Thomas Letendre
email: letendre@math.cnrs.fr.

Martin Puchol
email: martin.puchol@math.cnrs.fr.

Variance of the volume of random real algebraic submanifolds II

Keywords: Random submanifolds, Kac-Rice formula, linear statistics, Kostlan-Shub-Smale polynomials, Bergman kernel, real projective manifold, Wiener-Itō expansion Mathematics Subject Classification 2010: 14P99, 32A25, 53C40, 60G15, 60G57

Let X be a complex projective manifold of dimension n defined over the reals and let M be its real locus. We study the vanishing locus Zs d in M of a random real holomorphic section s d of E ⊗ L d , where L → X is an ample line bundle and E → X is a rank r Hermitian bundle, r ∈ {1, . . . , n}. We establish the asymptotics of the variance of the linear statistics associated with Zs d , as d goes to infinity. These asymptotics are of order d r-n 2 . As a special case, we get the asymptotic variance of the volume of Zs d .

. Second, we show that the leading constant in our asymptotics is positive. This last result is proved by studying the Wiener-Itō expansion of the linear statistics associated with the common zero set in RP n of r independent Kostlan-Shub-Smale polynomials.

Introduction

In recent years, the study of random submanifolds has been a very active research area [START_REF]C ∞ scaling asymptotics for the spectral projector of the Laplacian[END_REF][START_REF]Expected topology of random real algebraic submanifolds[END_REF][START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF] Marinucci | Non-universality of nodal length distribution for arithmetic random waves[END_REF][START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF][START_REF] Sarnak | Topologies of nodal sets of random band limited functions, Advances in the theory of automorphic forms and their L-functions[END_REF]. There exist several models of random submanifolds, built on the following principle. Given M a dimension n ambient manifold and r ∈ {1, . . . , n}, we consider the common zero set of r independent random functions on M . Under some technical assumption, this zero set is almost surely a codimension r smooth submanifold.

In this paper, we are interested in a model of random real algebraic submanifolds in a projective manifold. It was introduced in this generality by Gayet and Welschinger in [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF] and studied in [START_REF]Expected topology of random real algebraic submanifolds[END_REF][START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF] Letendre | Expected volume and Euler characteristic of random submanifolds[END_REF][START_REF]Variance of the volume of random real algebraic submanifolds[END_REF], among others. This model is the real counterpart of the random complex algebraic submanifolds considered by Bleher, Shiffman and Zelditch [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF][START_REF] Shiffman | Distribution of zeros of random and quantum chaotic sections of positive line bundles[END_REF][START_REF]Number variance of random zeros on complex manifolds II: smooth statistics[END_REF].

Framework. Let us describe more precisely our framework. More details are given in Sect. 2, below. Let X be a smooth complex projective manifold of dimension n 1. Let L be an ample holomorphic line bundle over X and let E be a rank r ∈ {1, . . . , n} holomorphic vector bundle over X . We assume that X , E and L are endowed with compatible real structures and that the real locus of X is not empty. We denote by M this real locus which is a smooth closed (i.e. compact without boundary) manifold of real dimension n.

Let h E and h L denote Hermitian metrics on E and L respectively, which are compatible with the real structures. We assume that h L has positive curvature ω, so that ω is a Kähler form on X . This ω induces a Riemannian metric g on X , hence on M . Let us denote by |dV M | the Riemannian volume measure on M induced by g.

Related works and novelty of the main results. This paper extends the results of [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF], by the first-named author. In [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Thm. 1.6], our main result (Thm. 1.6 above) was proved for r < n and α ∈ (0, α 0 ), where α 0 ∈ (0, 1) is some explicit constant depending on n and r. The main novelty in Thm. 1.6 is that it covers the case of maximal codimension (r = n), that is the case where Z d is almost surely a finite subset of M . This case was not considered in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF] because of additional singularities arising in the course of the proof, which caused it to fail when r = n.

An important contribution of the present paper is that we prove new estimates (see Lem. 5.26, 5.28 and 5.29) for operators related to the Bergman kernel of E ⊗ L d , which is the correlation kernel of the random field (s d (x)) x∈M . These estimates are one of the key improvements that allow us to prove Thm. 1.6 in the case r = n. They also allow us to consider α ∈ (0, 1) instead of α ∈ (0, α 0 ). Finally, the use of these estimates greatly clarifies the proof of Thm. 1.6 in the case r < n, compared to the proof given in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]. For this reason, we give the proof of Thm. 1.6 in the general case r ∈ {1, . . . , n} and not only for r = n. This does not lengthen the proof.

The second main contribution of this article is the proof of the positivity of the leading constant in Thm. 1.6 (cf. Thm. 1.8). This result did not appear in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]. Since the leading constant in Thm. 1.6 is universal, when r = n one can deduce Thm. 1.8 from results of Dalmao [START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF] (if r = n = 1) and Armentano-Azaïs-Dalmao-León [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF] (if r = n 2). In [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF], the authors proved Thm. 1.6 in the special case where Z d is the zero set in RP n of n independent Kostlan-Shub-Smale polynomials (see Sect. 6.1 below). Their results include the positivity of the leading constant, hence implies Thm.1.8 in this case. To the best of our knowledge, Thm. 1.8 is completely new for r < n.

Note that when n = r = 1, our setting covers the case of the binomial polynomials on C with standard Gaussian coefficients. Much more is known in this case, including variance estimates for the number of real zeros of non-Gaussian ensembles of real polynomials (see [START_REF] Tao | Local universality of zeroes of random polynomials[END_REF]).

Our proof of Thm. 1.8 uses the Wiener-Itō expansion of the linear statistics associated with the field (s d (x)) x∈M . This kind of expansion has been studied by Slud [START_REF] Slud | Multiple Wiener-Itō integral expansions for level-crossing-count functionals[END_REF] and Kratz-León [START_REF] Kratz | Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes[END_REF][START_REF]Central limit theorems for level functionals of stationary Gaussian processes and fields[END_REF]. It was used in a random geometry context in [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF][START_REF] Dalmao | Phase singularities in complex arithmetic random waves[END_REF][START_REF] Marinucci | Non-universality of nodal length distribution for arithmetic random waves[END_REF]. In [START_REF] Dalmao | Phase singularities in complex arithmetic random waves[END_REF][START_REF] Marinucci | Non-universality of nodal length distribution for arithmetic random waves[END_REF], the authors used these Wiener chaos techniques to prove Central Limit Theorems for the volume of the zero set of Arithmetic Random Waves on the two-dimensional flat torus (see also [START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF] in an algebraic setting). In [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF], these methods where used to prove Thm. 1.8 when r = n.

In the related setting of Riemannian Random Waves, Canzani and Hanin [START_REF] Canzani | Local integral statistics for monochromatic random waves[END_REF] obtained recently an asymptotic upper bound for the variance of the linear statistics. To the best of our knowledge, in this Riemannian setting, the precise asymptotics of the variance of the volume of random submanifolds are known only when the ambient manifold is S 2 (cf. [START_REF] Wigman | Fluctuations of the nodal length of random spherical harmonics[END_REF]) or T 2 (cf. [START_REF] Dalmao | Phase singularities in complex arithmetic random waves[END_REF][START_REF] Krishnapur | Nodal length fluctuations for arithmetic random waves[END_REF]). We refer to the introduction of [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF] for more details about related works.

About the proofs. The proof of Thm. 1.6 broadly follows the lines of the proof of [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Thm. 1.6]. The random section s d defines a centered Gaussian field (s d (x)) x∈M , whose correlation kernel is E d , the Bergman kernel of E ⊗ L d (see Sect. 2.4). Thanks to results of Dai-Liu-Ma [START_REF] Dai | On the asymptotic expansion of the Bergman kernel[END_REF] and Ma-Marinescu [START_REF]Exponential estimate for the asymptotics of Bergman kernels[END_REF], we know that this kernel decreases exponentially fast outside of the diagonal ∆ = {(x, y) ∈ M 2 | x = y} and that it admits a universal local scaling limit close to ∆ (see Sect. 3 for details).

By an application of Kac-Rice formulas (cf. Thm. 5.1 and 5.5), we can express the covariance of |dV d | , φ 1 and |dV d | , φ 2 as a double integral over M × M of φ 1 (x)φ 2 (y) times a density function D d (x, y) that depends only on E d . Our main concern is to understand the asymptotics of the integral of D d (x, y), as d → +∞.

Thanks to the exponential decay of the Bergman kernel, we can show that the leading term in our asymptotics is given by the integral of D d over a neighborhood ∆ d of ∆, of typical size 1 √ d (see Prop. 5.22). Changing variables so that we integrate on a domain of typical size independent of d leads to the apparition of a factor d -n 2 . Besides, D d takes values of order d r on ∆ d (see Prop. 5.25). This explains why the asymptotic variance is of order d r-n 2 in Thm. 1.6. The behavior of E d allows to prove that D d admits a universal local scaling limit on ∆ d . The main difficulty in our proof is to show that the convergence to this scaling limit is uniform on ∆ d (see Prop. 5.25 for a precise statement). This difficulty comes from the fact that D d is singular along ∆, just like almost everything in this problem. This is where our proof differs from [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]. In [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF], the uniform convergence of D d to its scaling limit on ∆ d is not established, and one has to work around this lack of uniformity. This yields a complicated proof that fails when r = n. Here, we manage to prove this uniform convergence, thanks to some new key estimates (see Lem. 5.26, 5.28 and 5.29) that form the technical core of the paper. This allows us to both improve on the results of [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF] and simplify their proof.

As we explained, our proof relies on two properties of the Bergman kernel E d : namely, the existence of a scaling limit around any point at scale 1 √ d , and its exponential decay outside of the diagonal. These features are also exhibited by Bergman kernels in other settings such as those of [START_REF] Bayraktar | Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights[END_REF] or [START_REF] Berman | Bergman kernels and equilibrium measures for line bundles over projective manifolds[END_REF], so one might hope to generalize our results to these settings, at least in the bulk. Unfortunately, we also need a precise understanding of the scaling limit of E d , which is possible in our framework because it is universal (it only depends on n) and invariant under isometries (see Sect. [START_REF] Bayraktar | Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights[END_REF] for more details). As far as we know, it is much more complicated to study this scaling limit in other settings (such as those of [START_REF] Bayraktar | Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights[END_REF] and [START_REF] Berman | Bergman kernels and equilibrium measures for line bundles over projective manifolds[END_REF]), so we do not pursue this line of inquiry in the present paper and leave it for future research.

Let us now discuss the proof of Thm. 1.8. One would expect to prove this by computing a good lower bound for I n,r , directly from its expression (see Eq. (1.2)). To the best of our knowledge this approach fails, and we have to use subtler techniques.

Since the leading constant in (1.1) only depends on n and r, we can focus on the case of the volume of Z d (where φ 1 = 1 = φ 2 ) in a particular geometric setting. We consider the common real zero set of r independent Kostlan-Shub-Smale polynomials in RP n (see Sect. 6.1 for details). This allows for explicit computations since the Bergman kernel is explicitly known in this setting. Moreover, the distribution of these polynomials is invariant under the action of O n+1 (R) on RP n , which leads to useful independence proprieties that are not satisfied in general.

In this framework, we adapt the strategy of [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF] to the case r < n. First, we compute the Wiener-Itō expansion of the volume of Z d . That is, we expand Vol (Z d ) as q∈N Vol (Z d ) [q], where the convergence is in the space of L 2 random variables on our probability space, and Vol (Z d ) [q] denotes the q-th chaotic component of Vol (Z d ). In particular, Vol (Z d ) [0] is the expectation of Vol (Z d ) and the (Vol (Z d ) [q]) q∈N are pairwise orthogonal L 2 random variables. Hence,

Var(Vol (Z d )) = q 1
Var(Vol (Z d ) [q]) .

The chaotic components of odd order of Vol (Z d ) are zero, but we prove that Var(Vol (Z d ) [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF]) is equivalent to d r-n 2 C as d → +∞, where C > 0 (see Lem. 6.17). This implies Thm. 1.8.

Outline of the paper. In Sect. 2 we describe precisely our framework and the construction of the random measures |dV d |. We also introduce the Bergman kernel of E ⊗ L d and prove that it is the correlation kernel of (s d (x)) x∈M . In Sect. 3, we recall estimates for the Bergman kernel, and its scaling limit. Sect. 4 is dedicated to the study of the Bargmann-Fock process, that is the Gaussian centered random process on R n whose correlation function is:

(w, z) -→ exp - 1 2 w -z 2 .
This field is the local scaling limit of the random field (s d (x)) x∈M , in a sense to be made precise below. Sect. 5 and 6 are concerned with the proofs of Thm.

Random real algebraic submanifolds

In this section, we introduce the main objects we will be studying throughout this paper. We first recall some basic definitions in Sect. 2.1. In Sect. 2.2, we introduce our geometric framework. In Sect. 2.3, we describe our model of random real algebraic submanifolds. Finally, we relate these random submanifolds to Bergman kernels in Sect. 2.4.

Random vectors

Let us recall some facts about random vectors (see for example [21, appendix A]). In this paper, we only consider centered random vectors, so we give the following definitions in this setting.

Let X 1 and X 2 be centered random vectors taking values in Euclidean (or Hermitian) vector spaces V 1 and V 2 respectively, then we define their covariance operator as:

Cov(X 1 , X 2 ) : v -→ E[X 1 v , X 2 ] from V 2 to V 1 . For all v ∈ V 2 , we set v * = • , v ∈ V * 2 . Then Cov(X 1 , X 2 ) = E[X 1 ⊗ X * 2 ] is an element of V 1 ⊗ V * 2 .
Let X be a centered random vector in a Euclidean space V . The variance operator of X is defined as

Var(X) = Cov(X, X) = E[X ⊗ X * ] ∈ V ⊗ V * .
Let Λ be a nonnegative self-adjoint operator on (V, • , • ), we denote by X ∼ N (Λ) the fact that X is a centered Gaussian vector with variance operator Λ. This means that the characteristic function of X is ξ → exp -1 2 Λξ , ξ . Finally, we say that X ∈ V is a standard Gaussian vector if X ∼ N (Id), where Id is the identity operator on V .

If Λ is positive, the distribution of X ∼ N (Λ) admits the density:

x → 1 √ 2π N det(Λ) exp - 1 2 Λ -1 x , x (2.1)
with respect to the normalized Lebesgue measure of V , where N = dim(V ). Otherwise, X takes values in ker(Λ) ⊥ almost surely, and it admits a similar density as a variable in ker(Λ) ⊥ .

General setting

Let us introduce more precisely our geometric framework. Let X be a smooth complex projective manifold of positive complex dimension n. Let c X be a real structure on X , i.e. an anti-holomorphic involution. The real locus of (X , c X ) is the set M of fixed points of c X . In the following, we assume that M is non-empty. It is known that M is a smooth closed submanifold of X of real dimension n (see [32, chap. 1]).

Let E → X be a holomorphic vector bundle of rank r ∈ {1, . . . , n}, we denote by π E its bundle projection. Let c E be a real structure on E, compatible with c X in the sense that c X

• π E = π E • c E and c E is fiberwise C-anti-linear. Let h E be a Hermitian metric on E such that c ⋆ E (h E ) = h E .
A Hermitian metric satisfying this condition is said to be real. Similarly, let L → X be an ample holomorphic line bundle equipped with a compatible real structure c L and a real Hermitian metric h L .

We assume that (L, h L ) has positive curvature, that is its curvature form ω is Kähler. Recall that, if ζ 0 is a local non-vanishing holomorphic section of L, then ω = 1 2i ∂∂ ln (h L (ζ 0 , ζ 0 )) locally. This Kähler form defines a Riemannian metric g on X (see [START_REF] Ph | Principles of algebraic geometry[END_REF]Sect. 0.2] for example). In turn, g induces a Riemannian volume measure on X and on any smooth submanifold of X . We denote by dV X = ω n n! the Riemannian volume form on (X , g). Similarly, let |dV M | denote the Riemannian measure on (M, g).

Let d ∈ N, we endow E ⊗ L d with the real structure c d = c E ⊗ (c L ) d , which is compatible with c X , and the real Hermitian metric

h d = h E ⊗ h d L . Let Γ(E ⊗ L d
) denote the space of smooth sections of E ⊗ L d , we define a Hermitian inner product on Γ(E ⊗ L d ) by:

∀s 1 , s 2 ∈ Γ(E ⊗ L d ), s 1 , s 2 = X h d (s 1 (x), s 2 (x)) dV X . (2.2)
Remark 2.1. In this paper, • , • will either denote the inner product of a Euclidean (or Hermitian) space or the duality pairing between a Banach space and its topological dual. Which one should be clear from the context.

We say that a section s ∈ Γ(E ⊗ L d ) is real if it is equivariant for the real structures, that is:

c d • s = s • c X . We denote by RΓ(E ⊗ L d ) the real vector space of real smooth sections of E ⊗ L d . The restriction of • , • to RΓ(E ⊗ L d
) is a Euclidean inner product. Note that, despite their name, real sections are defined on the whole complex locus X and that the Euclidean inner product is defined by integrating over X , not just M .

Let x ∈ M , then the fiber (E ⊗ L d ) x is a dimension r complex vector space and the restriction of c d to this space is a C-anti-linear involution. We denote by R(E ⊗ L d ) x the set of fixed points of this involution, which is a real r-dimensional vector space. Then, R(E ⊗ L d ) → M is a rank r real vector bundle and, for any

s ∈ RΓ(E ⊗ L d ), the restriction of s to M is a smooth section of R(E ⊗ L d ) → M .
Let H 0 (X , E ⊗ L d ) denote the space of global holomorphic sections of E ⊗ L d . This space is known to be finite-dimensional (compare [START_REF] Ma | Holomorphic Morse inequalities and Bergman kernels[END_REF]Thm. 1.4.1]). Let N d denote the complex dimension of H 0 (X , E ⊗ L d ). We denote by:

RH 0 (X , E ⊗ L d ) = s ∈ H 0 (X , E ⊗ L d ) c d • s = s • c X the space of global real holomorphic sections of E ⊗ L d . The restriction of the inner product (2.2) to RH 0 (X , E ⊗ L d ) (resp. H 0 (X , E ⊗ L d ))
makes it into a Euclidean (resp. Hermitian) space of real (resp. complex) dimension N d .

Random real algebraic submanifolds

This section is concerned with the definition of the random submanifolds we consider and some related random variables.

Let d ∈ N and s ∈ RH 0 (X , E ⊗ L d ), we denote by Z s the real zero set s -1 (0) ∩ M of s. If the restriction of s to M vanishes transversally, then Z s is a smooth closed submanifold of codimension r of M (note that this includes the case where Z s is empty). In this case, we denote by |dV s | the Riemannian volume measure on Z s induced by g. In the following, we consider |dV s | as the continuous linear form on (C 0 (M ), • ∞ ) defined by:

∀φ ∈ C 0 (M ), |dV s | , φ = x∈Zs φ(x) |dV s | .
Definition 2.2 (compare [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF]). We say that

RH 0 (X , E ⊗ L d ) is 0-ample if, for any x ∈ M , the evaluation map ev d x : s → s(x) from RH 0 (X , E ⊗ L d ) to R E ⊗ L d x is surjective. Lemma 2.3. There exists d 1 ∈ N, depending only on X , E and L, such that for all d d 1 , RH 0 (X , E ⊗ L d ) is 0-ample.
Proof. This can be deduced from the Riemann-Roch formula, for example. It is also a by-product of the computations of the present paper and will be proved later on, see Cor. 5.11 below.

