Thomas Letendre
email: thomas.letendre@ens-lyon.fr.

Expected volume and Euler characteristic of random submanifolds

Keywords: Euler characteristic, Riemannian random wave, spectral function, random polynomial, real projective manifold, ample line bundle, Bergman kernel, Gaussian field Mathematics Subject Classification 2010: 14P25, 32A25, 34L20, 60D05, 60G60

In a closed manifold of positive dimension n, we estimate the expected volume and Euler characteristic for random submanifolds of codimension r ∈ {1, . . . , n} in two different settings. On one hand, we consider a closed Riemannian manifold and some positive λ. Then we take r independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than λ and consider the random submanifold defined as the common zero set of these r functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as λ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle L and a rank r holomorphic vector bundle E that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of E ⊗ L d as d goes to infinity. The same techniques apply to both settings.

Introduction

Zeros of random polynomials were first studied by Bloch and Pòlya [START_REF] Bloch | On the roots of certain algebraic equations[END_REF] in the early 30s. About ten years later, Kac [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF] obtained a sharp asymptotic for the expected number of real zeros of a polynomial of degree d with independent standard Gaussian coefficients, as d goes to infinity. This was later generalized to other distributions by Kostlan in [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF]. In particular, he introduced a normal distribution on the space of homogeneous polynomials of degree d -known as the Kostlan distribution -which is more geometric, in the sense that it is invariant under isometries of CP 1 . Bogomolny, Bohigas and Leboeuf [START_REF] Bogomolny | Quantum chaotic dynamics and random polynomials[END_REF] showed that this distribution corresponds to the choice of d independent roots, uniformly distributed in the Riemann sphere.

In higher dimension, the question of the number of zeros can be generalized in at least two ways. What is the expected volume of the zero set? And what is its expected Euler characteristic? More generally, one can ask what are the expected volume and Euler characteristic of a random submanifold obtained as the zero set of some Gaussian field on a 1 Riemannian manifold. In this paper, we provide an asymptotic answer to these questions in the case of Riemannian random waves and in the case of real algebraic manifolds.

Let us describe our frameworks and state the main results of this paper. See section 2 for more details. Let (M, g) be a closed (that is compact without boundary) smooth Riemannian manifold of positive dimension n, equipped with the Riemannian measure |dV M | associated to g (defined below (2.1)). This induces a L 2 -inner product on C ∞ (M ) defined by:

(1.1) ∀φ, ψ ∈ C ∞ (M ), φ , ψ = x∈M φ(x)ψ(x) |dV M | .
It is well-known that the subspace V λ ⊂ C ∞ (M ) spanned by the eigenfunctions of the Laplacian associated to eigenvalues smaller than λ has finite dimension. Let 1 r n and let f (1) , . . . , f (r) ∈ V λ be independent standard Gaussian vectors, then we denote by Z f the zero set of f = (f (1) , . . . , f (r) ). Then, for λ large enough, Z f is almost surely a submanifold of M of codimension r (see section 2 below) and we denote by Vol (Z f ) its Riemannian volume for the restriction of g to Z f . We also denote by χ(Z f ) its Euler characteristic.

Theorem 1.1. Let (M, g) be a closed Riemannian manifold of dimension n. Let V λ be the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues smaller than λ. Let f (1) , . . . , f (r) be r independent standard Gaussian vectors in V λ , with 1 r n. Then the following holds as λ goes to infinity:

E[Vol (Z f )] = λ n + 2 r 2
Theorem 1.3. Let X be a complex projective manifold of dimension n defined over the reals and r ∈ {1, . . . , n}. Let L be an ample holomorphic Hermitian line bundle over X and E be a rank r holomorphic Hermitian vector bundle over X , both equipped with real structures compatible with the one on X . Let s be a standard Gaussian vector in RH 0 (X , E ⊗ L d ).

Then the following holds as d goes to infinity:

E[Vol (Z s )] = d r 2 Vol (RX ) Vol (S n-r ) Vol (S n ) + O d r 2 -1 .
Theorem 1.4. Let X be a complex projective manifold of dimension n defined over the reals and r ∈ {1, . . . , n}. Let L be an ample holomorphic Hermitian line bundle over X and E be a rank r holomorphic Hermitian vector bundle over X , both equipped with real structures compatible with the one on X . Let s be a standard Gaussian vector in RH 0 (X , E ⊗ L d ).

