 2x : σ ∈ RH x /RH 2x → ∇σ |x ∈ T * x X ⊗ L d x , and eval 3x : σ ∈ RH 2x /RH 3x → ∇ 2 σ |x ∈ Sym 2 (T * x X) ⊗ L d x .
The following Proposition 6 is a real analogue of Proposition 4. ||iσ 0 +c * (iσ 0 )|| and check in the same way that d -n h d ( σ ⊥ 0 , σ ⊥ 0 ) |x d converges to a positive constant as d grows to infinity. But the quotient σ 0 (x d ) σ 0 (x d ) and thus

σ ⊥ 0 (x d ) σ ⊥ 0 (x d ) converge to i as d grows to infinity. Likewise, σ 0 , σ 0 = σ + c * σ, iσ -ic * σ ||σ + c * σ||||iσ -ic * σ|| = 2ℜ(i σ, c * σ ) ||σ + c * σ||||iσ -ic * σ|| = O( 1 d ), so that σ ⊥ 0 , σ ⊥ 0 = O( 1 d
) and d -n 2 eval x as well as its inverse have bounded norms and determinants when d grows to infinity. The remaining cases are obtained by similar modifications of the proof of Corollary 1. 2

Proof of the main results

Proof of Theorem 6

The proof goes along the same lines as the one of Theorem 3. We begin with the following analogue of Corollary 1.

Corollary 2 Under the hypotheses of Proposition 3, we assume moreover that the manifold X and Hermitian bundle L are real. Then,

E R ( ν σ , χ ) = 1 #R σ d n X χω n + 1 #R σ n-1 k=0 ( i 2π ) n-k d k X ∂ ∂χ ∧ ω k ∧ RH 0 (X;L d )\R ∆ d λ x (∂ ∂λ x ) n-1-k dµ(σ),
where

λ x = log d 2 |f σ,x | 2 + n-1 i=1 |∂f σ,x (v i ) + f σ,x ∂g x (v i )| 2 .
Proof. The result follows after integration over RH 0 (X;

L d ) \ R ∆ d of
k ∈ {0, • • • , n-1} the integral RH 0 (X;L d )\R ∆ d λ x (∂ ∂λ x ) k dµ(σ) is uniformly bounded on the support of ∂ ∂χ d by a O(d k (log d) 2 /||v 1 || 2 ) since 1/||v 1 || 2 is integrable over X. The space RH 0 (X; L d ) \ R ∆ d is
equipped with its Gaussian measure, but the integrand only depends on the two-jets of sections σ ∈ RH 0 (X; L d ) at the point x ∈ X \(RX ∪Crit(p)). Since the Gaussian measure is a product measure, writing RH 0 (X; L d ) as the product of the space RH 3x introduced in §3.2 with its orthogonal complement RH ⊥ 3x , we deduce that it is enough to prove this uniform bound for the integral RH ⊥ 3x \R ∆ d λ x (∂ ∂λ x ) k dµ(σ). By Proposition 6, the evaluation maps eval x , eval 2x and eval 3x of jets up to order two at x provide an isomorphism between RH ⊥ 3x and L

d x ⊕ (T * x X ⊗ L d x ) ⊕ (Sym 2 (T * x X) ⊗ L d x )
. In particular, this implies that the space RH ⊥ 3x is also a complement of the space H 3x introduced in §2.3.1. By Proposition 4, these evaluation maps factor through an isomorphism I -1 d between RH ⊥ 3x and the orthogonal complement H ⊥ 3x of H 3x in H 0 (X; L d ). By Proposition 4 and Proposition 6, the Jacobian of I d is bounded independently of d, while the Gaussian measure µ R is bounded from above on RH ⊥ 3x by the measure (I d ) * µ up to a positive dilation of the norm, independent of d. After a change of variables given by this isomorphism I d and after the one given by the dilation, it becomes enough to prove that the integral H

0 (X;L d )\ ∆ d λ x (∂ ∂λ x ) k dµ(σ) is uniformly bounded on the support of χ d by a O(d k (log d) 2 /||v 1 || 2 )
. The latter follows from Proposition 5. Let us identify a neighborhood V of RX in X with the cotangent bundle of RX. We can assume that V \RX does not contain any critical point of p. Let χ : X → [0, 1] a be function of class C 2 satisfying χ = 1 outside of a compact subset of V and χ = 0 in a neighborhood of RX. For every d > 0, let χ d : X → [0, 1] be the function which equals one outside of V and whose restriction to V writes in local coordinates (q, p)