Let us now consider a random section in RH 0 (X , E ⊗L d ). Recall that RH 0 (X , E ⊗L d ), endowed with the inner product (2.2), is a Euclidean inner product of dimension N d . Let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Proof. Since d d 1 , RH 0 (X , E ⊗ L d ) is 0-ample. By a transversality argument (see [START_REF] Letendre | Expected volume and Euler characteristic of random submanifolds[END_REF]Sect. 2.6] for details), this implies that the restriction of s to M vanishes transversally for almost every s ∈ RH 0 (X , E ⊗ L d ) (with respect to the Lebesgue measure). Thus, almost surely, s d restricted to M vanishes transversally, and its zero set is a smooth closed submanifold of codimension r.

From now on, we only consider the case d d

The correlation kernel

For any d ∈ N, the random section s d ∈ RH 0 (X , E ⊗ L d ) defines a centered Gaussian process (s d (x)) x∈X . In this section, we recall the relation between the distribution of this process and the Bergman kernel of E ⊗ L d .

Recall that

(E ⊗ L d ) ⊠ (E ⊗ L d ) * stands for the bundle P ⋆ 1 E ⊗ L d ⊗ P ⋆ 2 E ⊗ L d *
over X × X , where P 1 (resp. P 2 ) denotes the projection from X × X onto the first (resp. second) X factor. The distribution of (s d (x)) x∈X is characterized by its covariance kernel, that is the section of

(E ⊗ L d ) ⊠ (E ⊗ L d ) * defined by: (x, y) → Cov(s d (x), s d (y)) = E[s d (x) ⊗ s d (y) * ]. Definition 2.5. Let E d denote the Bergman kernel of E ⊗ L d → X , that is the Schwartz kernel of the orthogonal projection from Γ(E ⊗ L d ) onto H 0 (X , E ⊗ L d ).
Let (s 1,d , . . . , s N d ,d ) be an orthonormal basis of RH 0 (X , E ⊗ L d ), then it is also an orthonormal basis of H 0 (X , E ⊗ L d ). Recall that, E d is given by:

E d : (x, y) -→ N d i=1 s i,d (x) ⊗ s i,d (y) * .
This shows that E d is a real global holomorphic section of (E ⊗ L d ) ⊠ (E ⊗ L d ) * . The following proves that the distribution of (s d (x)) x∈X is totally described by E d . Proposition 2.6 (compare [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF], Prop. 2.6). Let d ∈ N and let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Then, for all x and y ∈ X , we have:

Cov(s d (x), s d (y)) = E d (x, y).
∀w ∈ T y X , ∂ ♯ y E d (x, y) • w * = ∂ y E d (x, y) • w. Similarly, let ∂ x ∂ ♯ y E d (x, y) ∈ T * x X ⊗ T y X ⊗ E ⊗ L d x ⊗ E ⊗ L d *
y be defined by:

∀(v, w) ∈ T x X × T y X , ∂ x ∂ ♯ y E d (x, y) • (v, w * ) = ∂ x ∂ y E d (x, y) • (v, w).
The following corollary was proved in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Cor. 2.13].

Corollary 2.7. Let d ∈ N, let ∇ d be a metric connection on E ⊗ L d and let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Then, for all x and y ∈ X , we have:

Cov ∇ d x s, s(y) = E ∇ d x s ⊗ s(y) * = ∂ x E d (x, y), Cov s(x), ∇ d y s = E s(x) ⊗ ∇ d y s * = ∂ ♯ y E d (x, y), Cov ∇ d x s, ∇ d y s = E ∇ d x s ⊗ ∇ d y s * = ∂ x ∂ ♯ y E d (x, y).

Estimates for the Bergman kernel

In this section, we recall useful estimates for the Bergman kernels. In Sect. 3.1 we recall the definition of a preferred trivialization of E ⊗ L d → X . Then we state near-diagonal and off-diagonal estimates for a scaled version of E d in Sect. 3.2 and 3.3. Let d ∈ N, let us define a preferred trivialization of E ⊗ L d in a neighborhood of any point of M . The properties of this trivialization were studied in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Sect. 3.1]. Recall that the metric g on X is induced by the curvature of (L, h L ). Since, h L is compatible with the real structures, c X is an isometry of (X , g) (see [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Sect. 2.1] for details).

Real normal trivialization

Let R > 0 be such that 2R is less than the injectivity radius of X . Let x ∈ M , then the exponential map exp x : B TxX (0, 2R) → B X (x, 2R) defines a chart around x such that:

d x c X = (exp x ) -1 • c X • exp x .
In particular, since T x M = ker(d x c X -Id), we have that exp x induces a diffeomorphism from B TxM (0, 2R) to B M (x, 2R) that coincides with the exponential map of (M, g) at x.

We can now trivialize E ⊗ L d over B X (x, 2R), by identifying each fiber with E ⊗ L d x by parallel transport along geodesics, with respect to the Chern connection of E ⊗ L d . This defines a bundle map

ϕ x : B TxX (0, 2R) × E ⊗ L d x → E ⊗ L d /BX (x,2R)
that covers exp x . We call this trivialization the real normal trivialization of E ⊗ L d around x.

Definition 3.2. A connection ∇ d on E ⊗ L d → X is said to be real if for every smooth section s we have:

∀y ∈ X , ∇ d y (c d • s • c X ) = c d • ∇ d cX (y) s • d y c X . Such a connection induces a connection on R E ⊗ L d → M by restriction. Recall that c d denotes the real structure of E ⊗ L d . Let c d,x denote its restriction to E ⊗ L d x , then (d x c X , c d,x ) is a C-anti-linear involution of B TxX (0, 2R) × E ⊗ L d
x which is compatible with the real structure on the first factor. Since the Chern connection of E ⊗ L d is real (see [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Lem. 3.4]), the real normal trivialization is well-behaved with respect to the real structures, in the sense that for all z ∈ B TxX (0, 2R) and

ζ ∈ E ⊗ L d x , c d (ϕ x (z, ζ)) = ϕ x (d x c X • z, c d,x (ζ)) .
Thus, ϕ x can be restricted to a bundle map

B TxM (0, 2R) × R E ⊗ L d x → R E ⊗ L d /BM (x,2R)
that covers exp x . Finally, it is known (cf. [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Sect. 3.1]) that ϕ x is a unitary trivialization, i.e. its restriction to each fiber is an isometry. Similarly, its restriction to the real locus is an orthogonal trivialization of R E ⊗ L d /BM (x,2R) . The point is the following. The usual differentiation for maps from T x X to E ⊗ L d

x defines locally a connection ∇ d on E ⊗ L d /BX (x,2R) via the real normal trivialization. Since this trivialization is well-behaved with respect to both the real and the metric structures, ∇ d is a real metric connection. Then, by a partition of unity argument, there exists a global real metric connection ∇ d on E ⊗ L d that agrees with ∇ d on B X (x, R), i.e. ∇ d is trivial in the real normal trivialization, in a neighborhood of x. The existence of such a connection will be useful in the proof of our main theorem.

Near-diagonal estimates

In this section, we state estimates for a scaled version of the Bergman kernel in a neighborhood of the diagonal of M × M . As in the previous section, let R > 0 be such that 2R is less than the injectivity radius of X . Let x ∈ M , then the real normal trivialization ϕ x induces a trivialization of

E ⊗ L d ⊠ E ⊗ L d * over B X (x, 2R) × B X (x, 2R) that covers exp x × exp x . This trivialization agrees with the real normal trivialization of E ⊗ L d ⊠ E ⊗ L d * around (x, x).
In the normal chart exp x , the Riemannian measure dV X admits a positive density with respect to the Lebesgue measure of (T x X , g x ), denoted by κ : B TxX (0, 2R) → R + . Then, in the chart exp x : B TxM (0, 2R) → B M (x, 2R), the density of |dV M | with respect to the Lebesgue measure of (T x M, g x ) is √ κ. Let us identify E d with its expression in the real normal trivialization of E ⊗ L d ⊠ E ⊗ L d * around (x, x). Thus, the restriction of E d to the real locus is a map from

T x M × T x M to End R E ⊗ L d x .
Then, by [START_REF] Dai | On the asymptotic expansion of the Bergman kernel[END_REF]Thm. 4.18'] (see also [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Thm. 3.5] for a statement with the same notations as the present paper) we get the following estimate for E d and its derivatives of order at most 6.

Theorem 3.3 (Dai-Liu-Ma

). There exist C 1 and C 2 > 0, such that ∀k ∈ {0, 1, . . . , 6}, ∀d ∈ N * , ∀x ∈ M , ∀w, z ∈ B TxM (0, R),

D k (w,z)   E d (w, z) - d π n exp -d 2 z -w 2 κ(w) κ(z) Id R(E⊗L d )x   C 1 d n+ k 2 -1 1 + √ d( w + z ) 2n+12 exp -C 2 √ d z -w + O d -∞ ,
where D k is the k-th differential for a map from

T x M × T x M to End R E ⊗ L d
x , the norm on T x M is induced by g x and the norm on

(T * x M ) ⊗k ⊗ End E ⊗ L d
x is induced by g x and (h d ) x . The notation O(d -∞ ) means that, for any l ∈ N, this term is O d -l with a constant that does not depend on x, w, z nor d.

Recall that x is fixed. We denote by the e d the following scaled version of the Bergman kernel around x:

∀w, z ∈ B TxM 0, 2R √ d , e d (w, z) = π d n E d exp x w √ d , exp x z √ d . (3.1)
We consider e d as a map with values in End R E ⊗ L d x using the real normal trivialization around x. Note that e d highly depends on x, even if this is not reflected in the notation. In the following, the base point x will always be clear from the context.

Let ξ : R n × R n → R be defined by ξ(w, z) = exp -1 2 wz 2 , where • is the usual Euclidean norm. Let x ∈ M , any isometry from T x M to R n allows us to see ξ as a map from T x M × T x M to R. Let b n be a positive constant depending only on n and whose value will be chosen later on. Then, using the same notations as in Thm. 3.3 we get the following.

Proposition 3.4. Let α ∈ (0, 1), then there exists C > 0, depending only on α, n and the geometry of X , such that ∀k ∈ {0, 1, . . . , 6}, ∀d ∈ N * , ∀x ∈ M , ∀w, z ∈ B TxM (0, b n ln d), we have:

D k (w,z) e d -D k (w,z) ξ Id R(E⊗L d )x Cd -α .
Proof. First, we apply Thm. 3.3 for the scaled kernel e d . This yields that ∀k ∈ {0, 1, . . . , 6},

∀d ∈ N * , ∀x ∈ M , ∀w, z ∈ B TxM (0, b n ln d): D k (w,z) e d (w, z) - ξ(w, z) κ(w) κ(z) Id R(E⊗L d )x C 1 d (1 + 2b n ln d) 2n+12 + O d -∞ = O d -α , where κ : z → κ z √ d .
Since we used normal coordinates to define κ, the following relations hold uniformly on B TxX (0, R):

κ(z) = 1 + O z 2 , D z κ = O( z ) and ∀k ∈ {2, . . . , 6}, D k z κ = O(1).
By compactness, these estimates can be made independent of x ∈ M . Then, we get the following estimates for κ and its derivatives, uniformly in x ∈ M and z ∈ B TxM (0, b n ln d):

κ(z) = 1 + O (b n ln d) 2 d , D z κ = O b n ln d d and ∀k ∈ {2, . . . , 6}, D k z κ = O 1 d . Therefore, ∀k ∈ {0, 1, . . . , 6}, ∀d ∈ N * , ∀x ∈ M , ∀w, z ∈ B TxM (0, b n ln d): D k (w,z) ξ(w, z) κ(w) κ(z) -D k (w,z) ξ = O d -α .
We will use the expressions of some of the partial derivatives of ξ. Let us choose any orthonormal basis of T x M and denote by ∂ xi (resp. ∂ yi ) the partial derivative with respect to the i-th component of the the first (resp. second) variable. Lemma 3.5. Let i, j ∈ {1, . . . , n}, for all w = (w 1 , . . . , w n ) and z = (z 1 , . . . , z n ) ∈ T x M we have:

∂ xi ξ(w, z) = -(w i -z i ) exp - 1 2 w -z 2 , ∂ yj ξ(x, y) = (w j -z j ) exp - 1 2 w -z 2 ,
and

∂ xi ∂ yj ξ(x, y) = (δ ij -(w i -z i )(w j -z j )) exp - 1 2 w -z 2 ,
where δ ij equals 1 if i = j and 0 otherwise.

Proof. This is given by a direct computation.

Off-diagonal estimates

Finally, let us recall estimates quantifying the long range decay of the Bergman kernel E d . These estimates were proved by Ma and Marinescu in [START_REF]Exponential estimate for the asymptotics of Bergman kernels[END_REF]Thm. 5].

Let S be a smooth section of R E ⊗ L d ⊠ R E ⊗ L d * and let x, y ∈ M . We denote by S(x, y) C k the maximum of the norms of S and its derivatives of order at most k at (x, y) ∈ M ×M , where the derivatives of S are computed with respect to the connection induced by the Chern connection of E ⊗ L d and the Levi-Civita connection on M . The norms are the natural ones induced by h d and g.

Theorem 3.6 (Ma-Marinescu).

There exist d 0 ∈ N * , and positive constants C 1 and C 2 such that, for all k ∈ {0, 1, 2}, ∀d d 0 , ∀x, y ∈ M , we have:

E d (x, y) C k C 1 d n+ k 2 exp -C 2 √ d ρ g (x, y) ,
where ρ g (•, •) denotes the geodesic distance in (M, g).

Properties of the limit distribution

The estimates of Sect. 3.2 show that the family of random fields (s d (x)) x∈M has a local scaling limit around any point x ∈ M , as d goes to infinity. Moreover, the limit field does not depend on x. The limit is a Gaussian centered random process from R n to R r whose correlation kernel is e ∞ : (w, z) → ξ(w, z)I r , where I r stands for the identity of R r and ξ was defined in Sect. 3.2. This limit process is known as the Bargmann-Fock process. The goal of this section is to establish some properties of the Bargmann-Fock process. These results will be useful in the next section to prove that, for d large enough, the local behavior of s d around any given x ∈ M is the same as that of the limit process.

In the following, we denote by (s(z)) z∈R n a copy of the Bargmann-Fock process. Since ξ is smooth, we can assume the trajectories of s to be smooth. Note that s is both stationary and isotropic. Moreover, since e ∞ = ξI r , the field s is just a tuple of r independent identically distributed centered Gaussian fields whose correlation kernel is ξ.

Variance of the values

The first thing we want to understand about s is the distribution of (s(0), s(z)) ∈ R r ⊕ R r for any z ∈ R n . In the following, we canonically identify R r ⊕ R r with R 2 ⊗ R r .

Let z ∈ R n , then (s(0), s(z)) is a centered Gaussian vector in R 2 ⊗ R r whith variance operator:

Θ(z) = e ∞ (0, 0) e ∞ (0, z) e ∞ (z, 0) e ∞ (z, z) = ξ(0, 0)I r ξ(0, z)I r ξ(z, 0)I r ξ(z, z)I r = 1 e -1 2 z 2 e -1 2 z 2 1 ⊗ I r . (4.1) Let Q = 1 √ 2 1 -1 1 1 ∈ O 2 (R) denote the rotation of angle π 4 in R 2 .
We can explicitly diagonalize Θ(z) as follows.

(Q ⊗ I r ) Θ(z) (Q ⊗ I r ) -1 = 1 -e -1 2 z 2 0 0 1 + e -1 2 z 2 ⊗ I r . Proof. Since (Q ⊗ I r ) -1 = Q t ⊗ I r , by Eq. (4.1)
, it is enough to notice that:

Q 1 e -1 2 z 2 e -1 2 z 2 1 Q t = 1 -e -1 2 z 2 0 0 1 + e -1 2 z 2 . Lemma 4.2. For all z ∈ R n , det (Θ(z)) = 1 -e -z 2 r
. In particular, the distribution of

(s(0), s(z)) is non-degenerate for all z ∈ R n \ {0}.
Proof. We take the determinant of both sides in Eq. (4.1).

Variance of the 1-jets

Let us now study the variance structure of the 1-jets of s. For any z ∈ R n , we know that (s(0), s(z), d 0 s, d z s) is a centered Gaussian vector in:

R r ⊕ R r ⊕ ((R n ) * ⊗ R r ) ⊕ ((R n ) * ⊗ R r ) ≃ (R ⊕ R ⊕ (R n ) * ⊕ (R n ) * ) ⊗ R r .
Our goal in this section is to better understand its variance operator Ω(z). In the following, we write Ω(z) by blocks according to the previous splitting. Let ∂ x (resp. ∂ y ) denote the partial derivative with respect to the first (resp. second) variable for maps from R n × R n to End(R r ). Let us also define ∂ ♯ y as in Sect. 2.4. Then, we have:

Ω(z) =      e ∞ (0, 0) e ∞ (0, z) ∂ ♯ y e ∞ (0, 0) ∂ ♯ y e ∞ (0, z) e ∞ (z, 0) e ∞ (z, z) ∂ ♯ y e ∞ (z, 0) ∂ ♯ y e ∞ (z, z) ∂ x e ∞ (0, 0) ∂ x e ∞ (0, z) ∂ x ∂ ♯ y e ∞ (0, 0) ∂ x ∂ ♯ y e ∞ (0, z) ∂ x e ∞ (z, 0) ∂ x e ∞ (z, z) ∂ x ∂ ♯ y e ∞ (z, 0) ∂ x ∂ ♯ y e ∞ (z, z)      = Ω ′ (z) ⊗ I r , (4.2)
where

Ω ′ (z) =      ξ(0, 0) ξ(0, z) ∂ ♯ y ξ(0, 0) ∂ ♯ y ξ(0, z) ξ(z, 0) ξ(z, z) ∂ ♯ y ξ(z, 0) ∂ ♯ y ξ(z, z) ∂ x ξ(0, 0) ∂ x ξ(0, z) ∂ x ∂ ♯ y ξ(0, 0) ∂ x ∂ ♯ y ξ(0, z) ∂ x ξ(z, 0) ∂ x ξ(z, z) ∂ x ∂ ♯ y ξ(z, 0) ∂ x ∂ ♯ y ξ(z, z)      .
Let ∂ ∂x1 , . . . , ∂ ∂xn be any orthonormal basis of R n such that z = z ∂ ∂x1 and let (dx 1 , . . . , dx n ) denote its dual basis. Let (e 1 , e 2 ) denote the canonical basis of R 2 , we denote by B z the following orthonormal basis of

R 2 ⊗ (R ⊕ (R n ) * ) ≃ R ⊕ R ⊕ (R n ) * ⊕ (R n ) * : B z = (e 1 ⊗ 1, e 2 ⊗ 1, e 1 ⊗ dx 1 , e 2 ⊗ dx 1 , . . . , e 1 ⊗ dx n , e 2 ⊗ dx n ). Lemma 4.3. For any z ∈ R n , the matrix of Ω ′ (z) in the basis B z is:   Ω( z 2 ) 0 0 1 e -1 2 z 2 e -1 2 z 2 1 ⊗ I n-1   ,
where I n-1 is the identity matrix of size n -1 and, for all t 0, we set:

Ω(t) =     1 e -1 2 t 0 - √ te -1 2 t e -1 2 t 1 √ te -1 2 t 0 0 √ te -1 2 t 1 (1 -t)e -1 2 t - √ te -1 2 t 0 (1 -t)e -1 2 t 1     . ( 4

.3)

Proof. A direct computation yields the result, using the fact that z = ( z , 0, . . . , 0) in the basis ∂ ∂x1 , . . . , ∂ ∂xn . Recall that the partial derivatives of ξ are given by Lem. 3.5.