Then, if nr is even, the following holds as d goes to infinity:

E[χ (Z s )] = (-1) n-r 2 d n 2 Vol (RX ) Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1 ) + O d n 2 -1 .
In the case of random eigenfunctions of the Laplacian, Theorem 1.1 was already known to Bérard [START_REF] Bérard | Volume des ensembles nodaux des fonctions propres du laplacien, Bony-Sjöstrand-Meyer seminar[END_REF] for hypersurfaces. See also [34, thm. 1] where Zelditch shows that, in the case of hypersurfaces,

n + 2 λ E[Z f ] -----→ λ→+∞ Vol S n-1 Vol (S n ) |dV M |
in the sense of the weak convergence of measures. He also proves a similar result in the case of band limited eigenfunctions. Let us discuss Theorems 1.3 and 1.4 when X is CP n with the standard real structure induced by the conjugation in C n+1 , E is the trivial bundle X × C r and L = O(1) is the hyperplane bundle with its usual metric. Then RX = RP n and ω is the Fubini-Study metric on CP n , normalized so that it is the quotient of the Euclidean metric on the sphere S 2n+1 . Besides, RH 0 (X , L d ) is the space of real homogeneous polynomials of degree d in n + 1 variables, and Z s is the common real zero set of r independent such polynomials.

In this setting, Kostlan [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF] proved that, for any d 1,

E[Vol (Z s )] = d r 2 Vol RP n-r .
See also the paper [START_REF] Shub | Complexity of Bezout's theorem II: volumes and probabilities[END_REF] by Shub and Smale, where they compute the expected number of common real roots for a system of n polynomials in n variables. The expected Euler characteristic of a random algebraic hypersurface of degree d in RP n was computed by Podkorytov [START_REF] Podkorytov | The Euler characteristic of a random algebraic hypersurface[END_REF]. Both Kostlan's and Podkorytov's results were generalized by Bürgisser. In [START_REF] Bürgisser | Average volume, curvatures, and Euler characteristic of random real algebraic varieties[END_REF], he computed the expected volume and Euler characteristic of a submanifold Z s of RP n defined as the common zero set of r standard Gaussian polynomials P 1 , . . . , P r of degree d 1 , . . . , d r respectively. In particular, when these polynomials have the same degree d and nr is even, he showed that:

(1.

2)

E[χ (Z s )] = d r 2 n-r 2 p=0 (1 -d) p Γ p + r 2 p!Γ r 2 ,
where Γ denotes Euler's gamma function. Theorems 1.3 and 1.4 agree with these previous results.

Recently, Gayet and Welschinger computed upper and lower bounds for the asymptotics of the expected Betti numbers of random real algebraic submanifolds of a projective manifold, see [START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF]Expected topology of random real algebraic submanifolds[END_REF]. This relies on sharp estimates for the expected number of critical points of index i ∈ {0, . . . , n -r} of a fixed Morse function p : RX → R restricted to the random Z s . More precisely, let N i (Z s ) denote the number of critical points of index i of p /Zs , let Sym(i, n-r-i) denote the open cone of symmetric matrices of size nr and signature (i, nri) and let dν denote the standard Gaussian measure on the space of symmetric matrices. Gayet and Welschinger show [16, thm. 3.1.2] that:

(1.3) E[N i (Z s )] ∼ d→+∞ d π n 2 Vol (RX ) (n -1)! (n -r)! e R (i, n -r -i) 2 r-1 Γ r 2 ,
where e R (i, nri) =

Sym(i,n-r-i)

|det(A)| dν(A).

One can indirectly deduce Theorem 1.4 from this result and from [START_REF] Bürgisser | Average volume, curvatures, and Euler characteristic of random real algebraic varieties[END_REF] in the following way. By Morse theory:

E[χ(Z s )] = n-r i=0 (-1) i E[N i (Z s )] ∼ d→+∞ C n,r d n 2 Vol (RX )
where C n,r is a universal constant depending only on n and r. Specifying to the case of RP n , equation (1.2) gives the value of C n,r . Gayet and Welschinger also proved a result similar to (1.3) for hypersurfaces in the case of Riemannian random waves, see [START_REF] Gayet | Betti numbers of random nodal sets of elliptic pseudodifferential operators[END_REF]. It gives the order of growth of E[χ(Z f )] in Theorem 1.2, for r = 1.

In their book [START_REF] Taylor | Random fields and geometry[END_REF], Taylor and Adler compute the expected Euler characteristic of the excursion sets of a centered, unit-variance Gaussian field f on a smooth manifold M . This expectation is given in terms of the Lipschitz-Killing curvatures of M for the Riemannian metric g f induced by f , see [31, thm. 12.4.1]. One can deduce from this result the expected Euler characteristic of the random hypersurface f -1 (0), always in terms of the Lipschitz-Killing curvatures of (M, g f ). It might be possible to deduce Theorems 1.2 and 1.4 from this result, in the case of hypersurfaces, when the Gaussian field (f (x)) x∈M (resp. (s(x)) x∈RX ) has unit variance, but one would need to estimate the Lipschitz-Killing curvatures of (M, g f ) (resp. (M, g s )) as λ (resp. d) goes to infinity.

In a related setting, Bleher, Shiffman and Zelditch [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF] computed the scaling limit of the k-points correlation function for a random complex submanifold of a complex projective manifold. See also [START_REF]Universality and scaling of zeros on symplectic manifolds, Random matrix models and their applications[END_REF] in a symplectic framework. Our proofs of Theorems 1.2 and 1.4 use the same formalism as these papers, adapted to our frameworks.