∈ V ≃ T * RX → χ(q, √ d log d p) ∈ [0, 1]
, where q is the coordinate along RX and p the coordinate along the fibers of T * RX. This sequence (χ d ) d>0 converges to the constant function 1 in L 1 (X, R) as d grows to infinity, while the norm

||∂ ∂χ d || L 1 (X,R) is a O(( log d √ d ) n-2 ). Moreover, for every x ∈ Supp(χ d ), d(x, c X (x)) > 2 log d √ d
, so that when n = 2, Theorem 6 applies. From Proposition 2, we thus deduce that

RH 0 (X;L d )\R ∆ d ( x∈Rσ χ d (x))dµ R (σ) = #R σ + O(d(log d) 2 ). Moreover, for every σ ∈ RH 0 (X; L d ) \ R ∆ d and every d > 0, we have #Crit(p |RCσ ) ≤ #R σ -x∈Rσ χ d (x), so that after integration, RH 0 (X;L d )\R ∆ d #Crit(p |RCσ )dµ R (σ) = O(d(log d) 2 ),
hence the result when n = 2.

When n > 2, we apply Theorem 6 to the function χ and deduce that

RH 0 (X;L d )\R ∆ d ( x∈Rσ χ(x))dµ R (σ) = ( X χdx)#R σ + O(d n-1 (log d) 2 ).
The expectation of the number of real critical points satisfies now the bound

RH 0 (X;L d )\R ∆ d #Crit(p |RCσ )dµ R (σ) ≤ (1 - X χdx)#R σ + O(d n-1 (log d) 2 ).
Changing the function χ if necessary, the difference (1-X χdx) can be made as small as we want. We thus deduce from Proposition 2 that d -n RH 0 (X;L d )\R ∆ d #Crit(p |RCσ )dµ R (σ) converges to zero as d grows to infinity and the result.

Proof of Theorem 5

We will prove Theorem 5 by induction on the dimension n of (X, c X ). When n = 2, Theorem 5 is a consequence of Theorem 4. Let us assume now that n > 2 and that (X, c X ) is the product of a real curve (Σ, c Σ ) by a product of curves (F, c F ) of dimension n -1. We denote by p : (X, c X ) → (Σ, c Σ ) the projection onto the first factor. Again, for every section σ ∈ RH 0 (X, L d ) \ R ∆ d , the restriction of p to RC σ satisfies the hypotheses of Lemma 1, so that b * (RC σ ; Z/2Z) ≤ 4b * (RF ∩ RC σ ; Z/2Z) + #Crit(p |RCσ ).

Let us bound from above each term of the right hand side of the latter by a O(d n 2 (log d) n ). Every connected component R of RX has a neighborhood in X biholomorphic to a product of annuli in C and thus satisfies the conditions of Definition 6. Indeed, every complex annulus has a non vanishing holomorphic vector field. By product, every component of RF has a neighborhood in F trivialized by n -1 vector fields v 1 , • • • , v n-1 . We deduce a trivialization of ker dp = T F in the neighborhood of R. This open neighborhood can be completed into an atlas adapted to p with open sets disjoints from RX. For every d > 0, set θ d = 1 -χ d , where (χ d ) d>0 is the sequence of functions introduced in §3.3.2. Corollary 2 applies to

θ d whose L 1 norm is a O(( log d √ d ) n ) whereas ||∂ ∂θ d || L 1 (X,R) is a O(( log d √ d ) n-2
), so that the conclusions of Theorem 6 also hold for θ d . As a consequence,