Let z ∈ R n , and recall that z * ∈ (R n ) * was defined as z * = • , z , where

• , • is the canonical scalar product of R n . We denote by z * ⊗ z ∈ End((R n ) * ) the map η → η(z)z * . Then Lem. 4.3 shows that, as an operator on R ⊕ R ⊕ (R n ) * ⊕ (R n ) * : Ω ′ (z) =      1 e -1 2 z 2 0 -e -1 2 z 2 z e -1 2 z 2 1 e -1 2 z 2 z 0 0 e -1 2 z 2 z * I n e -1 2 z 2 (I n -z * ⊗ z) -e -1 2 z 2 z * 0 e -1 2 z 2 (I n -z * ⊗ z) I n      , (4.4)
where z * is to be understood as the constant map t → z * from R to (R n ) * , z is to be understood as the evaluation at the point z from (R n ) * to R and I n is the identity of R n . Indeed both sides of (4.4) have the same matrix in the basis B z .

We will now diagonalize explicitly Ω(z), as we did for Θ(z) in the previous section. The main step is to diagonalize Ω(z). Definitions 4.4. We denote by v 1 , v 2 , v 3 , v 4 and a the following functions from [0, +∞) to R:

v 1 : t -→1 -e -1 2 t   t 2 -1 + t 2 2   , v 2 : t -→1 -e -1 2 t   t 2 + 1 + t 2 2   , v 3 : t -→1 + e -1 2 t   t 2 -1 + t 2 2   , v 4 : t -→1 + e -1 2 t   t 2 + 1 + t 2 2   , a : t -→ 1 -t 2 1 + t 2 2 .
Note that, for all t 0, |a(t)| 1, so that the following makes sense.

Definitions 4.5. Let b + : t → 1 + a(t) and b -: t → 1a(t) from [0, +∞) to R. For all t 0, let us also denote:

P (t) = 1 2     b -(t) -b -(t) -b + (t) -b + (t) b + (t) -b + (t) b -(t) b -(t) b -(t) b -(t) -b + (t) b + (t) b + (t) b + (t) b -(t) -b -(t)     .
One can check that, for all t 0, P (t) is an orthogonal matrix.

Lemma 4.6. For every t ∈ [0, +∞), we have:

P (t) Ω(t)P (t) -1 =     v 1 (t) 0 0 0 0 v 2 (t) 0 0 0 0 v 3 (t) 0 0 0 0 v 4 (t)     .
Proof. See Appendix A.

Corollary 4.7. Let z ∈ R n , identifying Ω ′ (z) with its matrix in B z , we have:

P ( z 2 ) 0 0 Q ⊗ I n-1 Ω ′ (z) P ( z 2 ) 0 0 Q ⊗ I n-1 -1 =       v1( z 2 ) 0 0 0 0 v2( z 2 ) 0 0 0 0 v3( z 2 ) 0 0 0 0 v4( z 2 ) 0 0 1-e -1 2 z 2 0 0 1+e -1 2 z 2 ⊗ I n-1       .
By Eq. ( 4.2), we get a diagonalization of Ω(z) by tensoring each factor by I r in Corollary 4.7.

Lemma 4.8. For all z ∈ R n \ {0}, we have det (Ω(z)) > 0, that is the distribution of the random vector (s(0), s(z), d 0 s, d z s) is non-degenerate.

Proof. See Appendix A.

Conditional variance of the derivatives

The next step is to study the conditonal distribution of (d 0 s, d z s) given that s(0) = 0 = s(z), for any z ∈ R n \ {0}. Recall that (s(0), s(z), d 0 s, d z s) is a centered Gaussian vector with variance Ω(z) (see Eq. (4.2)). Moreover, if z = 0, the distribution of (s(0), s(z)) is non-degenerate by Lem. 4.2. Thus,

(d 0 s, d z s) given that s(0) = 0 = s(z) is a centered Gaussian vector in ((R n ) * ⊕ (R n ) * )⊗R r with variance operator: Λ(z) = ∂x∂ ♯ y e∞(0,0) ∂x∂ ♯ y e∞(0,z) ∂x∂ ♯ y e∞(z,0) ∂x∂ ♯ y e∞(z,z)
-∂xe∞(0,0) ∂xe∞(0,z) ∂xe∞(z,0) ∂xe∞(z,z) e∞(0,0) e∞(0,z) e∞(z,0) e∞(z,z)

-1 ∂ ♯ y e∞(0,0) ∂ ♯ y e∞(0,z) ∂ ♯ y e∞(z,0) ∂ ♯ y e∞(z,z)
.

By Equations (4.2) and (4.4), for all z ∈ R n \ {0}, we have Λ(z) = Λ ′ (z) ⊗ I r , where:

Λ ′ (z) =   I n -e -z 2 1-e -z 2 z * ⊗ z e -1 2 z 2 I n - 1 1-e -z 2 z * ⊗ z e -1 2 z 2 I n - 1 1-e -z 2 z * ⊗ z I n -e -z 2 1-e -z 2 z * ⊗ z   . (4.5)
As in the previous section, let us denote by ∂ ∂x1 , . . . , ∂ ∂xn an orthonormal basis of R n such that z = z ∂ ∂x1 and let (dx 1 , . . . , dx n ) denote its dual basis. Let (e 1 , e 2 ) denote the canonical basis of R 2 , we define B ′ z to be the following orthonormal basis of

R 2 ⊗ (R n ) * ≃ (R n ) * ⊕ (R n ) * : B ′ z = (e 1 ⊗ dx 1 , e 2 ⊗ dx 1 , . . . , e 1 ⊗ dx n , e 2 ⊗ dx n ).
Lemma 4.9. For any z ∈ R n \ {0}, the matrix of Λ ′ (z) in the basis

B ′ z is:   Λ( z 2 ) 0 0 1 e -1 2 z 2 e -1 2 z 2 1 ⊗ I n-1   ,
where, for all t > 0, we set:

Λ(t) =   1 -te -t 1-e -t e -1 2 t 1 -t 1-e -t e -1 2 t 1 -t 1-e -t 1 -te -t 1-e -t   . (4.6) Proof. Since z = z ∂ ∂x1 , we have z * ⊗ z = z 2 dx 1 ⊗ ∂ ∂x1 . Hence, the matrix of z * ⊗ z in (dx 1 , . . . , dx n ) is: z 2 0 0 0 .
Then the conclusion follows from Eq. (4.5).

Remark 4.10. We can extend continuously Λ at t = 0 by setting Λ(0) = 0. This yields continuous extensions of Λ ′ and Λ at z = 0. Note that Λ(0) is not the variance operator of (d 0 s, d 0 s) given that s(0) = 0.

Definitions 4.11. Let u 1 and u 2 denote the following functions from R to R:

u 1 : t -→ 1 -e -t + te -1 2 t 1 + e -1 2 t u 2 : t -→      1 -e -t -te -1 2 t 1 -e -1 2 t if t = 0, 0 if t = 0.
Once again, we will need an explicit diagonalization of Λ(z). Such a diagonalization is given by the following lemma, once we tensor each factor by I r . Lemma 4.12. Let z ∈ R n , identifying Λ ′ (z) with its matrix in B ′ z , we have:

(Q ⊗ I n ) Λ ′ (z) (Q ⊗ I n ) -1 =    u1( z 2 ) 0 0 u2( z 2 ) 0 0 1-e -1 2 z 2 0 0 1+e -1 2 z 2 ⊗ I n-1    .
Proof. By Lem. 4.9, we only need to check that, for all t 0,

Q Λ(t)Q t = u 1 (t) 0 0 u 2 (t) .
Lemma 4.13. For all z ∈ R n \ {0}, we have det (Λ(z)) > 0, i.e. the distribution of (d 0 s, d z s) given that s(0

) = 0 = s(z) is non-degenerate. Proof. See Appendix A. By Lem. 4.8, Ω(z) is a positive self-adjoint operator on (R ⊕ R ⊕ (R n ) * ⊕ (R n ) * ) ⊗ R r , for all z ∈ R n \ {0}
, and so is its inverse. Hence, Ω(z) -1 admits a unique positive square root, that we denote by Ω(z) -1 2 . Similarly, by Lem. 4.13, Λ(z) is a positive self-adjoint operator on Lemma 4.14. The map z -→ 0 Λ(z)

((R n ) * ⊕ (R n ) * ) ⊗ R
1 2 Ω(z) -1 2 is bounded on R n \ {0}. Proof. See Appendix A.

Finiteness of the leading constant

The goal of this section, it to prove that the constant I n,r defined by Eq. (1.2) and appearing in Thm. 1.6 is well-defined and finite. Definition 4.15. Let n ∈ N * and r ∈ {1, . . . , n}, for every t > 0 we set:

D n,r (t) = E det ⊥ (X(t)) det ⊥ (Y (t)) (1 -e -t ) r 2 -(2π) r Vol (S n-r ) Vol (S n ) 2 ,
where (X(t), Y (t)) is the centered Gaussian vector in M rn (R) × M rn (R) defined in Def. 1.5.

By the definition of I n,r (see Eq. (1.2)), we have:

I n,r = 1 2 +∞ 0 D n,r (t)t n-2 2
dt.

Hence, we have to prove that t → D n,r (t)t n-2 2

is integrable on (0, +∞), which boils down to computing the asymptotic expansions of E det ⊥ (X(t)) det ⊥ (Y (t)) as t → 0 and as t → +∞.

Let us now relate this to the Bargmann-Fock process (s(z)) z∈R n .

(0) = 0 = s(z), form a random vector in M rn (R) × M rn (R) which is distributed as X( z 2 ), Y ( z 2 ) .
Proof. Let us denote by X(z) and Y (z) the matrices of d 0 s and d z s in the bases ∂ ∂x1 , . . . , ∂ ∂xn and (ζ 1 , . . . , ζ r ), given that s(0) = 0 = s(z). We denote by X ij (z) (resp. Y ij (z)) the coefficients of X (resp. Y ) for i ∈ {1, . . . , r} and j ∈ {1, . . . , n}. By Lem. 4.9, the couples X ij , Y ij are centered Gaussian vectors in R 2 which are independant from one another. Moreover, the variance matrix of

X ij (z), Y ij (z) equals Λ( z 2 ) if j = 1 and 1 e -1 2 z 2 e -1 2 z 2 1 otherwise. By Def. 1.5, this is exactly saying that X(z), Y (z) is distributed as X( z 2 ), Y ( z 2 ) .
Lemma 4.17. Let n ∈ N * and r ∈ {1, . . . , n}. Then, as t → 0, we have the following:

E det ⊥ (X(t)) det ⊥ (Y (t)) ∼      (n -1)! (n -r -1)! if r < n, n! 2 t if r = n.
Proof. See Appendix A.

Lemma 4.18. For all n ∈ N * and r ∈ {1, . . . , n}, we have the following as t → +∞:

E det ⊥ (X(t)) det ⊥ (Y (t)) = (2π) r Vol (S n-r ) Vol (S n ) 2 + O te -t 2 Proof. See Appendix A.
Lem. 4.17

t n-2 2 D n,r (t) =      O 1 √ t as t → 0, O e -t 4
as t → +∞.

In particular,

I n,r = 1 2 +∞ 0 D n,r (t)t n-2 2
dt is well-defined and finite.

5 Proof of Theorem 1.6

This section is concerned with the proof of our main result (Thm. 1.6). Recall that X is a compact Kähler manifold of complex dimension n 1 defined over the reals and that M denotes its real locus, assumed to be non-empty. Let E → X be a rank r ∈ {1, . . . , n} real Hermitian vector bundle and L → X be a real Hermitian line bundle whose curvature form is ω, the Kähler form of X . We assume that E and L are endowed with compatible real structures. For all d ∈ N, we still denote by E d the Bergman kernel of E ⊗ L d . Finally, let s d denote a standard Gaussian vector in RH 0 (X , E ⊗ L d ), whose real zero set is denoted by Z d , and let |dV d | denote the Riemannian volume measure on Z d .

In Sect. 5.1 we recall Kac-Rice formulas and use them to derive an integral expression of Var(|dV d |). Sect. 5.2 is concerned with the study of some relevant random variables related to (s d (x)) x∈M . Finally, we conclude the proof in two steps, in Sect. 5.3 and 5.4.

Kac-Rice formulas

In this section, we use Kac-Rice formulas to derive an integral expression of Var(|dV d |). Classical references for this material are [1, chap. 11.5] and [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]Thm. 6.3]. Since our probability space is the finite-dimensional vector space RH 0 (X , E ⊗ L d ), it is possible to derive Kac-Rice formulas under weaker hypothesis than those given in [START_REF] Adler | Random fields and geometry[END_REF] and [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]. This uses Federer's coaera formula and the so-called double fibration trick, see [21, App. C] and the references therein. The first Kac-Rice formula we state (Thm. 5.1 below) was proved in [21, Thm. 5.3] and the second (Thm. 5.5 below) was proved in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Thm. 4.4].

Recall that the Jacobian det ⊥ (L) of an operator L was defined in Def. 1.4, that d 1 was defined in Lem. 2.3 and that a connection is said to be real if it satisfies the condition given in Def. 3.2.

s d ∼ N (Id) in RH 0 (X , E ⊗ L d ).
Then for all φ ∈ C 0 (M ) we have:

E x∈Z d φ(x) |dV d | = (2π) -r 2 x∈M φ(x) |det ⊥ (ev d x )| E det ⊥ ∇ d x s d s d (x) = 0 |dV M | . (5.1)
The expectation on the right-hand side of Eq. (5.1) is to be understood as the conditional expectation of

det ⊥ ∇ d x s d given that s d (x) = 0. Notation 5.2. Let ∆ = {(x, y) ∈ M 2 | x = y} denote the diagonal of M 2 .
Definition 5.3. Let d ∈ N and let (x, y) ∈ M 2 \ ∆ we denote by ev d x,y the evaluation map:

ev d x,y : RH 0 (X , E ⊗ L d ) -→ R E ⊗ L d x ⊕ R E ⊗ L d y . s -→ (s(x), s(y)) .
Lemma 5.4. There exists d 2 ∈ N, such that for all (x, y) ∈ M 2 \ ∆, ev d x,y is surjective. This was proved in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Prop. 4.2] in the case r < n, using Kodaira's embedding theorem. The proof can be adapted verbatim to the case r n. We will give an alternative proof using only estimates on the Bergman kernel (Lem. 5.23 and 5.26), see p. 24 below.

s d ∼ N (Id) in RH 0 (X , E ⊗ L d ).
Then for all φ 1 and φ 2 ∈ C 0 (M ) we have:

E (x,y)∈(Z d ) 2 \∆ φ 1 (x)φ 2 (y) |dV d | 2 = 1 (2π) r (x,y)∈M 2 \∆ φ 1 (x)φ 2 (y) det ⊥ ev d x,y E det ⊥ ∇ d x s d det ⊥ ∇ d y s d ev d x,y (s d ) = 0 |dV M | 2 .
(5.2)

Here, |dV M | 2 (resp. |dV d | 2 )
D d (x, y) =   E det ⊥ ∇ d x s d det ⊥ ∇ d y s d s d (x) = 0 = s d (y) det ⊥ ev d x,y - E det ⊥ ∇ d x s d s d (x) = 0 E det ⊥ ∇ d y s d s d (y) = 0 |det ⊥ (ev d x )| det ⊥ ev d y   .
Remark 5.7. Note that D d does not depend on the choice of ∇ d . Indeed, we only consider derivatives of s d at points where it vanishes.

Proposition 5.8. For all d max(d 1 , d 2 ), we have for any φ 1 , φ 2 ∈ C 0 (M ):

Var(|dV d |) (φ 1 , φ 2 ) = 1 (2π) r M 2 \∆ φ 1 (x)φ 2 (y)D d (x, y) |dV M | 2 + δ rn E[ |dV d | , φ 1 φ 2 ] ,
where δ rn equals 1 if r = n and 0 otherwise.

Proof. This was proved in [22, Sect. 4.2] for r < n, the case r = n requires an extra argument.

The following proof if valid for any r ∈ {1, . . . , n}. Let φ 1 and φ 2 ∈ C 0 (M ), we have:

Var(|dV d |) (φ 1 , φ 2 ) = E[ |dV d | , φ 1 |dV d | , φ 2 ] -E[ |dV d | , φ 1 ] E[ |dV d | , φ 2 ] . ( 5
|dV d | , φ 1 |dV d | , φ 2 = (x,y)∈(Z d ) 2 φ 1 (x)φ 2 (y) |dV d | 2 = (x,y)∈(Z d ) 2 \∆ φ 1 (x)φ 2 (y) |dV d | 2 + δ rn x∈Z d φ 1 (x)φ 2 (x) |dV d | , almost surely. Hence E[ |dV d | , φ 1 |dV d | , φ 2 ] = E (x,y)∈(Z d ) 2 \∆ φ 1 (x)φ 2 (y) |dV d | 2 + δ rn E[ |dV d | , φ 1 φ 2 ] . (5.4)
We apply Thm. 5.5 to the first term on the right-hand side of Eq. (5.4). Similarly, we apply Thm. 5.

1 to E[ |dV d | , φ i ] for i ∈ {1, 2}
. This yields the result by Eq. (5.3).

By Thm. 1.2, if r = n, for all φ 1 , φ 2 ∈ C 0 (M ) we have:

E[ |dV d | , φ 1 φ 2 ] = d n 2 2 Vol (S n ) M φ 1 φ 2 |dV M | + φ 1 ∞ φ 2 ∞ O d n 2 -1 .
Hence, in order to prove Thm. 1.6, we have to show that, for any n ∈ N * and r ∈ {1, . . . , n}:

M 2 \∆ φ 1 (x)φ 2 (y)D d (x, y) |dV M | 2 = d r-n 2 M φ 1 φ 2 |dV M | Vol S n-1 I n,r + φ 1 ∞ φ 2 ∞ O d r-n 2 -α + φ 1 ∞ ̟ φ2 C β d -β O d r-n 2 , (5.5)
where α, β, C β and I n,r are as in Thm. 1.6. This is done in two steps. The mass of the integral on the left-hand side of Eq. (5.5) concentrates in a neighborhood of ∆ of typical size 1

∆ d = (x, y) ∈ M 2 ρ g (x, y) < b n ln d √ d ,
where, ρ g is the geodesic distance in (M, g).