We now sketch the proofs of our main results in the Riemannian setting. The real algebraic case is similar. The first step is to express Vol (Z f ) (resp. χ(Z f )) as the integral of some function on Z f . In the case of the volume this is trivial, and the answer is given by the Chern-Gauss-Bonnet theorem (see section 4.2 below) in the case of the Euler characteristic. Then we use the Kac-Rice formula (see Theorem 5.3) which allows us to express E[Vol (Z f )] (resp. E[χ(Z f )]) as the integral on M of some explicit function that only depends on the geometry of M and on the covariance function of the smooth Gaussian field defined by f . It turns out that the covariance function of the field associated to the r independent standard Gaussian functions f (1) , . . . , f (r) in V λ is given by the spectral function of the Laplacian. In the algebraic case, the covariance function is given by the Bergman kernel of E ⊗ L d . This was already used in [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF][START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF].

Then, our results follow from estimates on the spectral function of the Laplacian (resp. the Bergman kernel) and their derivatives (see section 3). In the case of random waves, the estimates we need for the spectral function were proved by Bin [START_REF] Bin | Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold[END_REF], generalizing results of Hörmander [START_REF] Hörmander | The spectral function of an elliptic operator[END_REF]. In the algebraic case, much is known about the Bergman kernel [START_REF] Berman | A direct approach to Bergman kernel asymptotics for positive line bundles[END_REF][START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF][START_REF] Ma | Holomorphic Morse inequalities and Bergman kernels[END_REF][START_REF] Zelditch | Szegö kernels and a theorem of Tian[END_REF] but we could not find the estimates we needed in codimension higher than 1 in the literature. These estimates are established in section 3.3 using Hörmander-Tian peak sections. Peak sections were already used in this context in [START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF]Expected topology of random real algebraic submanifolds[END_REF], see also [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF]. The author was told by Steve Zelditch, after this paper was written, that one can deduce estimates for the Bergman kernel in higher codimension from the paper [START_REF] Berman | A direct approach to Bergman kernel asymptotics for positive line bundles[END_REF] by Berman, Berndtsson and Sjöstrand. This paper is organized as follows. In section 2 we describe how our random submanifolds are generated and the setting of the main theorems. Section 3 is dedicated to the estimates we need for the spectral function of the Laplacian and the Bergman kernel. In section 4 we derive an integral formula for the Euler characteristic of a submanifold. The main theorems are proved in section 5, and we deal with two special cases in section 6: the flat torus and the real projective space. For these examples, it is possible to compute expectations for fixed λ (resp. d) and we recover the results of Kostlan and Bürgisser. Three appendices deal respectively with: some standard results about Gaussian vectors, a rather technical proof we postponed until the end, and a derivation of the Kac-Rice formula using Federer's coarea formula.

Acknowledgements. I am thankful to Damien Gayet for his help and support in the course of this work. I would also like to thank Benoît Laslier for answering many of my questions about probability theory.

Random submanifolds

This section is concerned with the precise definition of the random submanifolds we consider. The first two subsections explain how we produce them in a quite general setting. The third one introduces the covariance kernel, which characterizes their distribution. We also describe the distribution induced on the bundle of 2-jets in terms of this kernel. Then we describe what we called Riemannian random waves, before explaining how to adapt all this in the real algebraic case. This kind of random submanifolds has already been considered by Bérard [START_REF] Bérard | Volume des ensembles nodaux des fonctions propres du laplacien, Bony-Sjöstrand-Meyer seminar[END_REF], Zelditch [START_REF]Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory[END_REF] and Nicolaescu [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF] in the Riemannian case, and by Gayet and Welschinger in the real algebraic case, see [START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF]Expected topology of random real algebraic submanifolds[END_REF][START_REF]Lower estimates for the expected Betti numbers of random real hypersurfaces[END_REF]. See also [START_REF] Bürgisser | Average volume, curvatures, and Euler characteristic of random real algebraic varieties[END_REF][START_REF] Edelman | How many zeros of a random polynomial are real?[END_REF][START_REF] Lerario | Statistics on Hilbert's 16th problem[END_REF] in special cases. From now on, we fix some r ∈ {1, . . . , n} that we think of as the codimension of our random submanifolds. Let

General setting

• , • denote the L 2 -scalar product on C ∞ (M, R r ) induced by |dV M |: for any f 1 and f 2 ∈ C ∞ (M, R r ), (2.2) f 1 , f 2 = x∈M f 1 (x) , f 2 (x) |dV M | ,
where the inner product on the right-hand side is the standard one on R r .

Notation 2.1. Here and throughout this paper • , • will always denote the inner product on the concerned Euclidean or Hermitian space.

Let V be a subspace of C ∞ (M, R r ) of finite dimension N . For any f ∈ V , we denote by Z f the zero set of f . Let D denote the discriminant locus of V , that is the set of f ∈ V that do not vanish transversally.