RH 0 (X;L d )\R ∆ d ( x∈Rσ θ d (x))dµ R (σ) = O(d n 2 (log d) n ). Since for every σ ∈ RH 0 (X; L d ) \ R ∆ d and every d > 0, #Crit(p |RCσ ) ≤ x∈Rσ θ d (x), we deduce that #Crit(p |RCσ ) is a O(d n 2 (log d) n ).
It remains to prove that the same holds for the integral RH 0 (X;L d )\R ∆ d b * (RF ∩ RC σ ; Z/2Z)dµ R (σ). If the space of integration was the space of real sections of the restriction L |F , this would follow from Theorem 5 in dimension n -1. We will thus reduce the space of integration to this one. Let us write (F, c compare with Corollary 1. We then proceed as in the proof of Theorem 4, noting that the latter integral only depends on the two-jets of sections σ ∈ RH 0 (X; L d )\R ∆ d . Let us thus decompose RH 0 (X; L d ) as the product of the subspace R H 3x of sections whose two-jet at x ∈ F of their restriction to F vanishes, with its orthogonal complement R H ⊥ 3x . We get

F ) = (Σ 2 , c σ 2 ) × (Y, c Y ) where (Σ 2 , c σ 2 ) is a smooth real curve and (Y, c Y ) a (n -2)-dimensional product of curves. From Lemma 1, for every σ ∈ RH 0 (X; L d ) \ R ∆
RH 0 (X;L d )\R ∆ d λ x (∂ ∂λ x ) n-2-k dµ R (σ) = R H ⊥ 3x \ ∆ d λ x (∂ ∂λ x ) n-2-k dµ R (σ)
for every k. Likewise, the space of sections RH 0 (F ; L d |F ) of the restriction L |F decomposes into the product RH 3x × RH ⊥ 3x . From Proposition 6, the map

σ ∈ R H ⊥ 3x → σ F = 1 √ d pr ⊥ (σ |F ) ∈ RH ⊥ 3x ,
composition of the restriction map to F , a contraction by √ d and projection pr ⊥ onto RH ⊥ 3x , as well as its inverse are of bounded norms and determinants. This map in fact asymptotically coincides with the isomorphism R H

⊥ 3x → L d x ⊕ T * x F ⊗ L d x ⊕ Sym 2 (T * x F ) ⊗ L d
x given by the evaluation maps, composed with the inverse of the isomorphism

RH ⊥ 3x → L d x ⊕ T * x F ⊗ L d x ⊕ Sym 2 (T * x F ) ⊗ L d
x given by the evaluation maps, see Proposition 5. From Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], the restricted section σ |F is indeed asymptotically orthogonal to RH 3x . Hence, there is a constant C > 0 such that

|| R H ⊥ 3x \ ∆ d λ x (∂ ∂λ x ) n-2-k dµ R (σ)|| ≤ C|| RH ⊥ 3x \ ∆ d (log d + λ x )(∂ ∂λ x ) n-2-k dµ R (σ F )||,
since f √ dσ F ,x = √ df σ F ,x , so that asymptotically, λ x (σ F ) = log d+λ x (σ) |F . Proceeding as in the proof of Theorem 4, the latter right hand side is a O(d n-2-k (log d) 3 ). We deduce that Hence, the result follows by recurrence over the dimension n. It comes from the fact that the number #R σ of critical points of our Lefschetz pencil does not coincide with the leading term ( X c 1 (L) n )d n given by Poincaré-Martinelli's formula, but is rather a polynomial of degree n given by Proposition 2 having ( X c 1 (L) n )d n as leading term. It would be of interest to identify every monomial of the latter with a term of Poincaré-Martinelli's formula or of any analytic formula. Anyway, because of this O( 1 d ||χ|| L 1 ) term in Theorem 6, we cannot use the function χ with support disjoint from RX which we used in the proof of Theorem 4 but rather have to use the function θ with support in a neighborhood of RX and which equals 1 on RX which we used in the proof of Theorem 5.

2. The use of Poincaré-Martinelli's formula with local trivializations (v 1 , • • • , v n-1 ) forces us to choose an atlas U on X and an associated partition of unity (ρ U ) U ∈U . As a consequence, even if a function θ on X equals one in a neighborhood of RX, so that ∂ ∂θ has support disjoint from RX, this is not true for the functions ρ U θ, so that Theorem 6 or Corollary 2 cannot be used. Recall that on RX, or near RX, at a smaller scale than log d √ d , the two peaks of a real peak section interfere so that results of Proposition 6 on evaluation maps no more hold. This forces us to have a neighborhood of each connected component of RX on which the vector fields v 1 , • • • , v n are globally defined, as it is the case for products of curves for instance.