In Sect. 5.3 below, we show that, in Eq. (5.5), the integral over M 2 \ ∆ d only contributes what turns out to be an error term. We refer to this term as the far off-diagonal term. In Sect. 5.4 we complete the proof of (5.5) by studying the near-diagonal term, that is the integral of φ 1 (x)φ 2 (y)D d (x, y) over ∆ d \ ∆. This turns out to be the leading term.

Expression of some covariances

In order to prove (5.5), we need to study the distribution of the random variables appearing in the definition of D d (see Def. 5.6). The purpose of this section is to introduce several variance operators that will appear in the proof. In the following ∇ d denotes a real connection on E ⊗ L d → X .

Uncorrelated terms

First of all, let us consider the distribution of s d (x) for any x ∈ M . Since

s d ∼ N (Id) and ev d x is linear (see Def. 2.2), s d (x) is a centered Gaussian vector in R E ⊗ L d x with variance operator ev d x (ev d x ) * = E d (x, x).
Lemma 5.10. For all x ∈ M , we have det

⊥ ev d x = d π rn 2 (1 + O(d -1
)), where the error term is independent of x.

Proof. We have det

⊥ ev d x = det(E d (x, x))
x ∈ M , ev d x is surjective, that is (s d (x)) is non-degenerate.
Then, let d ∈ N and x ∈ M , we denote by j d

x : s → (s(x), ∇ d x s) the evaluation of the 1-jet of a section at the point x. The distribution of the random vector

(s d (x), ∇ d x s d ) is a centered Gaussian in R E ⊗ L d x ⊕ R E ⊗ L d x ⊗ T *
x M with variance operator:

j d x (j d x ) * = E j d x (s d ) ⊗ j d x (s d ) * = E[s d (x) ⊗ s d (x) * ] E s d (x) ⊗ (∇ d x s d ) * E (∇ d x s d ) ⊗ s d (x) * E (∇ d x s d ) ⊗ (∇ d x s d ) * = E d (x, x) ∂ ♯ y E d (x, x) ∂ x E d (x, x) ∂ x ∂ ♯ y E d (x, x)
.

If d d 1 , then s d (x)
is non-degenerate and the distribution of ∇ d x s d given that s d (x) = 0 is a centered Gaussian whose variance equals

∂ x ∂ ♯ y E d (x, x) -∂ x E d (x, x) (E d (x, x)) -1 ∂ ♯ y E d (x, x
). By Thm. 3.3, this variance equals

d n+1 π n Id R(E⊗L d ) x ⊗T * x M +O(d -1
) as d goes to infinity and the error does not depend on x.

Remark 5.12. If (s, x) is such that s(x) = 0, then ∇ d

x s does not depend on the connection ∇ d . This explains why the distribution of ∇ d

x s d given that s d (x) = 0, in particular its variance, does not depend on ∇ d . Lemma 5.13. For every x ∈ M , we have:

E det ⊥ ∇ d x s d s d (x) = 0 = d n+1 π n r 2 (2π) r 2 Vol (S n-r ) Vol (S n ) 1 + O d -1 ,
where the error term is independent of x.

Proof. This was proved in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Lem. 4.7] for r < n. The proof is the same here.

Correlated terms far from the diagonal

Let us now focus on variables where non trivial correlations may appear in the limit. Let d ∈ N, for all (x, y) ∈ M 2 \ ∆, the random vector ev d x,y (s d ) = (s d (x), s d (y)) is a centered Gaussian vector with variance operator:

ev d x,y (ev d x,y ) * = E ev d x,y (s d ) ⊗ ev d x,y (s d ) * = E d (x, x) E d (x, y) E d (y, x) E d (y, y) , (5.6)
where we decomposed this operator according to the direct sum R E ⊗ L d x ⊕ R E ⊗ L d y . Definition 5.14. For all d ∈ N, for all (x, y) ∈ M 2 \ ∆, we denote by:

Θ d (x, y) = π d n E d (x, x) E d (x, y) E d (y, x) E d (y,
R E ⊗ L d x ⊕ R E ⊗ L d y ⊕ R E ⊗ L d x ⊗ T * x M ⊕ R E ⊗ L d y ⊗ T * y M ,
whose variance operator j d x,y (j d x,y ) * equals:

E j d x,y (s d ) ⊗ j d x,y (s d ) * =     E d (x, x) E d (x, y) ∂ ♯ y E d (x, x) ∂ ♯ y E d (x, y) E d (y, x) E d (y, y) ∂ ♯ y E d (y, x) ∂ ♯ y E d (y, y) ∂ x E d (x, x) ∂ x E d (x, y) ∂ x ∂ ♯ y E d (x, x) ∂ x ∂ ♯ y E d (x, y) ∂ x E d (y, x) ∂ x E d (y, y) ∂ x ∂ ♯ y E d (y, x) ∂ x ∂ ♯ y E d (y, y)     .
Definition 5.15. For all d ∈ N, for all (x, y) ∈ M 2 \ ∆, we denote by:

Ω d (x, y) = π d n      E d (x, x) E d (x, y) d -1 2 ∂ ♯ y E d (x, x) d -1 2 ∂ ♯ y E d (x, y) E d (y, x) E d (y, y) d -1 2 ∂ ♯ y E d (y, x) d -1 2 ∂ ♯ y E d (y, y) d -1 2 ∂ x E d (x, x) d -1 2 ∂ x E d (x, y) d -1 ∂ x ∂ ♯ y E d (x, x) d -1 ∂ x ∂ ♯ y E d (x, y) d -1 2 ∂ x E d (y, x) d -1 2 ∂ x E d (y, y) d -1 ∂ x ∂ ♯ y E d (y, x) d -1 ∂ x ∂ ♯ y E d (y, y)      ,
the variance operator of the centered Gaussian vector:

π d n 2 s d (x), s d (y), 1 √ d ∇ d x s d , 1 √ d ∇ d y s d .
Let us now assume that d d 2 , so that the distribution of (s d (x), s d (y)) is non-degenerate. Then the distribution of ∇ d

x s, ∇ d y s given that s d (x) = 0 = s d (y) is a centered Gaussian with variance operator:

∂x∂ ♯ y E d (x,x) ∂x∂ ♯ y E d (x,y) ∂x∂ ♯ y E d (y,x) ∂x∂ ♯ y E d (y,y) - ∂xE d (x,x) ∂xE d (x,y) ∂xE d (y,x) ∂xE d (y,y) E d (x,x) E d (x,y) E d (y,x) E d (y,y) -1 ∂ ♯ y E d (x,x) ∂ ♯ y E d (x,y) ∂ ♯ y E d (y,x) ∂ ♯ y E d (y,y)
.

Definition 5.16. For all d d 2 , for all (x, y) ∈ M 2 \ ∆, we set:

Λ d (x, y) = π n d n+1 ∂ x ∂ ♯ y E d (x, x) ∂ x ∂ ♯ y E d (x, y) ∂ x ∂ ♯ y E d (y, x) ∂ x ∂ ♯ y E d (y, y) - ∂ x E d (x, x) ∂ x E d (x, y) ∂ x E d (y, x) ∂ x E d (y, y) E d (x, x) E d (x, y) E d (y, x) E d (y, y) -1 ∂ ♯ y E d (x, x) ∂ ♯ y E d (x, y) ∂ ♯ y E d (y, x) ∂ ♯ y E d (y, y)
, which is the variance of the Gaussian vector

π n d n+1 1 2 ∇ d x s d , ∇ d y s d given that s d (x) = 0 = s d (y).
Remark 5.17. Once again, Λ d (x, y) is independent of the choice of ∇ d , and so is the distribution of

(∇ d x s d , ∇ d y s d ) given that s d (x) = 0 = s d (y).
On the other hand, the distribution of s d (x), s d (y), ∇ d

x s d , ∇ d y s d heavily depends on ∇ d , and so does Ω d (x, y). Hence, we will need to specify a choice of ∇ d at some point when dealing with Ω d .

Correlated terms close to the diagonal

Θ d (z) = Θ d x, exp x z √ d , seen as an operator on R E ⊗ L d x ⊕ R E ⊗ L d
x via the real normal trivialization centered at x. Remark 5.19. Beware that Θ d (z) depends on x, even if this is not reflected in the notation. However, we will show that the limit of Θ d (z) as d → +∞ does not depend on x.

Recall that e d was defined by Eq. (3.1) as a map from

T x M × T x M to End R E ⊗ L d
x . The definitions of Θ d (x, y) (Def.

Ω d (z) = Ω d x, exp x z √ d , seen as an operator on (R ⊕ R ⊕ T * x M ⊕ T * x M ) ⊗ R E ⊗ L d
x via the real normal trivialization centered at x. Let ∇ d be a real connection on E ⊗ L d such that, in the real normal trivialization around x, this connection coincides over the ball B TxX (0, R) with the usual differentiation for maps from T x X to E ⊗ L d

x . The existence of such a connection was established at the end of Sect. 3.1. Then, by Def. 5.15 and 5.20, we have for all d d 3 , for all x ∈ M and for all z ∈ B TxM (0, b n ln d) \ {0}:

Ω d (z) =     e d (0, 0) e d (0, z) ∂ ♯ y e d (0, 0) ∂ ♯ y e d (0, z) e d (z, 0) e d (z, z) ∂ ♯ y e d (z, 0) ∂ ♯ y e d (z, z) ∂ x e d (0, 0) ∂ x e d (0, z) ∂ x ∂ ♯ y e d (0, 0) ∂ x ∂ ♯ y e d (0, z) ∂ x e d (z, 0) ∂ x e d (z, z) ∂ x ∂ ♯ y e d (z, 0) ∂ x ∂ ♯ y e d (z, z)     . ( 5
Λ d (z) = Λ d x, exp x z √ d ,
seen as an operator on

(T * x M ⊕ T * x M ) ⊗ R E ⊗ L d
x via the real normal trivialization around x. Then, for all d max(d 2 , d 3 ), for all x ∈ M and all z ∈ B T *

x M (0, b n ln d) \ {0} we have:

Λ d (z) = ∂ x ∂ ♯ y e d (0, 0) ∂ x ∂ ♯ y e d (0, z) ∂ x ∂ ♯ y e d (z, 0) ∂ x ∂ ♯ y e d (z, z) - ∂ x e d (0, 0) ∂ x e d (0, z) ∂ x e d (z, 0) ∂ x e d (z, z) Θ d (z) -1 ∂ ♯ y e d (0, 0) ∂ ♯ y e d (0, z) ∂ ♯ y e d (z, 0) ∂ ♯ y e d (z, z)
.

Far off-diagonal term

In this section, we state that the far off-diagonal term in Eq. (5.5) only contributes an error term. This was already proved in [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF] for r < n. The proof is the same for r = n, so we refer to [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF] for the proof. Lem. 5.23 below is used in the proof of Prop. 5.22 but is also of independent interest for our purpose.

Proposition 5.22. Let φ 1 , φ 2 ∈ C 0 (M ), then we have the following as d → +∞:

M 2 \∆ d φ 1 (x)φ 2 (y)D d (x, y) |dV M | 2 = φ 1 ∞ φ 2 ∞ O d r-n 2 -1 ,
where the error term is independent of (φ 1 , φ 2 ).

Lemma 5.23. For every (x, y) ∈ M 2 \ ∆ d , we have:

Θ d (x, y) = π d n E d (x, x) 0 0 E d (y, y) Id +O d -n 2 -1 ,
where the error term is independent of (x, y)

∈ M 2 \ ∆ d . Proof. Since (x, y) ∈ M 2 \ ∆ d ,
Θ d (x, y) = π d n E d (x, x) 0 0 E d (y, y) + O d -n 2 -1 .
The results follows from the fact that the leading term is Id +O(d -1 ), by Thm. 3.3.

Near-diagonal term

In this section, we conclude the proof of Thm. 1.6, up to the technical lemmas whose proofs were postponed until Appendices A and B.

D d (x, z) = d -r D d x, exp x z √ d .
Recall that D n,r was defined by Def. 4.15. The main result of this section is the following.

Proposition 5.25. Let α ∈ (0, 1), then for all x ∈ M , for all z ∈ B TxM (0, b n ln d) \ {0} we have:

D d (x, z) = D n,r z 2 1 + O d -α + O d -α ,
where the error terms do not depend on (x, z).

First, let us prove that Prop. 5.8, 5.22 and 5.25 together imply Thm. 1.6.

Proof of Thm. 1.6. The main point is to compute the asymptotics of the near-diagonal term in Eq. (5.5). Let us fix α ∈ (0, 1), β ∈ 0, 1 2 and φ 1 , φ 2 ∈ C 0 (M ). Let x ∈ M , recall that √ κ is the density of |dV M | with respect to the Lebesgue measure, in the exponential chart centered at x, where κ was introduced in Sect. 3.2. Then, by a change of variable y = exp x z √ d , we have:

∆ d \∆ φ 1 (x)φ 2 (y)D d (x, y) |dV M | 2 = d r-n 2 x∈M φ 1 (x) z∈BT x M (0,bn ln d) φ 2 exp x z √ d D d (x, z)κ z √ d 1 2
dz |dV M | . (5.9)

As we already discussed in Sect. 3.2, κ(z) = 1 + O( z 2 ) and the error term is independent of x.

Hence, κ z

√ d 1 2 = 1 + O (ln d) 2 d
, and by Prop. 5.25, for all γ ∈ (α, 1),

z∈BT xM (0,bn ln d) φ 2 exp x z √ d D d (x, z)κ z √ d 1 2 dz = z∈BT xM (0,bn ln d) φ 2 exp x z √ d D n,r ( z 2 ) dz 1 + O d -γ + φ 2 ∞ O (ln d) n d γ .
(5.10)

Since γ > α, (ln d) n d -γ = O(d -α ). Similarly, there exists C β > 0 such that b n ln d √ d C β d -β
for all d ∈ N * . Then we have: 5.12), the innermost integral on the right-hand side of Eq. (5.9) equals:

z∈BT x M (0,bn ln d) φ 2 exp x z √ d -φ 2 (x) D n,r ( z 2 ) dz ̟ φ2 C β d -β z∈BT x M (0,bn ln d) D n,
φ 2 (x) Vol S n-1 I n,r + ̟ φ2 C β d -β O(1) + φ 2 ∞ O(d -α ),
and the error terms are independent of x ∈ M and (φ 1 , φ 2 ). Finally, by Eq. (5.9),

∆ d \∆ φ 1 (x)φ 2 (y)D d (x, y) |dV M | 2 = d r-n 2 M φ 1 φ 2 |dV M | Vol S n-1 I n,r + φ 1 ∞ ̟ φ2 C β d -β O(d r-n 2 ) + φ 1 ∞ φ 2 ∞ O(d r-n 2 -α ).
We conclude the proof by combining this last relation with Prop. 5.8, 5.22 and, in the case r = n, Thm. 1.2 for φ 1 φ 2 .

The remainder of this section is mostly dedicated to the proof of Prop. 5.25. We will deduce this proposition from several technical lemmas stated below.

Let x ∈ M , then, any choice of an isometry between T x M and R n and an isometry between R E ⊗ L d

x and R r allows us to see the Bargmann-Fock process (s(z)) z∈R n , studied in Sect. 4, as a smooth Gaussian process from T x M to R E ⊗ L d

x . The distribution of this process does not depend on our choice of isometries. Thus, in the following, we can consider Θ(z) and Θ d (z) (resp. Ω(z) and Ω d (z), resp. Λ(z) and Λ d (z)) as operators on the same space.

Lemma 5.26. Let α ∈ (0, 1), then for all x ∈ M , for all z ∈ B TxM (0, b n ln d) \ {0} we have:

Θ(z) -1 2 Θ d (z)Θ(z) -1 2 = Id +O d -α ,
where the error term does not depend on (x, z).

Proof. See Appendix B.

Remark 5.27. One could wonder why Lem. 5.26 does not state that Θ d (z) = Θ(z) (Id +O(d -α )), which would be somewhat simpler. First, note that this statement is not equivalent to Lem. 5.26, since some of the eigenvalues of Θ(z) converge to 0 as z → 0. In fact, this alternative statement turns out to be false in general. Moreover, even if Θ d (z) is a linear map, it represents a variance, i.e. something intrinsically bilinear. It is then quite natural to consider Θ(z)

-1 2 Θ d (z)Θ(z) -1 2 since this is how Θ d (z) transforms if we act on R E ⊗ L d x by Θ(z) 1 2
. This remark also applies to Lem. 5.28 and 5.29 below.

Let us forget about the proof of Prop. 5.25 for a minute, and prove the existence of d 2 (see Lem. 5.4) as a corollary of Lem. 5.23 and 5.26. Note that the proofs of these lemmas only rely on the estimates of Sect. 3, so there is no logical loop here.

Proof of Lem. 5.4. We want to prove that, as soon as d is large enough, ev d x,y is surjective for all (x, y) ∈ M 2 \ ∆, that is det ev

det (Θ d (x, y)) = det (Θ d (z)) = det (Θ(z)) 1 + O √ d = 1 -e -z 2 r 1 + O √ d , (5.14)
uniformly on ∆ d \ ∆. The result follows from Eq. (5.13) and (5.14).

We can now go back to the proof of Prop. 5.25.

Lemma 5.28. Let α ∈ (0, 1), then for all x ∈ M , for all z ∈ B TxM (0, b n ln d) \ {0} we have:

Ω(z) -1 2 Ω d (z)Ω(z) -1 2 = Id +O d -α ,
where the error term does not depend on (x, z).

Proof. See Appendix B.

Lemma 5.29. Let α ∈ (0, 1), then for all x ∈ M , for all z ∈ B TxM (0, b n ln d) \ {0} we have:

Λ(z) -1 2 Λ d (z)Λ(z) -1 2 = Id +O d -α ,
where the error term does not depend on (x, z).

Proof. Let α ∈ (0, 1). Let x ∈ M and z ∈ B TxM (0, b n ln d) \ {0}. By Def. 5.20, Ω d (z) is an operator on:

R 2 ⊗ R E ⊗ L d x ⊕ (T * x M ) 2 ⊗ R E ⊗ L d x .
Using this splitting, we write Ω d (z) by blocks as:

Ω d (z) = Θ d (z) Ω 1 d (z) * Ω 1 d (z) Ω 2 d (z) , thus defining Ω 1 d (z) and Ω 2 d (z). For d large enough, Θ d (z) is invertible and its Schur complement is Λ d (z) = Ω 2 d (z) -Ω 1 d (z)Θ d (z) -1 Ω 1 d (z) * . It is then known that Λ d (z) -1 is the bottom-right block of Ω d (z) -1 , that is: Λ d (z) -1 = 0 Id Ω d (z) -1 0 Id = 0 Id Ω(z) -1 + Ω(z) -1 2 O d -α Ω(z) -1 2 0 Id ,
where the second equality is given by Lem. 5.28 and the error term is independent of (x, z).

Similarly, Λ(z) is the Schur complement of Θ(z) in Ω(z), so that Λ(z) -1 = 0 Id Ω(z) -1 0 Id .