If f vanishes transversally, Z f is a (possibly empty) smooth submanifold of M of codimension r and we denote by |dV f | the Riemannian measure induced by the restriction of g to Z f . We also denote by Vol (Z f ) the volume of Z f and by χ(Z f ) its Euler characteristic. In the case r = n, when f / ∈ D, Z f is a finite set and |dV f | is the sum of the Dirac measures centered on points of Z f . We consider a random vector f ∈ V with standard Gaussian distribution. That is the distribution of f admits the density function:

(2.3) x → 1 (2π) N 2 exp - 1 2 x 2
with respect to the Lebesgue measure of V . Under some further technical assumptions on V (see section 2.2 below), f vanishes transversally almost surely. Hence, the random variables Vol (Z f ) and χ(Z f ) are well-defined almost everywhere, and it makes sense to compute their expectation.

For the convenience of the reader, we gathered the few results we need about Gaussian vectors in Appendix A. We introduce some notations here and refer to Appendix A for further details. In the sequel, we will denote by X ∼ N (m, Λ) the fact that the random vector X is distributed according to a Gaussian with mean m and variance Λ. A standard Gaussian vector is X ∼ N (0, Id). We will denote by dν N the standard Gaussian measure on a Euclidean space of dimension N , that is the measure with density (2.3) with respect to the Lebesgue measure.

The incidence manifold

Following [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF], we say that V is 0-ample if the map j 0

x : f → f (x) is onto for every x ∈ M . From now on, we assume that this is the case and we introduce an incidence manifold as Shub and Smale in [START_REF] Shub | Complexity of Bezout's theorem II: volumes and probabilities[END_REF] (see also [START_REF]Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF]Expected topology of random real algebraic submanifolds[END_REF]).

Let

F : (f, x) ∈ V × M → f (x) ∈ R r and let ∂ 1 F and ∂ 2 F
denote the partial differentials of F with respect to the first and second variable respectively. For any (f,

x) ∈ V × M , ∂ 1 F (f, x) = j 0 x and ∂ 2 F (f, x) = d x f. (2.4)
We assumed j 0

x to be surjective for every x ∈ M , thus F is a submersion. Then Σ = F -1 (0) is a smooth submanifold of codimension r of V × M , called the incidence manifold, and for any (f 0 , x) ∈ V × M :

T (f0,x) Σ = {(f, v) ∈ V × T x M | f (x) + d x f 0 • v = 0}.
We set π 1 : Σ → V and π 2 : Σ → M the projections from Σ to each factor. A vector f ∈ V is in the range of d (f0,x) π 1 if and only if there exists some v ∈ T x M such that (f, v) ∈ T (f0,x) Σ, that is f (x) is in the range of d x f 0 . Since V is 0-ample, the map j 0

x is onto, and d x f 0 is surjective if and only if d (f0,x) π 1 is. Thus, the discriminant locus D is exactly the set of critical values of π 1 . By Sard's theorem, D has measure 0 in V , both for the Lebesgue measure and for dν N , and f vanishes transversally almost surely.

We equip Σ with the restriction of the product metric on V × M . Then, whenever f / ∈ D,

(π 1 ) -1 (f ) = {f } × Z f is isometric to Z f ,
hence we will identify these sets. Similarly, we will identify (π 2 ) -1 (x) = ker(j 0

x ) × {x} with the subspace ker(j 0 x ) of V .

The covariance kernel

In this subsection we introduce the Schwarz kernel and covariance function associated to our space of random functions. It turns out (see Proposition 2.3 below) that these objects are equal. The first to use this fact were Bleher, Shiffman and Zelditch in the case of complex projective manifolds [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF] and in the case of symplectic manifolds [START_REF]Universality and scaling of zeros on symplectic manifolds, Random matrix models and their applications[END_REF]. In the setting of Riemannian random waves this was used by Zelditch [START_REF]Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory[END_REF] and Nicolaescu [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF].

In C ∞ (M, R r ) equipped with the L 2 -inner product (2.2), the orthogonal projection onto V can be represented by its Schwartz kernel, denoted by E. That is there exists a unique E : M × M → R r ⊗ R r such that for any smooth f : M → R r , the projection of f onto V is given by: (2.5)

x

→ E(x, •) , f = y∈M E(x, y) , f (y) |dV M | .
In the previous formula, the inner product on the right-hand side is the usual one on R r , acting on the second factor of R r ⊗ R r . The kernel E has the following reproducing kernel property:

(2.6) ∀f ∈ V, ∀x ∈ M, f (x) = E(x, •) , f .
If (f 1 , . . . , f N ) is any orthonormal basis of V , one can check that E is defined by:

(2.7)

E : (x, y) → N i=1 f i (x) ⊗ f i (y).
This proves that E is smooth. Besides, for all x ∈ M , E(x, x) is in the span of {ζ⊗ζ | ζ ∈ R r }, and for all ζ ∈ R r :

(2.8)

E(x, x) , ζ ⊗ ζ = N i=1 f i (x) ⊗ f i (x) , ζ ⊗ ζ = N i=1 ( f i (x) , ζ ) 2 0.
This last equation shows that we can check the 0-amplitude condition for V on its kernel.