3. Our Lefschetz pencils do not produce Morse functions from RX to R, but rather from RX to RP 1 . As a consequence, the total Betti number of RX is not bounded from above by the number of critical points of this pencil, one has to take into account also the total Betti number of a fiber, see Lemma 1, and thus prove the result by induction on the dimension. Such a fiber of the pencil becomes then submitted to the same constraints as X.

4. The weak convergence of the measure given by Theorems 3 and 6 was only proved at bounded distance of the critical set of the Lefschetz pencil in dimensions n > 2. It is actually not hard to prove it on the critical set for n = 3 but seems less clear to us for n > 3. This is another obstacle since the pencils have real critical points in general, which have to be approached by the supports of our test functions θ. Note finally that what is the exact value of the expectation in Theorem 4 remains a mystery, as well as what happens below this expectation. Is there any exponential rarefaction below this expectation similar to the one observed in [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF]? Is indeed the expectation a constant times d n 2 as soon as RX is non empty?

3 .

 3 The bound d n 2 (log d) n of Theorem 1 indeed appears to be the volume of a log d √ d -neighborhood of the real locus for the metric induced by the curvature form of L d .

  since we may assume that U 0 ∩ U ∞ retracts onto two fibers of p. Now, the restriction of p to U 0 is a Morse function taking values in I 0 and having the same critical points as p. By the Morse Lemma, b * (U 0 ) ≤ b * (F ) + #Crit(p |RX ). This proves the first part of Lemma 1. If p : X CP 1 is a real Lefschetz pencil with base locus B, we denote by X → X the blow-up of B in X and by p : X → CP 1 the induced Lefschetz fibration. From what has just been proved, we know that b

  Every atlas of X becomes adapted in the sense of Definition 4 after refinement. Let x be a point in X. If x is a regular point of p, by the implicit function theorem it has a neighborhood biholomorphic to the model (r) of Definition 4. If x is a base point (resp. a critical point), it has by definition (resp. by the holomorphic Morse Lemma) a neighborhood biholomorphic to the model (b) (resp. (c)) of Definition 4.

0equals 1 +

 1 || is a O( 1 d ), where ||σ Hx 0 || 2 = σ Hx 0 , σ Hx 0 denotes the L 2 -norm of σ Hx 0 . From the vanishing of the product σ 0 -σ Hx 0 , σ Hx 0 we deduce that ||σ Hx 0 || is a O( 1 d ). It follows that the norm of σ 0 -σ Hx 0 O( 1 d 2 ) and we set

2

 2

2. 4 Proposition 5

 45 Proof of Theorem 3 Under the hypotheses and notations of Proposition 3, for all k ∈ {0, • • • , n -1} and x ∈ U, the integral

2

 2

2. 4 . 1

 41 Proof of Proposition 5 outside of the base and critical loci of p

  2x and |h 0 (x)| = d h d (σ 0 , σ 0 ) |x = c d ||σ 0 ||, where ||σ 0 || = σ 0 , σ 0 and by Proposition 4, c d d -n+2 2 remains bounded

n+2 2 l

 2 0 ), O(d n+1 2 l 1 +l ′ 1 ) and O(d n+2 2 l 2 ) respectively, where l ′

  For every d > 0, we denote by L d the d-th tensor power of L, by RH 0 (X; L d ) the space of global real holomorphic sections of L d , which are the sections σ ∈ H 0 (X; L d ) satisfying σ • c X = c L • σ, and by R∆ d = ∆ d ∩ RH 0 (X; L d ) the real discriminant locus.

n 2 ) 1 √d

 21 . When (X, c X ) is a product of smooth real projective curve for instance, we can improve the o(d n ) given by Theorem 4 by a O(d n 2 (log d) n ), being much closer to a O(d n 2 ) bound, see Theorem 5 below. This d n 2 can be understood as the volume of a neighborhood of the real locus RX in X for the volume form induced by the curvature of L d , where 1

4 . 5

 45 interfere, interpolating the extreme cases just discussed. We are now interested in the case d(x, c X (x)) > 2 log d √ d , where we can establish a real analogue of Proposition Lemma Let (L, c L ) be a real holomorphic Hermitian line bundle of positive curvature over a smooth real projective manifold (X, c X ) of positive dimension n. Let (x d ) d∈N * be a sequence of points such that d(x d , c X (x d )) > 2 log d √ d and let (σ d ) d∈N * be an associated sequence of sections given by Lemma 4 with p ′ = 2. Then, the Hermitian product σ d , c * σ d is a O( 1 d 2 ), so that the norm ||σ d + c * σ d || equals √ 2||σ d ||(1 + O(1/d 2 )).