Since by Lem. 4.14, 0 Λ(z)

1 2 Ω d (z) -1 2 is bounded, Λ(z) 1 2 Λ d (z) -1 Λ(z) 1 2 = Id +O(d -α
), and the error term still does not depend on (x, z).

Lemma 5.30. Let α ∈ (0, 1), let x ∈ M and z ∈ B TxM (0, b n ln d) \ {0}. Let X d (z) and Y d (z) be random vectors in T * x M ⊗ R E ⊗ L d x such that (X d (z), Y d (z)) ∼ N (Λ d (z)
). Then we have:

E det ⊥ (X d (z)) det ⊥ (Y d (z)) = E det ⊥ (X ∞ (z)) det ⊥ (Y ∞ (z)) 1 + O d -α
where (X ∞ (z), Y ∞ (z)) ∼ N (Λ(z)) and the error term does not depend on (x, z).

Proof. See Appendix B.

We conclude this section with the proof of Prop.

1 d r E det ⊥ ∇ d x s d det ⊥ ∇ d y s d s d (x) = 0 = s d (y) det ⊥ ev d x,y = E det ⊥ (X d (z)) det ⊥ (Y d (z)) det (Θ d (z)) 1 2
, where (X d (z), Y d (z)) ∼ N (Λ d (z)). By Lem. 5.26 and 5.30, this equals:

E det ⊥ (X ∞ (z)) det ⊥ (Y ∞ (z)) det (Θ(z)) 1 2 1 + O d -α ,
where (X ∞ (z), Y ∞ (z)) ∼ N (Λ(z)) and the error term does not depend on (x, z). By the definition of Λ(z

) (cf. Sect. 4.3) (X ∞ (z), Y ∞ (z)) is distributed as (d 0 s, d z s)
, where s is a copy of the Bargmann-Fock process from

T x M to R E ⊗ L d x .
Then, by Lem. 4.16,

E det ⊥ (X ∞ (z)) det ⊥ (Y ∞ (z)) = E det ⊥ X( z 2 ) det ⊥ Y ( z 2 ) ,
where X( z 2 ), Y ( z 2 ) was defined by Def. 1.5. Besides, det (Θ(z)) = 1e -z 2 r by Lem. 4.1, so that:

1 d r E det ⊥ ∇ d x s d det ⊥ ∇ d y s d s d (x) = 0 = s d (y) det ⊥ ev d x,y = D n,r z 2 + (2π) r Vol (S n-r ) Vol (S n ) 2 1 + O d -α .
Besides, Lem. 5.10 and 5.13,

1 d r E det ⊥ ∇ d x s d s d (x) = 0 |det ⊥ (ev d x )| E det ⊥ ∇ d y s d s d (y) = 0 |det ⊥ (ev d x )| = (2π) r Vol (S n-r ) Vol (S n ) 2 + O d -1
This yields the desired relation.

Proof of Theorem 1.8

The goal of this section is to prove that the leading constant in Thm. 1.6 is positive. Sect. 6.1 is concerned with the definition and proprieties of Kostlan-Shub-Smale polynomials. In Sect. 6.2 we recall some facts about Wiener chaoses. In Sect. 6.3 we compute the chaotic expansion of the linear statistics |dV d | , φ . Finally, we conclude the proof of Thm. 1.8 in Sect. 6.4.

Kostlan-Shub-Smale polynomials

In this section, we describe a special case of our real algebraic framework. This is the setting we will be considering throughout the proof of Thm. 1.8. A good reference for the complex algebraic material of this section is [START_REF] Ph | Principles of algebraic geometry[END_REF].

Definition

We choose X to be the complex projective space CP n with the real structure induced by the complex conjugation in C n+1 . The real locus of X is the real projective space RP n . We set L = O(1) → CP n as the hyperplane line bundle, that is the dual of the tautological line bundle: [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF][START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF][START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF][START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]

O(-1) = (ζ, x) ∈ C n+1 × CP n ζ ∈ x -→ CP n .

Recall that ample line bundles on CP

= α 0 + • • • + α n . We also define X α = X α0 0 • • • X αn n and α! = α 0 ! • • • α n !. Finally, if |α| = d, we denote by d α the multinomial coefficient d! α! . It is well-known (cf.
) that RH 0 (CP n , C r ⊗O(d)) is the space R hom d [X 0 , . . . , X n ]
r of tuples of real homogeneous polynomials of degree d in n+1 variables. The r terms R hom d [X 0 , . . . , X n ] in RH 0 (CP n , C r ⊗ O(d)) are pairwise orthogonal for the inner product (2.2). Besides, in restriction to one of these terms, (2.2) equals:

(P, Q) → x∈CP n h d (P (x), Q(x)) |dV CP n | = 1 π(d + n)! z∈C n+1 P (z)Q(z)e -z 2 dz. (6.1) An orthonormal basis of R hom d [X 0 , . . . , X n ] is then (d+n)! π n d! d α X α |α|=d . Hence, a standard Gaussian in RH 0 (CP n , C r ⊗ O(d)
) is a r-tuple of independent random polynomials of the form:

(d + n)! π n d! |α|=d a α d α X α , (6.2)
where the coefficients (a α ) |α|=d are independent real standard Gaussian variables. Such a random polynomial is called a Kostlan-Shub-Smale polynomial (KSS for short).

Correlation kernel

In this section, we study the distribution of the KSS polynomial (see Eq. ( 6.2)). In the setting of Sect. 6.

1.1, E d is the Bergman kernel of C r ⊗ O(d) → CP n .
Since the first factor is trivial, we have

E d = I r ⊗ E ′ d , where I r is the identity of C r and E ′ d is the Bergman kernel of O(d) → CP n . Note that E ′
d is the correlation kernel of the field s ′ d defined by one KSS polynomial, seen as a random section of O(d). By Eq. ( 6.2) we have:

E ′ d (x, y) = E[s ′ d (x) ⊗ s ′ d (y) * ] = (d + n)! π n d! |α|=d d α X α (x) ⊗ X α (y) *
Note that (6.1) is invariant under the action of the orthogonal group O n+1 (R) on the right. Hence, the distribution of KSS polynomials (6.2) and E ′ d are equivariant under this action. Since O n+1 (R) acts transitively on the couples of points of RP n at a given distance, E ′ d (x, y) only depends on the geodesic distance ρ g (x, y), and the same holds for derivatives. Loosely speaking, this implies the following, where derivatives are computed with respect to the Chern connection. These properties are very specific of the case of KSS polynomials. They will be useful in Sect. 6.3, to compute the Wiener-Itō expansion of |dV d | , φ . We do not give more details here, since Properties 1, 2 and 3 can also be deduced from the expression of E ′ d in local coordinates that we derive below.

Local expression of the kernel

Let x ∈ RP n , we want to compute the expression of E ′ d in some good coordinates around x. We could use the real normal trivialization, but the computations would be cumbersome. Instead, we use a slightly different trivialization. Since E ′ d is equivariant under the action O n+1 (R), we can assume that x = [1 : 0 :

• • • : 0]. We have a chart ψ x : (z 1 , . . . , z n ) -→ [1 : z 1 : • • • : z n ] from R n to B RP n x, π
2 . We trivialize O(d) over B RP n x, π 2 by identifying each fiber with O(d) x , by parallel transport with respect to the Chern connection ∇ d along curves of the form t → ψ x (tz) with z ∈ R n . Thanks to this trivialization, we can consider E ′ d as a map taking values in R. Recall that we defined a scaled version e d of the Bergman kernel E d by Eq. (3.1). The following is related without being an exact analogue. For all w, z ∈ R n , we set:

ξ d (w, z) = π n d! (d + n)! E ′ d (ψ x (w), ψ x (z)) . (6.3)
A computation in local coordinates yields the following lemma. The Chern connection ∇ d coincides at the origin with the usual differential in our trivialization. Hence taking the values at (0, 0) of the following expressions proves that s ′ d satisfies Properties 1, 2 and 3 (cf. Sect. 6.1.2). Lemma 6.2. Let d ∈ N * and let i, j ∈ {1, . . . , n}. Then for all w, z ∈ R n we have:

ξ d (w, z) =   1 + w , z 1 + w 2 1 + z 2   d , ∂ xi ξ d (w, z) = dξ d (w, z) z i 1 + w , z - w i 1 + w 2 , ∂ yj ξ d (w, z) = dξ d (w, z) w j 1 + w , z - z j 1 + z 2 , ∂ xi ∂ yj ξ d (w, z) = ξ d (w, z) dδ ij 1 + w , z - d 2 w i w j (1 + w , z )(1 + w 2 ) - d 2 z i z j (1 + w , z )(1 + z 2 ) + d 2 w i z j (1 + w 2 )(1 + z 2 ) + (d 2 -d)z i w j (1 + w , z ) 2 ,
where δ ij = 1 if i = j, and δ ij = 0 otherwise.

Hermite polynomials and Wiener chaos

In the setting of KSS polynomials, we consider

RH 0 (CP n , C r ⊗ O(d)) = R hom d [X 0 , . . . , X n ]
r , equipped with the inner product (6.1). For simplicty, in this section and the following, we denote by V d this Euclidean space and by dν d its standard Gaussian measure. With these notations, (V d , dν d ) is our probability space and we denote by L 1 (dν d ) (resp. L 2 (dν d )) the space of integrable (resp. square integrable) random variables on this space. Thm. 1.6 shows that for d large enough, for all φ ∈ C 0 (RP n ), |dV d | , φ ∈ L 2 (dν d ). The proof given in Sect. 5 shows that this is true for any d max(d 0 , d 1 , d 2 , d 3 ), in this framework this is true for any d ∈ N * . The idea of this section is to find a nice orthogonal decomposition of L 2 (dν d ). We will study |dV d | , φ thanks to this decomposition in Sect. 6.3. These techniques were already used in a similar context in [START_REF] Armentano | On the asymptotic variance of the number of real roots of random polynomial systems[END_REF][START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF][START_REF] Dalmao | Phase singularities in complex arithmetic random waves[END_REF][START_REF] Marinucci | Non-universality of nodal length distribution for arithmetic random waves[END_REF], for example. See [START_REF] Nourdin | Normal approximation with Malliavin calculus[END_REF] for background on the following material. Definition 6.3. For all k ∈ N, we denote by H k the k-th Hermite polynomial. These polynomials are defined recursively by: H 0 = 1, H 1 = X and, for all k ∈ N * , H k+1 (X) = XH k (X)-kH k-1 (X). Remark 6.4. Equivalently, one can define H k by: H 0 = 1 and ∀k ∈ N,

H k+1 = XH k -H ′ k . Lemma 6.5. Let k ∈ N, then H k is a polynomial of degree k which is even if k is even and odd if k is odd. Moreover, H 2k (0) = (-1) k (2k)! 2 k k! and H 2k+1 (0) = 0.
Proof. This is proved by induction, using the recursive definition of H k .

Let us denote by dµ N the standard Gaussian measure on R N . We also denote by L 2 (dµ N ) the space of square integrable functions with respect to dµ N . Recall that the family 1

√ k! H k k∈N is a
Hilbert basis of L 2 (dµ 1 ) (see [START_REF] Nourdin | Normal approximation with Malliavin calculus[END_REF]Prop. 1.4.2]). Similarly, in dimension N , the family:

N i=1 1 √ α i ! H αi (X i ) α ∈ N N
is a Hilbert basis of L 2 (dµ N ). The result in dimension 1 shows that this family is orthonormal. Then, one only needs to check that the space of polynomials in N variables is dense in L 2 (dµ N ).

For N = 1 this is proved in [28, Prop. 1.1.5], and the same proof works in any dimension. As in Sect. 5, we denote by s d a generic element of (V d , dν d ), that we think of as a standard Gaussian vector in

V d . Let η ∈ V * d , then η(s d ) ∈ L 2 (dν d ) is a real centered Gaussian variable. Moreover, for any η, η ′ ∈ V * d , we have E[η(s d )η ′ (s d )] = η , η ′ . Thus, V * d is canonically isometric to a subspace of L 2 (dν d ), via η → η(s d ).
From now on, we identify V * d with its image, so that

V * d ⊂ L 2 (dν d ) is a centered Gaussian Hilbert space.
Definition 6.6. Let (η 1 , . . . , η N d ) denote an orthonormal basis of V * d , that is the (η i (s d )) i∈{1,...,N d } are independent real standard Gaussian variables. For all q ∈ N, we define C d [q], the q-th Wiener chaos of the field s d , as the subspace of L 2 (dν d ) spanned by the orthogonal family:

N d i=1 H αi (η i ) α ∈ N N d , |α| = q . Remarks 6.7. • C d [0] is the space of constant random variables in L 2 (dν d ) and C d [1] = V * d . • We do not need to take the closure in the definition of C d [q] since it is finite-dimensional.
Lemma 6.8. The Wiener chaoses (C d [q]) q∈N of s d do not depend on the choice of the orthonormal basis (η 1 , . . . , η N d ) appearing in Def. 6.6.

Proof. Let (η 1 , . . . , η N d ) and (η ′ 1 , . . . , η ′ N d ) be two orthonormal basis of V * d . There exists an orthogonal transformation U of V * d such that, for all i ∈ {1, . . . , N d }, η ′ i = U (η i ). The situation being symmetric, we only have to prove that, for any β ∈ N N d such that |β| = q, N d i=1 H βi (η i ) is a linear combination of elements of the family:

N d i=1 H αi (U (η i )) α ∈ N N d , |α| = q .
Dropping the dependence on d, this amounts to proving that: if

X = (X 1 , . . . , X N ) ∈ R N and U ∈ O N (R) then, for all β ∈ N N such that |β| = q, N i=1 H βi (X i ) is a linear combination of the N i=1 H αi (U (X i )) |α|=q .
By [START_REF] Nourdin | Normal approximation with Malliavin calculus[END_REF]Prop. 1.4.2], we have:

∀t ∈ R n , α∈N N N i=1 H αi (X i ) t α α! = exp t , X - t 2 2 ,
where • , • is the standard inner product of R N and • the associated norm. The right-hand side being invariant under orthogonal transformation, we have:

∀t ∈ R n , α∈N N N i=1 H αi (U (X i )) (U (t)) α α! = α∈N N N i=1 H αi (X i ) t α α! .
Now, the components of U (t) are homogeneous polynomials of degree 1 in (t 1 , . . . , t N ). Hence (U (t)) α can only contribute terms of degree |α| to the sum. We conclude by identifying the coefficients of these power series of the variable t. Lemma 6.9. For all

d ∈ N * , q∈N C d [q] is dense in L 2 (dν d ).
Moreover, the terms of this direct sum are pairwise orthogonal.

Proof. The family

N d i=1 H αi (X i ) α∈N N d
being orthogonal, the (C d [q]) q∈N are pairwise orthogonal by definition. Let (s 1,d , . . . , s N d ,d ) be an orthonormal basis of V d . We have s d = a i s i,d , where the a i are independent N (1). For any i ∈ {1, . . . , N d }, let η i = • , s i,d , so that η i (s d ) = a i . Then, for any q ∈ N, C d [q] is spanned by the random variables

N d i=1 H αi (a i ) |α|=q .
Any square integrable functional of s d can be written as

F (a 1 , . . . , a N d ), with F ∈ L 2 (dµ N d ).
The conclusion follows, since the span of

N d i=1 H αi (X i ) α ∈ N N d is dense in L 2 (dµ N d ).
Notation 6.10. Let d ∈ N * and let A ∈ L 2 (dν d ). For any q ∈ N, we denote by A[q] the q-th chaotic component of A, that is its projection onto

C d [q]. Then, we have A = q∈N A[q] in L 2 (dν d ). By definition, A[0] = E[A]. Moreover, the C d [q]
being pairwise orthogonal, we have E[A[q]] = 0 for any q ∈ N * , and Var(A) = q∈N * Var(A[q]).

Wiener-Itō expansion of the linear statistics

Recall that we consider a standard Gaussian section

s d ∈ V d = R hom d [X 0 , . . . , X n ]
r and that |dV d | denotes the Riemannian measure of integration over its real zero set. Let us fix d ∈ N * and φ ∈ C 0 (RP n ). By Thm. 1.6,

|dV d | , φ ∈ L 2 (dν d ).
The goal of this section is to compute the chaotic expansion of these variables. For all q ∈ N, we denote

|dV d | [q] , φ for ( |dV d | , φ ) [q]. Since |dV d | , φ ∈ L 2 (dν d ), for any A ∈ L 2 (dν d ) we have (A |dV d | , φ ) ∈ L 1 (dν d ) and: E[A |dV d | , φ ] = E x∈Z d φ(x)A(s d ) |dV d | .
Even if A depends on s d , we can apply a Kac-Rice formula (cf. [START_REF]Variance of the volume of random real algebraic submanifolds[END_REF]Thm. 5.3]). Thus, we have:

E[A |dV d | , φ ] = (2π) -r 2 x∈RP n φ(x) |det ⊥ (ev d x )| E A det ⊥ ∇ d x s d s d (x) = 0 |dV RP n | ,
Recall that s d , is a tuple of independent KSS polynomials and that

E d = I r ⊗ E ′ d
, where E ′ d is the correlation kernel of one KSS polynomial. By Eq. (6.3) and Lem. 6.2 we have:

det ⊥ ev d x = det(E d (x, x)) 1 2 = det(E ′ d (x, x)) r 2 = (d + n)! π n d! r 2 . Denoting d (d+n)! π n d! -1 2 ∇ d x s d by L d (x) and (d+n)! π n d! -1
2 s d (x) by t d (x), we get:

E[A |dV d | , φ ] = d 2π r 2 x∈RP n φ(x)E A det ⊥ (L d (x)) t d (x) = 0 |dV RP n | . (6.4)
Let x ∈ RP n , without loss of generality we can assume that the coordinates on R n+1 are such that x = [1 : 0 :

• • • : 0]. Let ζ 0 (x) be one of the two unit vectors in RO(d) x , the other one being -ζ 0 (x). This gives an isomorphism between (R ⊕ T * x (RP n )) ⊗ O(d) x and R ⊕ T * x (RP n ), so that we can consider (t d (x), L d (x)) as an element of R r ⊕ (T * x RP n ) r . We denote by (t (1)
d (x), . . . , t (r)
d (x)) the components of t d (x) and by (L

(1) d (x), . . . , L (r) d (x)) those of L d (x). The couples (t (i) d (x), L (i) d (x)) are centered Gaussian vectors in R ⊕ T *
x RP n that are independent from one another. Moreover, by Lem. 6.2, for all i ∈ {1, . . . , r}, the variance operator of (t

(i) d (x), L (i) d (x)) is Id.