Lemma 2.2. V is 0-ample if and only if, for all x ∈ M and all ζ ∈ R r \ {0}, we have:

E(x, x) , ζ ⊗ ζ > 0.
That is if and only if E(x, x) is a positive-definite bilinear form on (R r ) * for any x ∈ M .

On the other hand, the standard Gaussian vector f ∈ V defines a smooth centered Gaussian field (f (x)) x∈M with values in R r . Its distribution is totally determined by its covariance function: (x, y) → Cov(f (x), f (y)) from M ×M to R r ⊗R r , where Cov(f (x), f (y)) stands for the covariance form of the random vectors f (x) and f (y) (cf. Appendix A).

Proposition 2.3. Let V be a finite-dimensional subspace of C ∞ (M, R r ) and E its Schwartz kernel. Let f ∼ N (0, Id) in V , then we have:

∀x, y ∈ M, Cov(f (x), f (y)) = E[f (x) ⊗ f (y)] = E(x, y).
Proof. Let x and y ∈ M , then the first equality is given by Lemma A.7. We will now show that E[f (x) ⊗ f (y)] satisfies condition (2.5) to prove the second equality. Let f 0 : M → R r be a smooth function and

x ∈ M , y∈M E[f (x) ⊗ f (y)] , f 0 (y) |dV M | = y∈M E[ f (x) ⊗ f (y) , f 0 (y) ] |dV M | = y∈M E[f (x) f (y) , f 0 (y) ] |dV M | = E[f (x) f , f 0 ] . If f 0 ∈ V ⊥ , this equals 0. If f 0 ∈ V , we have: E[f (x) f , f 0 ] = E[ E(x, •) , f f 0 , f ] = E(x, •) , f 0 = f 0 (x),
where we used the reproducing kernel property (2.6) both for f and f 0 and we applied Lemma A.8 to f ∼ N (0, Id). In both cases, x → E[f (x) f , f 0 ] is the projection of f 0 onto V , which shows the second equality in Proposition 2.3.

This tells us that the distribution of our Gaussian field is totally determined by the Schwartz kernel E. In our cases of interest, asymptotics are known for E and its derivatives, see section 3 below. This is what allows us to derive asymptotics for the expectation of the volume and Euler characteristic of Z f .

Random jets

Let ∇ M be the Levi-Civita connection on (M, g). For any smooth f : M → R r , we denote by ∇ 2 f = ∇ M df the Hessian of f . Let ∂ x (resp. ∂ y ) denote the partial derivative with respect to the first (resp. second) variable for maps from M × M to R r ⊗ R r . Likewise, we denote by ∂ x,x (resp. ∂ y,y ) the second partial derivative with respect to the first (resp. second) variable twice. As for the Hessian above, all the higher order derivatives are induced by ∇ M . Now, let f ∼ N (0, Id) in V . We will describe the distribution induced by f on the 2-jets bundle of M . Let x ∈ M , then we denote by J k

x (R r ) the space of k-jets of smooth functions from M to R r at the point x (we will only use k ∈ {0, 1, 2}). We already defined

j 0 x : C ∞ (M, R r ) -→ R r . f → f (x)
We define similarly,

j 1 x : C ∞ (M, R r ) → R r ⊗ (R ⊕ T * x M ) f → (f (x), d x f ) and j 2 x : C ∞ (M, R r ) → R r ⊗ (R ⊕ T * x M ⊕ Sym(T * x M )), f → (f (x), d x f, ∇ 2
x f ) where Sym(T *

x M ) denotes the space of symmetric bilinear forms on T * x M . The map j 2

x induces an isomorphism between J 2 x (R r ) and the image of j 2

x . In the sequel we will identify these spaces through j 2

x . Likewise, J 1 x (R r ) and the image of j 1

x will be identified through j 1

x . Lemma 2.4. Let V be a finite-dimensional subspace of C ∞ (M, R r ) and E its Schwartz kernel. Let f ∼ N (0, Id) in V and x ∈ M . Then j 2

x (f ) = (f (x), d x f, ∇ 2 x f ) is a centered Gaussian vector, and its variance form Var(j 2

x (f )) is characterized by:

Var (f (x)) = E[f (x) ⊗ f (x)] = E(x, x), (2.9) Var (d x f ) = E[∇ x f ⊗ ∇ x f ] = (∂ x ∂ y E)(x, x), (2.10) Var ∇ 2 x f = E ∇ 2 x f ⊗ ∇ 2 x f = (∂ x,x ∂ y,y E)(x, x), (2.11) Cov (f (x), d x f ) = E[f (x) ⊗ ∇ x f ] = (∂ y E)(x, x), (2.12) Cov f (x), ∇ 2 x f = E f (x) ⊗ ∇ 2 x f = (∂ y,y E)(x, x), (2.13) Cov d x f, ∇ 2 x f = E ∇ x f ⊗ ∇ 2 x f = (∂ x ∂ y,y E)(x, x). (2.14)
Proof. The first equality on each line is given by Lemmas A.4 and A.7. Then Proposition 2.3 gives the second equality in (2.9). The other equalities are obtained by taking partial derivatives of (2.9) With this lemma, we have described the distribution of j 2

x (f ) only in terms of E. Since j 0 x (f ) and j 1

x (f ) are the projections of j 2 x (f ) onto R r and R r ⊗ (R ⊕ T * x M ) respectively, their distributions are also characterized by Lemma 2.4.