1 / 2 

 12 By assumption, the balls B(x, log d √ d ) and B(c X (x), log d √ d ) are disjoint, so that by Lemma 4, these two last terms are O(1/d 2 )||σ d ||. Hence,

Proposition 6

 6 Let (L, c L ) be a real holomorphic Hermitian line bundle of positive curvature over a smooth complex projective manifold X of positive dimension n. Let(x d ) d∈N * be a sequence of points in X such that d(x d , c X (x d )) > 2 log d/ √ d. Then, the maps d -n 2 eval x d , d -n+1 2 eval 2x d and d -n+22 eval 3x d as well as their inverse have bounded norms and determinants, as long as d is large enough.Note that the evaluation maps eval x d , eval 2x d and eval 3x d of Proposition 6 are only R-linear.Proof of Proposition 6. The proof is analogous to the one of Lemma 5. Let σ 0 = σ+c * σ ||σ+c * σ|| be a real peak section given by Definition 8, where σ is a peak section given by x d and Definition 7. From Lemma 5, we know that ||σ + c * σ|| equals√ 2 + O(1/d 2 ). Moreover 2 log d √ d < d(x d , c X (x d )) and by Lemma 4, the L 2 -norm of c * σ in a neighborhood of x d is a O(1/d 2 ). From the mean inequality, we deduce the bound (see[START_REF] Hörmander | Notions of convexity[END_REF], Theorem 4.2.13 for instance)h d (c * σ, c * σ) |x d = O(1/d 2 ), so that by Lemma 2.1 of[START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], d -n h d (σ 0 , σ 0 ) |x d converges to a non negative constant as d grows to infinity. Denote by σ RHx 0 the orthogonal projection of σ 0 onto RH x . We proceed as in the proof of Proposition 4 to get that σ, since c is an isometry for the L 2 -Hermitian product. Hence, σ 0 , RHx 0 || , we deduce as in the proof of Proposition 4 that d -n h d (σ ⊥ 0 , σ ⊥ 0 ) |x d converges to a positive constant as d grows to infinity. Replacing σ by iσ, we define σ 0 = iσ 0 +c * (iσ 0 )

2

 2

3. 3 . 2

 32 Proof of Theorem 4Equip (X, c X ) with a real Lefschetz pencil p : X CP 1 and denote by F a regular fiber of p. For every section σ ∈ RH 0 (X;L d ) \ R ∆ d ,the restriction of p to RC σ satisfies the hypotheses of Lemma 1, so that b * (RC σ ; Z/2Z) ≤ b * (RF ∩ RC σ ; Z/2Z) + #Crit(p |RCσ ). By Smith-Thom's inequality, see Theorem 2, b * (RF ∩RC σ ; Z/2Z) ≤ b * (F ∩C σ ; Z/2Z) while from Lemma 3 applied to L |F , b * (F ∩ C σ ; Z/2Z) is a O(d n-1 ). As a consequence, we have to prove that the expectation of #Crit(p |RCσ ) is a o(d n ) and even a O(d(logd) 2 ) when n = 2.

1 F θ d ω n- 1 +F∂

 11 d , b * (RF ∩ RC σ ; Z/2Z) ≤ 4b * (RY ∩ RC σ ; Z/2Z) + #Crit(p 2|RF ∩RCσ ),where p 2 : F → Σ 2 is the projection onto the first factor. Denote, with a slight abuse of notation, by θ d the restriction of θ d to F . We haveRH 0 (X;L d )\R ∆ d #Crit(p 2|RF ∩RCσ )dµ R (σ) ≤ RH 0 (X;L d )\R ∆ d   x∈Crit(p |F ∩Cσ ) θ d (x)   dµ R (σ).After integration over RH 0 (X; L d )\R ∆ d of the relation given by Proposition 3 applied toF and v 1 , • • • , v n-2 , we deduce RH 0 (X;L d )\R ∆ d   x∈Crit(p |F ∩Cσ ) θ d (x)   dµ R (σ) ≤ d n-∂θ d ∧ ω k ∧ RH 0 (X;L d )\R ∆ d λ x (∂ ∂λ x ) n-2-k dµ R (σ), where λ x = log d 2 |f σ,x | 2 + n-2 i=1 |∂f σ,x (v i ) + f σ,x ∂g x (v i )| 2 ,