Let us choose any orthonormal basis of T *

x RP n , and denote by

(L i1 d (x), . . . , L in d (x)) the coordi- nates of L (i) d (x) in this basis, so that L ij d (x) 1 i r 1 j n
is the matrix of L d (x). Then,

t (i) d (x) 1 i r ⊔ L ij d (x) 1 i r, 1 j n
is a family of independent real standard Gaussian variables in L 2 (dν d ) and we can complete it into an orthonormal basis of C d [START_REF] Adler | Random fields and geometry[END_REF]. We denote by {S

(i) d (x) | r(n + 1) < i N d },
the last elements of such a basis. Below, we will work in the Hilbert basis of L 2 (dν d ) obtained by considering Hermite polynomials in these variables. Remark 6.11. We just used the fact that our random field satifies Properties 1, 2 and 3 of Sect. 6.1.2. This is what makes this computation specific to the case of KSS polynomials. Notation 6.12. Let α ∈ N r , β ∈ N r × N n and γ ∈ N N d -r(n+1) , we will use the following notations:

H α (t d (x)) = r i=1 H αi t (i) d (x) , H β (L d (x)) = 1 i r 1 j n H βij L ij d (x) , and
H γ (S d (x)) = N d i=r(n+1)+1 H γi S (i) d (x) .
We first expand det

⊥ (L d (x)) in L 2 (dν d ). Since det ⊥ (L d (x)
) only depends on the variables {L ij d (x) | 1 i r, 1 j n}, we have:

det ⊥ (L d (x)) = β∈N r ×N n B β H β (L d (x)) √ β! ,
where

B β = 1 √ β! E det ⊥ (L d (x)) H β (L d (x)) for all β ∈ N r × N n .
The coefficient B β only depends on the distribution of L d (x), which is a standard Gaussian for all x ∈ RP n . Hence B β is independent of x. These coefficients have several symmetries. Note that

det ⊥ L ij d (x) i,j
is invariant under the following operations:

• multiplying a whole column or a whole row by -1;

• permuting the rows or permuting the columns.

Since the Hermite polynomials of odd degrees are odd (cf. Lem. 6.5), the first point shows that B β = 0 whenever there exists i ∈ {1, . . . , r} such that n j=1 β ij is odd or there exists j ∈ {1, . . . , n} such that r i=1 β ij is odd. We denote by I the set of multi-indices β ∈ N r × N n such that for all i ∈ {1, . . . , r}, n j=1 β ij is even and, for all j ∈ {1, . . . , n}, r i=1 β ij is even. If |β| = 2, then the only way for β to belong to I is that there exists (i, j) such that β ij = 2, the other components of β being zero. The second point above shows that, in this case, the value of B β does not depend on the index (i, j) such that β ij = 2. Notation 6.13. Let B 2 denote the common value of the B β for β ∈ I such that |β| = 2.

We can also expand A ∈ L 2 (dν d ) as:

A = α,β,γ A α,β,γ (x) H α (t d (x)) √ α! H β (L d (x)) √ β! H γ (S d (x)) √ γ! , where A α,β,γ (x) = E A Hα(t d (x)) √ α! H β (L d (x)) √ β! Hγ (S d (x)) √ γ!
. Then, using the orthonormality properties of the Hermite polynomials, we get:

E A det ⊥ (L d (x)) t d (x) = 0 = α,β A α,β,0 (x)B β H α (0) √ α! , (6.5)
where the sum runs over multi-indices such that α ∈ 2N r (see Lem.6.5), and β ∈ I. Then, by Eq. (6.4) and (6.5), for any A ∈ C d [q], we have:

E[A |dV d | , φ ] = d 2π r 2 α∈2N r , β∈I |α|+|β|=q B β H α (0) √ α! E A x∈RP n φ(x) H α (t d (x)) √ α! H β (L d (x)) √ β! |dV RP n |
We have proved the following proposition.

Proposition 6.14. For all d ∈ N * , for all φ ∈ C 0 (RP n ), for all q ∈ N,

|dV d | [2q + 1] , φ = 0 and |dV d | [2q] , φ = d 2π r 2 x∈RP n φ(x) α∈2N r , β∈I |α|+|β|=2q B β H α (0) √ α! H α (t d (x)) √ α! H β (L d (x)) √ β! |dV RP n | . (6.6)
Remarks 6.15.

• Recall that the values of the t

• Since β∈I, |β|=p B β 1 √ β! H β (L d (x)) is the p-th chaotic component of det ⊥ (L d (x))
, it does not depend on our choice of an orthonormal basis of T *

x RP n . Hence, neither does the value of the sum on the right-hand side of Eq. (6.6), for any given x ∈ RP n .

• By [21, Lem. A.14], we have:

B 0 = E det ⊥ (L x (d)) = (2π) r Vol (S n-r )
Vol (S n ) .

Then, Prop. 6.14 for q = 0 shows that, in the setting of KSS polynomials, for all φ ∈ C 0 (RP n ),

E[ |dV d | , φ ] = d r 2 RP n φ |dV RP n | Vol (S n-r ) Vol (S n ) .
That is, in this case, the error term in Thm. 1.2 is zero for any d ∈ N * .

Let us conclude this section by writing

|dV d | [2]
, φ in a more explicit way. Lemma 6.16. For all d ∈ N * , for all φ ∈ C 0 (RP n ),

|dV d | [2] , φ = d r 2 Vol (S n-r ) 2n Vol (S n ) x∈RP n φ(x) L d (x) 2 -n t d (x) 2 |dV RP n | .
Proof. By Prop. 6.14 and Lem. 6.5, we have:

|dV d | [2] , φ = d 2π r 2 x∈RP n φ(x) - B 0 2 ( t d (x) 2 -r) + B 2 √ 2 ( L d (x) 2 -nr) |dV RP n | , (6.7)
where B 2 is defined by Ntn. 6.13. Since H 2 = X 2 -1, we have:

n √ 2B 2 = n j=1 E det ⊥ (L d (x)) H 2 (L 1j d (x)) = E det ⊥ (L d (x)) L (1)
d (x) 2 -nB 0 . It was proved in [21, App. B] that det ⊥ (L d (x)) is distributed as L (1) d (x) Z n-1 . . . Z n-r+1
, where L

(1) d (x), Z n-1 , . . . , Z n-r+1 are globally independent and, for all p ∈ {nr + 1, . . . , n -1}, Z p is a standard Gaussian vector in R p . Since L

B 0 = E det ⊥ (L d (x)) = E L (1) d (x) n-1 p=n-r+1 E[ Z p ] = (2π) r 2 Vol (S n-r ) Vol (S n ) ,
and

B 2 = 1 n √ 2 E L (1) d (x) 3 n-1 p=n-r+1 E[ Z p ] - B 0 √ 2 = B 0 √ 2 2π n Vol (S n ) Vol (S n+2 ) -1 = B 0 n √ 2 .
We plug these relations in Eq. (6.7) and this yields the result.

Conclusion of the proof

In this section, we finally prove Thm. 1.8. The key point is the following. Lemma 6.17. Let Z d be the common zero set of r independent Kostlan-Shub-Smale polynomials in RP n then we have the following as d goes to infinity:

Var(Vol (Z d ) [2]) ∼ d r-n 2 r 1 + 2 n π n 2
Vol (S n-r )

2
16 Vol (S n ) .

Let us prove that this lemma implies Thm. 1.8.

Proof of

d -r+ n 2 Var(Vol (Z d )) = Vol (RP n ) Vol S n-1 (2π) r I n,r + δ rn 2 Vol (S n ) + o (1)
.

On the other hand, as we explained as the end of Sect. 6.2,

d -r+ n 2 Var(Vol (Z d )) = d -r+ n 2 q∈N * Var(Vol (Z d ) [q]) d -r+ n 2 Var(Vol (Z d ) [2]) .
By Lem. 6.17, we get: We conclude this section by the proof of Lem. 6.17.

Vol S n-1 (2π) r I n,r + δ rn 2 Vol (S n ) r 8 1 + 2 n π n 2 Vol (S n-r ) Vol (S n ) 2 > 0.
Proof of Lem. 6.17. Recall that |dV d | is the Riemannian volume measure on Z d and that 1 is the unit constant function on RP n . By Lem. 6.16, we have:

Vol (Z d ) [2] = |dV d | [2] , 1 = d r 2 Vol (S n-r ) 2n Vol (S n ) x∈RP n L d (x) 2 -n t d (x) 2 |dV RP n | .
Since this is a centered variable, its variance equals:

d r Vol (S n-r ) 2n Vol (S n ) 2 (x,y)∈(RP n ) 2 E L d (x) 2 -n t d (x) 2 L d (y) 2 -n t d (y) 2 |dV RP n | 2 .
Using the invariance of the distribution of s d under isometries, we get that:

Var(Vol (Z d ) [2]) = d r Vol (S n-r ) 2 8n 2 Vol (S n ) J n,r (d), (6.8)
where, setting (cf. [16, p. 30]). By a change of variable y = [1 :

x 0 = [1 : 0 : • • • : 0], J n,r (d) = y∈RP n E L d (x 0 ) 2 -n t d (x 0 ) 2 L d (y) 2 -n t d (y) 2 |dV RP n | . Since B RP n x 0 , π 2 = {[1 : z 1 : • • • : z n ] ∈ RP n | z ∈ R n }
z 1 : • • • : z n ],
we have:

J n,r (d) = z∈R n F d (z)(1 + z 2 ) -n+1 2 dz, (6.9)
where

F d (z) = E L d (0) 2 -n t d (0) 2 L d (z) 2 -n t d (z) 2 .
Here, we denoted t d (z) instead of t d ([1 :

z 1 : • • • : z n ]) and L d (z) instead of L d ([1 : z 1 : • • • : z n ]).
Let us fix z ∈ R n and compute F d (z). Using once again the invariance under the action of O n+1 (R) on RP n , we can assume that z = ( z , 0, . . . , 0). Let ∂ ∂x1 , . . . , ∂ ∂xn denote the basis of the tangent space of RP n at [1 : z : 0 : • • • : 0] given by the partial derivatives in our chart ψ x0 (see Sect. 6.1.3). This basis is orthogonal, but

∂ ∂x1 = (1 + z 2 ) -1 and ∂ ∂xj = (1 + z 2 ) -1 2
for all j ∈ {2, . . . , n}. The random vectors t

(i) d (0), t (i) d (z), L i1 d (0), L i1 d (z), . . . , L in d (0), L in d (z) for i ∈ {1, .
. . , r} are independent equidistributed centered Gaussian vector in R 2n+2 . The previous relations, together with Lem. 6.2, show that their common variance matrix, by blocks of size 2 × 2, is:

          A d ( z 2 ) B d ( z 2 ) t 0 • • • 0 B d ( z 2 ) D d ( z 2 ) 0 • • • 0 0 0 C d ( z 2 ) . . . . . . . . . . . . 0 0 0 • • • 0 C d ( z 2 )           , (6.10)
where, for all t 0,

A d (t) = 1 (1 + t) -d 2 (1 + t) -d 2 1 , B d (t) = 0 √ dt(1 + t) -d 2 - √ dt(1 + t) -d 2 0 , C d (t) = 1 (1 + t) 1-d 2 (1 + t) 1-d 2 1 and D d (t) = 1 (1 + t -dt)(1 + t) -d 2 (1 + t -dt)(1 + t) -d 2 1 .
(6.11)

Using the independence and equidistribution of the couples t

(i) d (x), L (i)
d (x) , we have:

F d (z) = r   j,l E L 1j d (0) 2 L 1l d (z) 2 -n l E t (1)
d (0) 2 L 1l d (z) 2 -n j E L 1j d (0) 2 t (1)
d (z) 2 + n 2 E t (1)
d (0) 2 t (1)
d (z) 2   . If (X, Y ) is a centered Gaussian vector in R 2 such that Var(X) = 1 = Var(Y ),
∀t ∈ R, F d (t) = 1 + t d -d 1 + t d -t 2 + (n -1) 1 + t d -2nt + n 2 .
Then, by a change of variable t = d z 2 in Eq. (6.9),

J n,r (d) = d -n 2 r Vol S n-1 +∞ t=0 F d (t)t n-2 2 1 + t d -n+1 2
dt. (6.12)

F d (t)t n-2 2 1 + t d -n+1 2 -----→ d→+∞ t 2 -2t(n + 1) + n(n + 1) t n-2 2 e -t .
Moreover, for all d ∈ N * ,

F d (t)t n-2 2 1 + t d -n+1 2 1 + t d -d t n-2 2 4t 2 + (n + 1)(3t + n) . Let d 0 > n 2 + 2. Since 1 + t d -d is a non-increasing sequence of d, for all d d 0 , F d (t)t n-2 2 1 + t d -n+1 2 1 + t d 0 -d0 t n-2 2 4t 2 + (n + 1)(3t + n) ,
and the right-hand side is integrable as a function of t. By Lebesgue's Theorem, we have:

+∞ t=0 F d (t)t n-2 2 1 + t d n+1 2 dt -----→ d→+∞ +∞ 0 t 2 -2t(n + 1) + n(n + 1) t n-2 2 e -t dt = Γ n 2 + 2 , (6.13)
where Γ is Euler's Gamma function. The conclusion follows from Eq. (6.8), (6.12) and (6.13).

A Technical computations for Section 4

Before proving the technical lemmas of Sect. 4, we state several estimates that will be useful in this section and the next. Recall Def. The following hold as t goes to 0.

a(t) = 1 - t 2 - t 2 8 + O(t 3 ), (b + (t)) 2 = 2 - t 2 + O(t 2 ), (A.3) b + (t)b -(t) = √ t 1 + O(t 2 ) , (b -(t)) 2 = t 2 + t 2 8 + O(t 3 ), (A.4) u 1 (t) = t + O(t 2 ), u 2 (t) = t 2 12 + O(t 3 ), (A.5) v 1 (t) = 2 + O(t), v 2 (t) = t 3 48 + O(t 4 ), (A.6) v 3 (t) = t + O(t 2 ), v 4 (t) = 2 + O(t). (A.7)
Proof of Lem. 4.6. Recall that P is defined by Def. 4.5 and Ω by Eq. (4.3). One can check by a direct computation that, for any t ∈ [0, +∞),

P (t) = (A(t) ⊗ I 2 )σ(Q ⊗ I 2 ), where
A(t) = 1 √ 2 b -(t) -b + (t) b + (t) b -(t) , σ =     1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0     , and Q = 1 √ 2 1 -1 1 1 .
Moreover, these three matrices are orthogonal. Then we have:

σ(Q ⊗ I 2 ) Ω(t)(Q t ⊗ I 2 )σ t = σ     1 -e -1 2 t 0 0 - √ te -1 2 t 0 1 + e -1 2 t √ te -1 2 t 0 0 √ te -1 2 t 1 -(1 -t)e -1 2 t 0 - √ te -1 2 t 0 0 1 + (1 -t)e -1 2 t     σ t =     1 -e -1 2 t - √ te -1 2 t 0 0 - √ te -1 2 t 1 + (1 -t)e -1 2 t 0 0 0 0 1 + e -1 2 t √ te -1 2 t 0 0 √ te -1 2 t 1 -(1 -t)e -1 2 t     = I 4 + e -t 2 1 √ t √ t t -1 ⊗ -1 0 0 1 ,
where I 4 stands for identity matrix of size 4. Recalling the definitions of (v i (t)) 1 i 4 , b + (t) and b -(t) (see Def. 4.4 and 4.5). We conclude the proof by checking that:

A(t) 1 √ t √ t t -1 (A(t)) t =   t 2 -1 + t 2 2 0 0 t 2 + 1 + t 2 2   .
Proof of Lem. 4.8. Let z ∈ R n \ {0}, by Eq. (4.2) and Lem. 4.3, we have:

det (Ω(z)) = det (Ω ′ (z)) r = det Ω( z 2 ) r 1 -e -z 2 r(n-1)
and it is enough to prove that det Ω(t) > 0 whenever t > 0. By Lem. 4.6, we have:

∀t 0, det Ω(t) = v 1 (t)v 2 (t)v 3 (t)v 4 (t) = 1 -(t 2 + 2)e -t + e -2t = f (t), (A.8)
where the last equality defines f : [0, +∞) → R. We have f (0) = 0 and for all t > 0, f ′ (t) = e -t g(t)

where g(t) = t 2 -2t + 2 -2e -t . Then g(0) = 0 and ∀t > 0, g ′ (t) = 2(e -t -1 + t) > 0. Thus g is positive on (0, +∞) and so is f . Finally, we have ∀t > 0, det Ω(t) > 0.

Proof of Lem. 4.13. Let z ∈ R n \ {0}, as above we have:

det (Λ(z)) = det (Λ ′ (z)) r = det Λ( z 2 ) r 1 -e -z 2 r(n-1)
and it is enough to prove that det Λ(t) > 0 whenever t > 0. By Lem. 4.12, we have:

∀t > 0, det Λ(t) = u 1 (t)u 2 (t) = 1 -(t 2 + 2)e -t + e -2t 1 -e -t =
det Ω(t)

1e -t , by Eq. (A.8). We just proved that det Ω(t) is positive for every positive t. Hence the result.

Proof of Lem. 4.14. First, recall that Ω(z) = Ω ′ (z) ⊗ I r (see Eq. (4.2)) and Λ(z) = Λ ′ (z) ⊗ I r (see Eq. (4.5)). Hence, we only need to prove that the map z -→ 0 Λ ′ (z)

1 2 Ω ′ (z) -1 2 is bounded on R n \ {0}. Then, let z ∈ R n \ {0}, the matrix of Ω ′ (z) in the orthonormal basis B z of R 2 ⊗ (R ⊕ R n ) (see Sect.
′ (z) -1 2 in B z is:    Ω( z 2 ) -1 2 0 0 1 e -1 2 z 2 e -1 2 z 2 1 -1 2 ⊗ I n-1    .
where the (A ij ) and (B ij ) are globally independent real standard Gaussian variables, not depending on t. Note that by Lem. 4.13, the Λ ij (t) are positive for any t > 0. We deduce from Lem. 4.12 that for any i ∈ {1, . . . , r}, for all t > 0:

Λ i1 (t) = α(t) β(t) β(t) α(t) and ∀j ∈ {2, . . . , n}, Λ ij (t) = γ(t) δ(t) δ(t) γ(t) ,
where

α(t) = 1 2 u 2 (t) + u 1 (t) , γ(t) = 1 2 1 + e -1 2 t 2 + 1 -e -1 2 t 2 , (A.9) β(t) = 1 2 u 2 (t) -u 1 (t) , δ(t) = 1 2 1 + e -1 2 t 2 -1 -e -1 2 t 2 . (A.10)
We denote A j = (A 1j , . . . , A rj ) t the j-th column of A and similarly

B j = (B 1j , . . . , B rj ) t . Then, E det ⊥ (X(t)) det ⊥ (Y (t)) = E[Ψ(t, A, B)],
where

Ψ(t, A, B) = det ⊥ (α(t)A 1 + β(t)B 1 , γ(t)A 2 + δ(t)B 2 , . . . , γ(t)A n + δ(t)B n ) det ⊥ (β(t)A 1 + α(t)B 1 , δ(t)A 2 + γ(t)B 2 , . . . , δ(t)A n + γ(t)B n ) . (A.11) By (A.5), α(t) = 1 2 √ t + O(t) and β(t) = -1 2 √ t + O(t).
We extend continuously α, β, γ and δ by α(0) = 0 = β(0) and γ(0) = 1 √ 2 = δ(0). The function Ψ also extend continuously at t = 0. Then α, β, γ and δ are bounded functions on (0, 1] and Ψ is the square root of a polynomial of degree 4r in (A, B) whose coefficients are bounded functions of t. In particular, for all t ∈ (0, 1], Ψ(t, A, B) is dominated by a polynomial in (A, B) whose coefficients are independent of t. By Lebesgue's Theorem,

E det ⊥ (X(t)) det ⊥ (Y (t)) ---→ t→0 E[Ψ(0, A, B)] .
(A.12)

Let j ∈ {2, . . . , n}, we define X j = (X 1j , . . . , X rj ) t by X j = γ(0)A j + δ(0)B j = 1 √ 2 (A j + B j ). Then the (X ij ) with i ∈ {1, . . . , r} and j ∈ {2, . . . , n} are independent real standard Gaussian variables. Setting X 1 = (X 11 , . . . , X r1 ) t = 0, we have:

Ψ(0, A, B) = det ⊥ (X 1 , X 2 , . . . , X n ) 2 = det (X 1 , X 2 , . . . , X n )(X 1 , X 2 , . . . , X n ) t = 1 k1<•••<kr n det (X ikj ) 1 i,j r 2 ,
by the Cauchy-Binet formula. Let 1

k 1 < k 2 < • • • < k r n. If k 1 = 1
, the first column of (X ikj ) 1 i,j r is zero and its determinant equals 0. Otherwise,

E det (X ikj ) 1 i,j r 2 = σ,τ ∈Sr ε(σ)ε(τ ) r i=1 E X ik σ(i) X ik τ (i) = r!. (A.13)
Hence, if r < n, we have

E[Ψ(0, A, B)] = r! n -1 r = (n -1)! (n -r -1)! ,
and by Eq. (A.12), we proved Lem. 4.17 in this case. If r = n, we have E[Ψ(0, A, B)] = 0 and we must be more precise.