Riemannian random waves

In this section, we describe what we called Riemannian random waves, that is random linear combinations of eigenfunctions of the Laplacian.

Let ∆ denote the Laplace-Beltrami operator on the closed Riemannian manifold (M, g). Recall the following classical facts from the theory of elliptical operators, see [12, thm. 4.43].

Theorem 2.5.

1. The eigenvalues of ∆ : C ∞ (M ) → C ∞ (M ) can be arranged into a strictly increasing sequence of non-negative numbers (λ k ) k∈N such that λ k -----→ k→+∞ +∞.

2. The associated eigenspaces are finite-dimensional, and they are pairwise orthogonal for the L 2 -inner product (1.1) on C ∞ (M ) induced by g.

Let λ 0, then we denote by V λ the subspace of C ∞ (M ) spanned by the eigenfunctions of ∆ associated to eigenvalues that are less or equal to λ. Each f = (f (1) , . . . , f (r) ) ∈ (V λ ) r defines naturally a map from M to R r so that we can see (V λ ) r as a subspace of C ∞ (M, R r ). By Theorem 2.5, V λ is finite-dimensional so we can apply the construction of sections 2.1 to 2.4 to (V λ ) r .

Since we consider a product situation, it is possible to make several simplifications. First, the scalar product (2.2) on (V λ ) r is induced by the one on V λ . Thus f = (f (1) , . . . , f (r) ) is a standard Gaussian in (V λ ) r if and only if f (1) , . . . , f (r) are r independent standard Gaussian in V λ . For a f ∈ (V λ ) r satisfying this condition, f (1) , . . . , f (r) are independent, and so are their derivatives of any order. This means that, for every x ∈ M , the matrices of Var(f (x)), Var(d x f ) and Var(∇ 2

x f ) in the canonical basis of R r are block diagonal. Another way to say this is that the kernel of (V λ ) r is a product in the following sense. We denote by e λ : M × M → R the Schwartz kernel of the orthogonal projection onto V λ in C ∞ (M ) and by E λ the Schwartz kernel of (V λ ) r . The kernel e λ is the spectral function of the Laplacian and precise asymptotics are known for e λ and its derivatives, see section 3.

Let (ϕ 1 , . . . , ϕ N ) be an orthonormal basis of V λ . By (2.7),

(2.15)

e λ : (x, y) → N i=1 ϕ i (x)ϕ i (y).
Let (ζ 1 , . . . , ζ r ) denote the canonical basis of R r . The maps ϕ i ζ q : M → R r with 1 i N and 1 q r give an orthonormal basis of (V λ ) r and, for all x and y ∈ M ,

E λ (x, y) = r q=1 N i=1 (ϕ i (x)ζ q ) ⊗ (ϕ i (y)ζ q ) = e λ (x, y) r q=1 ζ q ⊗ ζ q .
Lemma 2.6. Let (ζ 1 , . . . , ζ r ) be any orthonormal basis of R r , for all x, y ∈ M ,

E λ (x, y) = e λ (x, y) r q=1 ζ q ⊗ ζ q .
An immediate consequence of this and Lemma 2.2 is that (V λ ) r is 0-ample if and only if e λ (x, x) > 0 for all x ∈ M . By (2.15), this is equivalent to V λ being base-point-free, that is for every x ∈ M , there exists f ∈ V λ such that f (x) = 0. Lemma 2.7. For all λ 0, (V λ ) r is 0-ample.

Proof. The constant functions on M are eigenfunctions of ∆ associated to the eigenvalue 0. Thus for every λ 0, V λ contains all constant functions on M , hence is base-point-free. By the above remark, (V λ ) r is then 0-ample.

The real algebraic setting

Let us now describe more precisely the real algebraic framework. The main difference with what we did previously is that we consider sections of a rank r vector bundle instead of maps to R r . The local picture is the same as in sections 2.1 to 2.5, so that we can adapt everything to this setting. But the formalism is a bit heavier. A classical reference for most of this material is [START_REF] Ph | Principles of algebraic geometry[END_REF].

Let X be a smooth complex projective manifold of complex dimension n. We equip X with a real structure, that is with an antiholomorphic involution c X . We assume that its real locus, the set of fixed points of c X , is not empty and we denote it by RX . Let L be an ample holomorphic line bundle over X equipped with a real structure c L compatible with the one on X . By this we mean that c X • π = π • c L , where π : L → X stands for the projection map onto the base space. Similarly, let E be a holomorphic vector bundle of rank r over X , with a compatible real structure c E .