1 2

 1 RH 0 (X;L d )\R ∆ d #Crit(p 2|RF ∩RCσ ) = O(d n-1 ||θ d || L 1 (F,R) ) +O(d n-2 (log d) 3 ||∂ ∂θ d || L 1 (F,R) ) = O(d n-(log d) n )).

3. 4

 4 Final remarks Several technical issues prevent us from improving Theorem 4 with a O(d n 2 (log d) n ) bound in general or a O(d n 2 ) bound. 1. First of all, Theorem 6, which is central in the proof of Theorem 4, contains a O( 1 d ||χ|| L 1 ) term.

5 .√ d might be improved by a 1 √d

 51 The log d √ d -scale which we use throughout the paper comes from Lemma 4 taken out from [18]. It ensures that outside of the ball of radius log d √ d , the L 2 -norm of peak sections is a O( 1 d 2p ′ ). This log d instead if a weaker upper bound for this L 2 -norm, such as a O(1), suffices. However, even with such a 1 √ d -scale, we would still have some log d term in our Theorem 4, because some log d term shows up in our estimates of integrals arising from Poincaré-Martinelli's formula, at the end of §2.4.1.

  ). Proof. Denote by X (resp. C σ d ) the blow-up of the base locus B (resp. B ∩ C σ d ) of p (resp. p |Cσ d ), so that X (resp. C σ d ) is equipped with a Lefschetz fibration induced by p : X → CP 1 (resp. p | Cσ d : C σ d → CP 1 ). Let F be a regular fiber of p transverse to C σ d and F be the corresponding fiber in X. By Proposition 1,

  2 dσ, where ||σ|| 2 = σ, σ and dσ denotes the Lebesgue measure associated to . Denote by ∆ d ⊂ H 0 (X; L d ) the discriminant locus, that is the set of sections of H 0 (X; L d ) which do not vanish tranversally. Likewise, denote by ∆ d ∈ H 0 (X; L d ) the union of ∆ d with the set of sections σ ∈ H 0 (X; L d ) such that either the restriction of p to C σ is not Lefschetz, or this vanishing locus C σ meets the critical set Crit(p). By Bertini's theorem (see for example Theorem 8.18 of[START_REF] Hartshorne | Algebraic geometry[END_REF]), ∆ d is a hypersurface of H 0 (X; L d ) as soon as d is large enough, which will be assumed throughout this article.

  Indeed, only the first term ∂f σ,x (v i ) ofh i = ∂f σ,x (v i ) + df σ,x ∂g x (v i ) dependson β 1 and by Proposition 4, ||∂f σ,x || 2 |x = h d (∇σ 1 , ∇σ 1 ) |x grows as ||σ 1 || times d

		n+1 2	while
	c d grows as d	n+2 2

[START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF]

) are affine in β 1 with coefficients bounded independently of d.

  where H x , H 2x and H 3x have been introduced in §2.3.1. Likewise, with a slight abuse of notation, denote by eval x , eval 2x and eval 3x the restrictions of the evaluation maps to the spaces RH 0 (X; L d )/RH x , RH x /RH 2x and RH 2x /RH 3x respectively, so that:

  the relation given by Proposition 3. 2 Proof of Theorem 6. We apply Corollary 2 to the functions χ d , d > 0 and use the notations of this corollary. The first term in the right hand side of the formula given by this corollary is X χ d dx + O( 1 d ||χ d || L 1 ) as follows from Proposition 2. It is thus enough to prove that for

The research leading to these results has received funding from the European Community's Seventh Framework Progamme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement n o [258204], as well as from the French Agence nationale de la recherche, ANR-08-BLAN-0291-02.