Let us now assume that r = n. Then, X and Y are square matrices and their Jacobians are simply the absolute values of their determinants. For all t > 0, we have:

Ψ(t, A, B) = t 2 det 2 t α(t)A 1 + 2 t β(t)B 1 , γ(t)A 2 + δ(t)B 2 , . . . , γ(t)A n + δ(t)B n det 2 t β(t)A 1 + 2 t α(t)B 1 , δ(t)A 2 + γ(t)B 2 , . . . , δ(t)A n + γ(t)B n . (A.14)
By Eq. (A.5), we have:

2 t α(t) = 1 √ 2 + O( √ t) and 2 t β(t) = -1 √ 2 + O( √ t).
We can apply the same kind of argument as above. By Lebesgue's Theorem:

2 t E[Ψ(t, A, B)] ---→ t→0 E[|det(Y 1 , X 2 , . . . , X n )| |det(-Y 1 , X 2 , . . . , X n )|] = E det(Y 1 , X 2 , . . . , X n ) 2 ,
where

Y 1 = (Y 11 , . . . , Y r1 ) t = 1 √ 2 (A 1 -B 1 ). Since Y 1 , X 2
, . . . , X n are independent N (Id) in R r , the same computation as Eq. (A.13) shows that:

E det(Y 1 , X 2 , . . . , X n ) 2 = r! = n!. Hence, if r = n, we have: E det ⊥ (X(t)) det ⊥ (Y (t)) = E[Ψ(t, A, B)] ∼ n!
2 t, as t → 0. Proof of Lem. 4.18. For any t > 0, let us denote by:

Λ(t) =    Λ(t) 0 0 1 e -1 2 t 2 e -1 2 t 2 1 ⊗ I n-1    ⊗ I r , (A.15)
the variance matrix of (X(t), Y (t)).

In the following, we denote by L = (X, Y ) a generic element of M rn (R) × M rn (R). We have:

E det ⊥ (X(t)) det ⊥ (Y (t)) = 1 (2π) rn det Λ(t) -1 2 det ⊥ (X) det ⊥ (Y ) exp - 1 2 Λ(t) -1 L , L dL. (A.16)
By Lem. 4.12, we have Λ(t) = Id +O te -t 2 as t → +∞. Then, det Λ(t)

-1 2 = 1 + O te -t 2 . Moreover, by the Mean Value Theorem, exp - 1 2 Λ(t) -1 L , L -e -1 2 L 2 = e -1 2 L 2 exp - 1 2 Λ(t) -1 -Id L , L -1 e -1 2 L 2 L 2 2 Λ(t) -1 -Id exp L 2 2 Λ(t) -1 -Id .
Then, since Λ(t) -1 = Id +O te -t

2

, this last term is smaller than e -1 4 L 2 L 2 2 Λ(t) -1 -Id for all t large enough. Hence,

det ⊥ (X) det ⊥ (Y ) exp - 1 2 Λ(t) -1 L , L -e -1 2 L 2 dL 1 2 Λ(t) -1 -Id det ⊥ (X) det ⊥ (Y ) L 2 e -1 4 L 2 dL = O te -t 2 .
Thanks to this relation and Eq. (A.16), we get that:

E det ⊥ (X(t)) det ⊥ (Y (t)) = E det ⊥ (X(∞)) det ⊥ (Y (∞)) + O te -t 2 , where (X(∞), Y (∞)) ∼ N (Id) in M rn (R) × M rn (R). Finally, by [21, Lem. A.14], E det ⊥ (X(∞)) det ⊥ (Y (∞)) = E det ⊥ (X(∞)) 2 = (2π) r Vol (S n-r ) Vol (S n ) 2 .

B Technical computations for Section 5

Proof of Lem. 5.26. Let α ∈ (0, 1), we want to prove that Θ(z)

-1 2 Θ d (z)Θ(z) -1 2 -Id = O(d -α ) uniformly for x ∈ M and z ∈ B TxM (0, b n ln d). Recall that Q = 1 √ 2 1 -1 1 1 . Since Q ∈ O 2 (R), it is equivalent to prove that: Q ⊗ Id R(E⊗L d ) x Θ(z) -1 2 (Θ d (z) -Θ(z)) Θ(z) -1 2 Q ⊗ Id R(E⊗L d ) x -1 = O(d -α ). (B.1)
Recall that e d was defined by Eq. (3.1) and that e ∞ = ξ Id R(E⊗L d ) x (see Sect. 4). For any d ∈ N, for all x ∈ M and for all w, z ∈ B TxM (0, b n ln d) we set: ε d (w, z) = e d (w, z)e ∞ (w, z). By Eq. (4.1) and (5.7) we have:

Θ d (z) -Θ(z) = ε d (0, 0) ε d (0, z) ε d (z, 0) ε d (z, z) .
Then, by Lem. 4.1, for all x ∈ M and z ∈ B TxM (0, b n ln d) \ {0} we have:

Q ⊗ Id R(E⊗L d ) x Θ(z) -1 2 (Θ d (z) -Θ(z)) Θ(z) -1 2 Q ⊗ Id R(E⊗L d ) x -1 = a d (z) b d (z) * b d (z) c d (z) ,
where

a d (z) = 1 2 1 -e -1 2 z 2 -1 (ε d (z, z) -ε d (z, 0) -ε d (0, z) + ε d (0, 0)), b d (z) = - 1 2 1 -e -z 2 -1/2 (ε d (z, z) -ε d (z, 0) + ε d (0, z) -ε d (0, 0)), and
c d (z) = 1 2 1 + e -1 2 z 2 -1 (ε d (z, z) + ε d (z, 0) + ε d (0, z) + ε d (0, 0)).
Let β ∈ (α, 1), by Prop. 3.4 we have D 2 (w,z) ε d Cd -β , where C is independent of x ∈ M and w, z ∈ B TxM (0, b n ln d). Then, a second order Taylor expansion around (0, 0) gives:

ε d (z, z) -ε d (z, 0) -ε d (0, z) + ε d (0, 0) C z 2 d -β . Since we consider z ∈ B TxM (0, b n ln d) and 1 -e -1 2 z 2 ∼ z 2 2
as z → 0, we have:

a d (z) C z 2 d -β 2 1 -e -1 2 z 2 = O (ln d) 2 d -β = O(d -α ),
where the error term does not depend on (x, z). We obtain Eq. (B.1) by reasonning similarly for b d (z) and c d (z).

Proof of Lem. 5.28. The idea of the proof is the same as that of Lem. 5.26 above. Let α ∈ (0, 1), we want to prove that:

Ω(z) -1 2 (Ω d (z) -Ω(z)) Ω(z) -1 2 = O d -α . (B.2)
Recall that we defined: ε d (w, z) = e d (w, z)e ∞ (w, z) for any x ∈ M and w, z ∈ B TxM (0, b n ln d).

We can express Ω d (z) -Ω(z) in terms of ε d and its derivatives. Then we write the matrix of the left-hand side of Eq. (B.2) in an orthonormal basis that diagonalizes Ω(z). The coefficients of this matrix are linear combinations of ε d and its derivatives. We will prove that they are O(d -α ) using Taylor expansions and the estimates of Sect. 3.3. The details are longer than in the proof of Lem. 5.26 for two reasons. First, the basis in which Ω(z) is diagonal now depends on z. Second, some of the eigenvalues of Ω(z) are O( z 6 ) as z → 0, so that we need to consider Taylor expansions of order 6 for some coefficients. In addition, the matrices involved are less easily described than in the proof of Lem. 5.26.

Recall that e d was defined by Eq. (3.1) and that e ∞ = ξ Id R(E⊗L d ) x (see Sect. 4). We expressed Ω(z) in terms of e ∞ in Eq. (4.2) and Ω d (z) in terms of e d in Eq. (5.8). As an operator on:

R E ⊗ L d x ⊕ R E ⊗ L d x ⊕ T * x M ⊗ R E ⊗ L d x ⊕ T * x M ⊗ R E ⊗ L d x , we have: Ω d (z) -Ω(z) =      ε d (0, 0) ε d (0, z) ∂ ♯ y ε d (0, 0) ∂ ♯ y ε d (0, z) ε d (z, 0) ε d (z, z) ∂ ♯ y ε d (z, 0) ∂ ♯ y ε d (z, z) ∂ x ε d (0, 0) ∂ x ε d (0, z) ∂ x ∂ ♯ y ε d (0, 0) ∂ x ∂ ♯ y ε d (0, z) ∂ x ε d (z, 0) ∂ x ε d (z, z) ∂ x ∂ ♯ y ε d (z, 0) ∂ x ∂ ♯ y ε d (z, z)      .
Let us choose an orthonormal basis ∂ ∂x1 , . . . , ∂ ∂xn of T x M such that z = z ∂ ∂x1 . We denote by (dx 1 , . . . , dx n ) its dual basis. We can then define a basis of R 2 ⊗ (R ⊕ T *

x M ) similar to B z (see Sect. 4.2). For any i ∈ {1, . . . , n}, we denote by ∂ xi (resp. ∂ yi ) the partial derivative with respect to the i-th component of first (resp. second) variable for maps from T x M ×T x M to End R E ⊗ L d

x . Then we can split Ω d (z) -Ω(z) according to the previous basis in the following way: Let us denote by P(z) the operator whose matrix in our basis is:

Ω d (z) -Ω(z) =       A d (z) B
P ( z 2 ) 0 0 Q ⊗ I n-1 ⊗ I r ,
where P was defined by Def. 4.5 and Q = 1 ), where the constant involved in this notation is independent of (x, z). The main difficulty comes the fact that v 2 and v 3 converge to 0 as z → 0 (see Eq. (A.6) and (A.7)).

d (z) = 1 4 v 2 ( z 2 ) -1 ×        (b -( z 2 )) 2 ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) + ∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) -b + ( z 2 )b -( z 2 ) (∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) + ∂ x1 ε d (0, z) -∂ x1 ε d (0, 0)) -b + ( z 2 )b -( z 2 ) ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) -∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) + (b + ( z 2 )) 2 (ε d (z, z) -ε d (z, 0) -ε d (0, z) + ε d (0, 0))        , (B.16) b (1) d (z) = 1 4 v 1 ( z 2 )v 3 ( z 2 ) -1 2 ×        -(b + ( z 2 )) 2 ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) -∂ x1 ∂ ♯ y1 ε d (0, 0) -b + ( z 2 )b -( z
d (z) = 1 4 v 2 ( z 2 )v 3 ( z 2 ) -1 2 ×        b + ( z 2 )b -( z 2 ) ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) -∂ x1 ∂ ♯ y1 ε d (0, 0) -(b + ( z 2
d (z) = 1 4 v 2 ( z 2 )v 4 ( z 2 ) -1 2 ×        -(b -( z 2 )) 2 ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) -∂ x1 ∂ ♯ y1 ε d (0, 0) + b + ( z 2 )b -( z 2 ) (∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) + ∂ x1 ε d (0, 0)) + b + ( z 2 )b -( z 2 ) ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) + ∂ ♯ y1 ε d (0, 0) -(b + ( z 2 )) 2 (ε d (z, z) -ε d (z, 0) + ε d (0, z) -ε d (0, 0))        , (B.
d (z) = 1 4 v 3 ( z 2 )v 4 ( z 2 ) -1 2 ×        -b + ( z 2 )b -( z 2 ) ∂ x1 ∂ ♯ y1 ε d (z, z) -∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) -(b -( z 2 )) 2 (∂ x1 ε d (z, z) + ∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) -∂ x1 ε d (0, 0)) + (b + ( z 2 )) 2 ∂ ♯ y1 ε d (z, z) -∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) + b + ( z 2 )b -( z 2
The term with the worst apparent singularity at z = 0 is a Recall also that we chose our coordinates so that z = ( z , 0, . . . , 0). Using Taylor expansions around (0, 0) for ε d and its derivatives, we get:

∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) + ∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) = 4∂ x1 ∂ ♯ y1 ε d (0, 0) + 2 z ∂ 2 x1 ∂ ♯ y1 ε d (0, 0) + ∂ x1 ∂ ♯

Lemma 2 . 4 .

 24 For every d d 1 , Z s d is almost surely a smooth closed codimension r submanifold of M .

Notation 3 . 1 .

 31 In the following, B A (a, R) denotes the open ball of center a and radius R in the metric space A.

1 2

 1 r and we denote by Λ(z) its positive square root.

Lemma 4 . 16 .

 416 Let z ∈ R n \ {0}. Let ∂ ∂x1 , . . . , ∂ ∂xn be an orthonormal basis of R n such that z = z ∂ ∂x1 and let (ζ 1 , . . . , ζ r ) be any orthonormal basis of R r . Then, the matrices of d 0 s and d z s in these bases, given that s

  and 4.18 and the definition of D n,r (Def. 4.15) allows to derive the following. Corollary 4.19. Let n ∈ N * and r ∈ {1, . . . , n}, then we have:

Theorem 5 . 1 (

 51 Kac-Rice formula 1). Let d d 1 , let ∇ d be any real connection on E ⊗ L d and let

Theorem 5 . 5 (

 55 Kac-Rice formula 2). Let d d 2 , let ∇ d be any real connection on E ⊗ L d and let

  stands for the product measure on M 2 (resp. (Z d ) 2 ) induced by |dV M | (resp. |dV d |). The expectation on the right-hand side of Eq. (5.2) is the conditional expectation of det ⊥ ∇ d x s d det ⊥ ∇ d y s d given that ev d x,y (s d ) = 0. Proof. This formula was proved in [22, Thm. 4.4] in the case 1 r < n. The hypothesis r < n does not appear in the proof and can be changed to r n without any other modification. Definition 5.6. Let d max(d 1 , d 2 ), and let ∇ d be any real connection on E ⊗ L d . We denote by D d : M 2 \ ∆ → R the map defined by:

√d.Definitions 5 . 9 . 1 C2 n 2 + 1

 59121 More precisely, let us now fix the value of the constant b n appearing in Prop. 3.4. We set b n = , where C 2 > 0 is the constant appearing in the exponential in Thm. 3.6. Moreover, for all d ∈ N * , we denote:

1 2 ,

 2 and by Thm. 3.3 E d (x, x) = d n n Id +O(d -1 ) . Corollary 5.11. There exists d 1 ∈ N, such that for all d d 1 , for all

y) , the variance of the centered Gaussian vector π d n 2 (

 2 s d (x), s d (y)). Note that, by Lem. 5.4, for all d d 2 , ev d x,y (ev d x,y ) * is non-singular, i.e. (s d (x), s d (y)) is nondegenerate and Θ d (x, y) is non-singular. Let d ∈ N and (x, y) ∈ M 2 \ ∆, we denote by j d x,y : s → s(x), s(y), ∇ d x s, ∇ d y s the evaluation of the 1-jets at (x, y). Then j d x,y (s d ) = s d (x), s d (y), ∇ d x s d , ∇ d y s d is a centered Gaussian vector in:

Finally, we needDefinition 5 . 18 .

 518 to consider the distribution of the 1-jets of s d at x and y ∈ M , when the distance between x and y is of order 1 √ d . As in Sect. 3, let R > 0 be such that 2R is less than the injectivity radius of X . There exists d 3 ∈ N such that, for all d d 3 , b n ln d √ d R. Let d d 3 and let (x, y) ∈ ∆ d \ ∆. Using the real normal trivialization of E ⊗ L d around x (see Sect. 3.1), we can see π d n 2 (s d (x), s d (y)) as a random vector in R E ⊗ L d x ⊕ R E ⊗ L d x . Since the distance from x to y is smaller than the injectivity radius of M , we can write y as exp x z √ d for some z ∈ T x M . Moreover z = √ d ρ g (x, y) < b n ln d. Let d d 3 , let x ∈ M and let z ∈ B TxM (0, b n ln d) \ {0}, we set:

Definition 5 . 20 .

 520 5.14) and e d show that, for all d d 3 , for all x ∈ M and for all z ∈ B TxM (0, b n ln d) \ {0}: Θ d (z) = e d (0, 0) e d (0, z) e d (z, 0) e d (z, z) . (5.7) We can define Ω d (z) and Λ d (z) similarly and express them in terms of e d and its derivatives. Let d d 3 , let x ∈ M and let z ∈ B TxM (0, b n ln d) \ {0}, we set:

. 8 )

 8 Definition 5.21. Let d max(d 2 , d 3 ), let x ∈ M and let z ∈ B T * x M (0, b n ln d) \ {0}, we set:

  we have ρ g (x, y) b n ln d √ d . With our choice of b n (see Def. 5.9), the error term in Thm. 3.6 is then O(d n-k 2 -1 ), uniformly on M 2 \ ∆ d . Thus, by Thm. 3.6,

Definition 5 . 24 .

 524 Let d max(d 1 , d 2 , d 3 ), let x ∈ M and let z ∈ B TxM (0, b n ln d) \ {0}, we define:

  d x,y (ev d x,y ) * = 0. By Eq. (5.6) and the definition of Θ d (Def. 5.14), det ev d x,y (ev d x,y ) * = d π 2rn det (Θ d (x, y)) , so we have to show that det (Θ d (x, y)) does not vanish on M 2 \ ∆, for d large enough. By Lem. 5.23 and Thm. 3.3, det (Θ d (x, y)) = 1 + O d -1 , (5.13) uniformly on M 2 \ ∆ d . Let (x, y) ∈ ∆ d \ ∆ and let us assume that d d 3 so that we can write y as exp x z √ d with z ∈ B TxM (0, b n ln d) \ {0}. Then, by Lem. 5.26 and 4.8,

5 . 25 .