Let h L and h E be real Hermitian metrics on L and E respectively, that is

c * L (h L ) = h L and c * E (h E ) = h E .
We assume that h L is positive in the sense that its curvature form ω is Kähler. Locally we have:

ω /Ω = 1 2i ∂ ∂ ln (h L (ζ, ζ))
where ζ is any non-vanishing local holomorphic section of L on the open set Ω ⊂ X . This form corresponds to a Hermitian metric g C = ω(•, i•) on X whose real part is a Riemannian metric g. We denote by dV X the volume form ω n n! on X . Remark 2.8. The normalization of ω is the one of [START_REF] Berman | A direct approach to Bergman kernel asymptotics for positive line bundles[END_REF][START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF], but differs from our references concerning peak sections [START_REF]Expected topology of random real algebraic submanifolds[END_REF][START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF]. This will cause some discrepancies with the latter two in section 3.2. With our convention, the Fubini-Study metric on RP n induced by the standard metric on the hyperplane line bundle O(1) is the quotient of the Euclidean metric on S n . Let d ∈ N, then the vector bundle E ⊗ L d comes with a real structure

c d = c E ⊗ c d L compatible with c X and a real Hermitian metric h d = h E ⊗ h d L .
We equip the space Γ(E ⊗ L d ) of smooth sections of E ⊗ L d with the L 2 Hermitian product defined by: (2. [START_REF]Expected topology of random real algebraic submanifolds[END_REF])

∀s 1 , s 2 ∈ Γ(E ⊗ L d ), s 1 , s 2 = X h d (s 1 , s 2 ) dV X .
We know from the vanishing theorem of Kodaira and Serre that the space H 0 (X , E ⊗ L d ) of global holomorphic sections of E ⊗ L d has finite dimension N d and that N d grows as a polynomial of degree n in d, when d goes to infinity. 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 real holomorphic sections of E ⊗L d , which has real dimension N d . The Hermitian product (2.16) induces a Euclidean inner product on RH 0 (X , E ⊗ L d ). Notice that we integrate on the whole of X , not only on the real locus, even when we consider real sections.

If s ∈ RH 0 (X , E ⊗ L d ) is such that its restriction to RX vanishes transversally, then its real zero set Z s = s -1 (0) ∩ RX is a (possibly empty) submanifold of RX of codimension r. We denote by |dV s | the Riemannian measure induced on this submanifold by the metric g.

As in section 2.2, we consider the incidence manifold:

(2.17)

Σ d = (s, x) ∈ RH 0 (X , E ⊗ L d ) × RX s(x) = 0 .
In this setting, Σ d is the zero set of the bundle map

F d : RH 0 (X , E ⊗ L d ) × RX → R(E ⊗ L d ) over RX defined by F d : (s, x) → s(x).
In a trivialization, the situation is similar to the one in section 2.2. Thus, if RH 0 (X , E ⊗ L d ) is 0-ample, Σ d is a smooth manifold equipped with two projection maps, π 1 and π 2 , onto RH 0 (X , E ⊗ L d ) and RX respectively. By Sard's theorem, the discriminant locus of RH 0 (X , E ⊗ L d ) then has measure 0 for any non-singular Gaussian measure, since it is the set of critical values of π 1 .

Remark 2.9. Here, by RH 0 (X , E ⊗ L d ) is 0-ample we mean that, for every x ∈ RX , the evaluation map j 0,d

x : s ∈ RH 0 (X , E ⊗ L d ) → s(x) ∈ R(E ⊗ L d ) x is onto.
Let ∇ d denote any real connection on E ⊗L d , that is such that for every smooth section s,

∇ d (c d • s • c X ) = c d • ∇ d s • dc X .
For example one could choose the Chern connection. We consider the vertical component ∇ d F d of the differential of F d , whose kernel is the tangent space of Σ d . For any (s 0 , x) ∈ Σ d the partial derivatives of F d are given by:

∂ d 1 F d (s 0 , x) = j 0,d x and ∂ d 2 F d (s 0 , x) = ∇ d x s 0 . (2.18)
Note that we only consider points of the zero section of E ⊗L d , hence all this does not depend on the choice of ∇ d .

Let P 1 (resp. P 2 ) denote the projection from X × X onto the first (resp. second) factor.