 525 Recall the definitions of D n,r (t) (Def. 4.15), D d (x, y) (Def. 5.6) and D d (x, z) (Def. 5.24). Proof of Prop. 5.25. Let us fix α ∈ (0, 1). Let x ∈ M and z ∈ B TxM (0, b n ln d) \ {0}, we set y = exp x z √ d . We have defined Θ d (z) and Λ d (z) so that:

Notations 6 . 1 .

 61 n are of the form O(d) = (O(1)) ⊗d with d ∈ N * (see[START_REF] Ph | Principles of algebraic geometry[END_REF] Sect. 1.1]). The complex conjugation and the usual Hermitian inner product of C n+1 induce compatible real and metric structures on O(-1), hence on O(1) by duality. The resulting Hermitian metric on O(1) is positive and its curvature is the Fubini-Study Kähler form on CP n . With our choice of normalization, the induced Riemannian metric is the quotient of the Euclidean metric on S 2n+1 ⊂ C n+1 . Finally, we choose E to be the rank r trivial bundle C r × CP n → CP n , with the compatible real and metric structures inherited from the standard ones in C r . Let α = (α 0 , . . . , α n ) ∈ N n+1 , we denote its length by |α|

d

  (x) and L ij d (x) depend on the choice of the unit vector ζ 0 (x), that we used to trivialize O(d) x . The only other choice of such a unit vector is -ζ 0 (x). Changing ζ 0 (x) to -ζ 0 (x) changes t

d

  (x) and L ij d (x) to -L ij d (x). Since we only consider multi-indices (α, β) such that |α| + |β| is even, the monomials appearing with a non-zero coefficient in H α (t d (x)) H β (L d (x)) have even total degree. Hence, the value of H α (t d (x)) H β (L d (x)) does not depend on the choice of ζ 0 (x).

( 1 )

 1 d (x) ∼ N (Id) in a Euclidean space of dimension n, we have:

Remark 6 .

 6 [START_REF] Kratz | Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes[END_REF]. In[START_REF] Dalmao | Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials[END_REF],Dalmao proved that for n = r = 1, we have Var(Vol (Z d )) ∼ σ 2 √ d with σ 2 ≃ 0.57 • • • . What we just said shows that σ 2 = 1 + I 1,1 , and the lower bound we get for this term in the proof of Thm. 1.8 is 3 8 √ π ≃ 0.21 • • • . Thus, asymptotically, chaotic components of order greater than 4 must contribute to the leading term of Var(Vol (Z d )).

  4.4, 4.5 and 4.11. The following hold as t goes to infinity. , 2}, u i (t) ----→ t→+∞ 1 and ∀i ∈ {1, 2, 3, 4}, v i (t) ----→

  4.2) is given by Lem. 4.3, and the matrix of Ω

A

  d (z) = ε d (0, 0) ε d (0, z) ε d (z, 0) ε d (z, z) , (B.4) ∀i ∈ {1, . . . , n}, B (i) d (z) = ∂ xi ε d (0, 0) ∂ xi ε d (0, z) ∂ xi ε d (z, 0) ∂ xi ε d (z, z) , (B.5) ∀i, j ∈ {1, . . . , n}, C (ij) d (z) = ∂ xi ∂ ♯ yj ε d (0, 0) ∂ xi ∂ ♯ yj ε d (0, z) ∂ xi ∂ ♯ yj ε d (z, 0) ∂ xi ∂ ♯ yj ε d (z, z) . (B.6)

√ 2 1 - 1 1 1 .

 111 Since P(z) is orthogonal, (B.2) is equivalent to the following:P(z)Ω(z) -1 2 (Ω d (z) -Ω(z)) Ω(z) -1 2 P(z) -1 = O d -α . (B.7)By Lem. 4.7, the matrix of P(z)Ω(z) r .On the other hand, by Eq. (B.3), P(z) (Ω d (z) -Ω(z)) P(z)

2 )-b + ( z 2 )b -( z 2 )

 222 t ⊗ Id , (B.8) ∀i ∈ {2, . . . , n}, B(i)d (z) C (i1) d (z) = (Q ⊗ Id) B (i) d (z) C (i1) d (z) P ( z 2 ) t ⊗ Id , (B.9) ∀i, j ∈ {2, . . . , n}, C (ij) d (z) = (Q ⊗ Id)C (ij) d (z)(Q t ⊗ Id). (B.10) ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) + ∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) -(b -( z 2 )) 2 (∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) + ∂ x1 ε d (0, z) -∂ x1 ε d (0, 0)) + (b + ( z 2 )) 2 ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) -∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) + b + ( z 2 )b -( z 2 ) (ε d (z, z)ε d (z, 0)ε d (0, z) + ε d (0, 0))

2 )b -( z 2 )

 22 2 ) (∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) + ∂ x1 ε d (0, 0)) b + ( z 2 )b -( z 2 ) ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) + ∂ ♯ y1 ε d (0, 0) -(b -( z 2 )) 2 (ε d (z, z)ε d (z, 0) + ε d (0, z)ε d (0, 0)) ∂ x1 ∂ ♯ y1 ε d (z, z) + ∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) -∂ x1 ∂ ♯ y1 ε d (0, 0) + (b -( z 2 )) 2 (∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) + ∂ x1 ε d (0, 0)) -(b + ( z 2 )) 2 ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) + ∂ ♯ y1 ε d (0, 0) b + ( z 2 )b -( z 2 ) (ε d (z, z)ε d (z, 0) + ε d (0, z)ε d (0, 0))

)) 2 (

 2 ∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) + ∂ x1 ε d (0, 0)) + (b -( z 2 )) 2 ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) + ∂ ♯ y1 ε d (0, 0) b + ( z 2 )b -( z 2 ) (ε d (z, z)ε d (z, 0) + ε d (0, z)ε d (0, 0))

  ( z 2 )) 2 ∂ x1 ∂ ♯ y1 ε d (z, z) -∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) + b + ( z 2 )b -( z 2 ) (∂ x1 ε d (z, z) + ∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) -∂ x1 ε d (0, 0)) + b + ( z 2 )b -( z 2 ) ∂ ♯ y1 ε d (z, z) -∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) + (b -( z 2 )) 2 (ε d (z, z) + ε d (z, 0) + ε d (0, z) + ε d (0, 0))

.

  ) (ε d (z, z) + ε d (z, 0) + ε d (0, z) + ε d (0, 0)) ( z 2 )) 2 ∂ x1 ∂ ♯ y1 ε d (z, z) -∂ x1 ∂ ♯ y1 ε d (z, 0) -∂ x1 ∂ ♯ y1 ε d (0, z) + ∂ x1 ∂ ♯ y1 ε d (0, 0) b + ( z 2 )b -( z 2 ) (∂ x1 ε d (z, z) + ∂ x1 ε d (z, 0) -∂ x1 ε d (0, z) -∂ x1 ε d (0, 0)) b + ( z 2 )b -( z 2 ) ∂ ♯ y1 ε d (z, z) -∂ ♯ y1 ε d (z, 0) + ∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) + (b + ( z 2 )) 2 (ε d (z, z) + ε d (z, 0) + ε d (0, z) + ε d (0, 0)) (B.23) We need to prove that each one of the terms (B.14) to (B.23) is a O(d -α

( 3 )

 3 d (see(B.16)). We will show below that a (3) d (z) = O(d -α ) uniformly in (x, z). The proofs that the other nine coefficients are O(d -α ) follow the same lines, and they are strictly easier technically. We leave them to the reader. By Eq. (A.6), v 2 ( z 2 ) ∼ z 6 48 as z → 0. Hence, we have to expand the second factor in (B.16) up to a O( z 6 ). Let β ∈ (α, 1), recall that, by Prop. 3.4, the partial derivatives of ε d of order up to 6 are O(d -β ) uniformly on B TxM (0, b n ln d) × B TxM (0, b n ln d).

y1 2 ε4+ z 5 O 5 O 2 ∂+ z 6 O

 25526 d (0, 0) + z 2 ∂ 3 x1 ∂ ♯ y1 ε d (0, 0) + ∂ 2 x1 ∂ ♯ y1 2 ε d (0, 0) + ∂ x1 ∂ ε d (0, 0) + z 4 O(d -β ), (B.24) ∂ x1 ε d (z, z) -∂ x1 ε d (z, 0) + ∂ x1 ε d (0, z) -∂ x1 ε d (0, 0) = 2 z ∂ x1 ∂ ♯ y1 ε d (0, 0) + z 2 ∂ 2 x1 ∂ ♯ y1 ε d (0, 0) + ∂ x1 ∂ (d -β ), (B.25) ∂ ♯ y1 ε d (z, z) + ∂ ♯ y1 ε d (z, 0) -∂ ♯ y1 ε d (0, z) -∂ ♯ y1 ε d (0, 0) = 2 z ∂ x1 ∂ ♯ y1 ε d (0, 0) + z 2 ∂ 2 x1 ∂ ♯ y1 ε d (0, 0) + ∂ x1 ∂ (d -β ), (B.26) ε d (z, z)ε d (z, 0)ε d (0, z) + ε d (0, 0) = z x1 ∂ ♯ y1 ε d (0, 0) + z (d -β ). (B.27) Now, we can combine Eq. (B.24), (B.25),(B.26) and (B.27) with the expansions around 0 of (b + ( z 2 )) 2 (cf. Eq. (A.3)), (b -( z 2 )) 2 and b + ( z 2 )b -( z 2 ) (cf. Eq. (A.4)). Using Prop. 3.4

  1.6 and Thm. 1.8 respectively. Note that in Sect.[START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF] we have to study in details the model of Kostlan-Shub-Smale polynomials, which is the simplest example of our general real algebraic setting. We conclude this paper by two appendices, App. A and App. B, in which we gathered the proofs of the technical lemmas of Sect.[START_REF] Bayraktar | Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights[END_REF] and Sect. 5 respectively. 2 Random real algebraic submanifolds 2.1 Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Random real algebraic submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The correlation kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Estimates for the Bergman kernel 3.1 Real normal trivialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Near-diagonal estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Off-diagonal estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variance of the values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Variance of the 1-jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Conditional variance of the derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Finiteness of the leading constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermite polynomials and Wiener chaos . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Wiener-Ito expansion of the linear statistics . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	4 Properties of the limit distribution
	4.1 A Technical computations for Section 4
	B Technical computations for Section 5

5 Proof of Theorem 1.6 5.1 Kac-Rice formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Expression of some covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Uncorrelated terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Correlated terms far from the diagonal . . . . . . . . . . . . . . . . . . . . . 5.2.3 Correlated terms close to the diagonal . . . . . . . . . . . . . . . . . . . . . 5.3 Far off-diagonal term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Near-diagonal term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Proof of Theorem 1.8 6.1 Kostlan-Shub-Smale polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Correlation kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Local expression of the kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2

  1 , so that Z s d is almost surely a random smooth closed submanifold of M of codimension r. For simplicity, we denote Z d = Z s d and |dV d | = |dV s d |. Let φ ∈ C 0 (M ) and d d 1 , the real random variable |dV d | , φ is called the linear statistic of degree d associated with φ. For example, |dV d | , 1 is the volume of Z d .

  Thus, the Bergman kernel of E ⊗ L d gives the correlations between the values of the random section s d . By taking partial derivatives of this relation, we obtain the correlations between the values of s d and its derivatives. More details about what follows can be found in [22, Sect. 2.3]. Let ∇ d be a metric connection on E ⊗L d , it induces a dual connection (∇ d ) * on (E ⊗L d ) * , which is compatible with the metric (cf. [16, Sect. 0.5]). We can then define a natural metric connection ∇ d 1 on P ⋆ 1 (E ⊗ L d ) → X × X whose partial derivatives are: ∇ d with respect to the first variable, and the trivial connection with respect to the second. Similarly, (∇ d ) * induces a metric connection ∇ d 2 on P ⋆ 2 (E ⊗ L d ) * and ∇ d 1 ⊗ Id + Id ⊗∇ d 2 is a metric connection on (E ⊗ L d ) ⊠ (E ⊗ L d ) * . We denote by ∂ x (resp. ∂ y ) the partial derivative of ∇ d

	(resp. second) variable. Let ∂ ♯ y E d (x, y) ∈ T y X ⊗ E ⊗ L d	1 ⊗ Id + Id ⊗∇ d 2 with respect to the first x ⊗ E ⊗ L d * y be defined by:

  .3) Since Z d has almost surely dimension nr, the diagonal in Z d × Z d is negligible if and only if r < n. Moreover, if r = n then both |dV d | and |dV d | 2 are counting measures. Then,

  -1 ), since 1 4 (b n ln d) 2 ln d for d large enough. By Eq. (5.10), (5.11) and (

	r ( z	2 ) dz, (5.11)
	where ̟ φ2 is the continuity modulus of φ 2 (see Def. 1.3). Besides, by Cor. 4.19,
	(bn ln d) 2

z∈BT xM (0,bn ln d)

D n,r ( z 2 ) dz = Vol S n-1 1 2 t=0 D n,r (t)t n-2 2 dt = Vol S n-1 I n,r + O e -1 4 (bn ln d) 2 ,

(5.12)

and the error term is O(d

1 .

 1 The variance of s ′ d (x) does not depend on x ∈ RP n . 2. For all x ∈ RP n , s ′ does not depend on i, nor on our choice of orthonormal basis, nor on x ∈ RP n .

	d (x) and ∇ d x s ′ d are independent.
	3. If ∂ ∂x1 , . . . , ∂ ∂xn is any orthonormal basis of T x RP n , then	∂s ′ d ∂xi (x) and	∂s ′ d ∂xj (x) are independent
	if i = j. Moreover, the variance of	∂s ′ d ∂xi (x)

  Thm. 1.8. Let us consider the common zero set Z d of r independent KSS polynomials in RP n , and denote by |dV d | the Riemannian volume measure on Z d . Let 1 be the unit constant function on RP n , we have |dV d | , 1 = Vol (Z d ) and, by Thm. 1.6,

  has full measure in RP n , we can restrict the above integral to this ball and use the local coordinates introduced in Sect. 6.1.3. These coordinates are centered at x 0 . Moreover, the density of |dV RP n | with respect to the Lebesgue measure in this chart is z → (1 + z 2 ) -n+1 2

Acknowledgments. We wish to thank Federico Dalmao and Maurizia Rossi for interesting discussions about Wiener-Itō expansions, and Xiaonan Ma for helpful discussions about Bergman kernels and for having encouraged our collaboration. Thomas Letendre also thanks Damien Gayet for suggesting this problem in the first place, and for his support.

Thomas Letendre was supported by the French National Research Agency through the ANR grants SpInQS (ANR-17-CE40-0011) and UniRaNDom (ANR-17-CE40-0008). Martin Puchol was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

1 d → X (see Eq. (2.

Contents 1 Introduction 1

Similarly, by Lem. 4.9, the matrix of 0 Λ ′ (z)

in B z is:

Hence, our problem reduces to proving that: t -→ 0 Λ(t)

Ω(t) -1 2 is bounded on (0, +∞). Recall that, for all t ∈ [0, +∞), P (t) ∈ O 4 (R) was defined by Def. 4.5. By Lem. 4.6 and 4.12, for all t ∈ (0, +∞) we have:

where:

By Lem. 4.8, for all t > 0, the (v i (t))

as to goes to 0. Hence for all i ∈ {1, . . . , 6}, m i is a bounded function from (0, +∞) to R, which concludes the proof.

Proof of Lem. 4.17. Recall that, for all t > 0, the couples (X ij (t), Y ij (t)) are independent centered Gaussian vectors in R 2 . We denote by Λ ij (t) the variance matrix of (X ij (t), Y ij (t)), which equals

otherwise (see Def. 1.5, Lem. 4.16 and Lem. 4.9). For all i ∈ {1, . . . , r}, j ∈ {1, . . . , n} and t > 0, we can write:

Then, in order to prove Eq. (B.7), we have to prove that:

Since these are heavy computations, we do not reproduce them in totality here. In the following, we give some details about the proof of (B.11), which is the most difficult of these three relations to establish. The proofs of (B.12) and (B.13) are similar and left to the fearless reader.

Let us focus on the proof of (B.11). We denote once again, we obtain:

where we used Eq. (A.2), (A.6) and the fact that z b n ln d. This concludes the proof for a

d . As we already explained, we proceed similarly for the other nine coefficients to get (B.11), and the same kind of computations yields (B.12) and (B.13).

Proof of Lem. 5.30. Let α ∈ (0, 1), x ∈ M and z ∈ B TxM (0, b n ln d) \ {0}. We will denote by

We also set χ(L) = det ⊥ (X) det ⊥ (Y ) . We have:

), and these estimates are uniform in (x, z). As in the proof of Lem. 4.18, by the Mean Value Theorem, for all L we have:

Recall that, by Lem. 4.12, the eigenvalues of the positive symmetric operator Λ(z) are u 1 ( z 2 ), u 2 ( z 2 ), 1 + exp -1 2 z 2 and 1exp -1 2 z 2 , with some multiplicities. These are bounded functions of z (see Eq. (A.2) and (A.5)). Hence, χ Λ(z) 1 2 L is the square root of a polynomial in L whose coefficients are bounded functions of z. Thus, the integral on the right-hand side of Eq. (B.29) is bounded, independently of (x, z). We get:

Finally, by Eq. (B.28), we find

, is a positive function of z. By Lem. 4.16,

and by Lem. 4.17 and 4.18, if r < n, this quantity admits positive limits when z goes to 0 or z goes to +∞. Thus, in this case,

) is bounded from below by positive constant, independent of (x, z). Then, Eq. (B.30) shows that:

and this concludes the proof for r < n.

If r = n, the leading term in Eq. (B.30) goes to 0 as z → 0, so that we need to be more precise. From now on, we assume that r = n. Let us assume for now that, in this case, we have:

as z → 0, where the constant involved in the O z 2 is uniform in (x, z). Then, proceeding as we did in the case r < n, we get the following equivalent of Eq. (B.30):

By Lem. 4.17,

as z → 0. Hence,

uniformly for x ∈ M and z 1. In the domain z

) is bounded from below by a positive constant independent of (x, z), and we proceed as in the case r < n, using Eq. (B.30). This yields the result for r = n.

To conclude the proof, we still have to prove that (B.31) holds when r = n. Let us write L = (A, B) and Λ(z)

We choose any orthonormal basis of R E ⊗ L d

x and an orthonormal basis of T x M such that the coordinates of z are ( z , 0, . . . , 0). We denote by (A ij ), (B ij ), (X ij (z)) and (Y ij (z)) ∈ M rn (R) the matrices of A, B, X(z) and Y (z) in these bases.

The matrix of Λ(z) in the basis defined by B ′ z (see Sect. 4.3) and our basis of

, where Λ was defined by Eq. (A.15). That is, using the same notations as in the proof of Lem. 4.17 (see Eq. (A.9) and (A.10)), for all i ∈ {1, . . . , r}:

Hence, we have:

where Ψ was defined by Eq. (A.11). Recall that Ψ satisfies (A.14) when r = n. As in the proof of Lem. 4.17 (cf. App. A), by Lebesgue's Theorem we have:

where A j (resp. B j ) denotes the j-th column of the matrix of A (resp. B) and L = (A, B). This limit is finite, which proves that (B.31) is satisfied and concludes the proof.