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 . Let E d denote the Schwartz kernel of the orthogonal projection from the space of real smooth sections of E ⊗ L d onto RH 0 (X , E ⊗ L d ). It is the unique section of (E ⊗ L d ) ⊠ (E ⊗ L d ) such that, for every real smooth section s of E ⊗ L d , the projection of s on RH 0 (X , E ⊗ L d ) is
given by:

x → E d (x, •) , s = y∈X h d (E d (x, y), s(y)) dV X .
Here, h d acts on the second factor of (E

⊗ L d ) x ⊗ (E ⊗ L d ) y .
The kernel E d satisfies a reproducing kernel property similar to (2.6):

(2.19) ∀s ∈ RH 0 (X , E ⊗ L d ), ∀x ∈ RX , s(x) = E d (x, •) , s ∈ R(E ⊗ L d ) x . If (s 1 , . . . , s N d ) is an orthonormal basis of RH 0 (X , E ⊗ L d ) then for all x and y ∈ X , (2.20) E d (x, y) = N d i=1 s i (x) ⊗ s i (y).
For any x ∈ RX , this shows

E d (x, x) is in the span {ζ ⊗ ζ | ζ ∈ R(E ⊗ L d ) x }.
We also get the analogue of Lemma 2.7.

Lemma 2.10.

RH 0 (X , E ⊗ L d ) is 0-ample if and only if for any x ∈ RX , E d (x, x) is a positive-definite bilinear form on (R(E ⊗ L d ) x ) * .
Let s be a standard Gaussian vector in RH 0 (X , E ⊗ L d ), then (s(x)) x∈RX defines a Gaussian field with values in E ⊗ L d and its covariance function is a section of (E ⊗ L d ) ⊠ (E ⊗ L d ). The same proof as for Proposition 2.3 gives the following.

Proposition 2.11. Let E d be the Schwartz kernel of RH 0 (X , E ⊗ L d ). Let s ∼ N (0, Id) in RH 0 (X , E ⊗ L d ), then we have: ∀x, y ∈ X , Cov(s(x), s(y)) = E[s(x) ⊗ s(y)] = E d (x, y).
Remark 2.12. The kernel E d is also the kernel of the orthogonal projection onto

H 0 (X , E⊗L d ) in Γ E ⊗ L d for the Hermitian inner product (2.16), that is the Bergman kernel of E ⊗ L d .
As in section 2.4, let ∇ RX denote the Levi-Civita connection on (RX , g). This connection and ∇ d induce the connection

∇ RX ⊗ Id + Id ⊗∇ d on T * (RX ) ⊗ (E ⊗ L d ). We denote by ∇ 2,d the second covariant derivative (∇ RX ⊗ Id + Id ⊗∇ d ) • ∇ d . Let x ∈ RX and let J k x (E ⊗ L d
) denote the space of real k-jets of real smooth sections of E ⊗ L d at x. On the space of smooth real sections of E ⊗ L d we define

j 1,d x : s → (s(x), ∇ d x s) and j 2,d x : s → (s(x), ∇ d x s, ∇ 2,d x s),
where ∇ d x s and ∇ 2,d x s are implicitly restricted to T x RX . These maps induce isomorphisms from J 1

x (E ⊗ L d ) to the image of

j 1,d x in R(E ⊗ L d ) x ⊗ (R ⊕ T * x RX ) and from J 2 x (E ⊗ L d ) to the image of j 2,d x in R(E ⊗ L d ) x ⊗ (R ⊕ T * x RX ⊕ Sym(T * x RX
)) respectively. Note that the above isomorphisms are not canonical since they depend on the choice of ∇ d .

We have the following equivalent of Lemma 2.4, with the same proof, and similar notations for the partial covariant derivatives. Lemma 2.13. Let E d denote the Bergman kernel of RH 0 (X , E ⊗ L d ) and let s be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Let x ∈ RX , then j 2,d x (s) is a centered Gaussian vector, and its variance form is characterized by:

Var (s(x)) = E[s(x) ⊗ s(x)] = E d (x, x), (2.21) Var ∇ d x s = E ∇ d x s ⊗ ∇ d x s = (∂ x ∂ y E d )(x, x), (2.22) Var ∇ 2,d x s = E ∇ 2,d x s ⊗ ∇ 2,d x s = (∂ x,x ∂ y,y E d )(x, x), (2.23) Cov s(x), ∇ d x s = E s(x) ⊗ ∇ d x s = (∂ y E d )(x, x), (2.24) Cov s(x), ∇ 2,d x s = E s(x) ⊗ ∇ 2,d x s = (∂ y,y E d )(x, x), (2.25) Cov ∇ d x s, ∇ 2,d x s = E ∇ d x s ⊗ ∇ 2,d x s = (∂ x ∂ y,y E d )(x, x). (2.26)

Estimates for the covariance kernels

We state in this section the estimates for the kernels described above and their first and second derivatives. These estimates will allow us to compute the limit distribution for the random 2-jets induced by the Gaussian field (f (x)) x∈M (resp. (s(x)) x∈RX ).

In the case of the spectral function of the Laplacian e λ , the asymptotics of section 3.1 were established by Bin [START_REF] Bin | Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold[END_REF], extending results of Hörmander [START_REF] Hörmander | The spectral function of an elliptic operator[END_REF]. In the algebraic case, Bleher, Shiffman and Zelditch used estimates for the related Szegö kernel, see [4, thm. 3.1]. In terms of the Bergman kernel, a similar result was established in [START_REF] Berman | A direct approach to Bergman kernel asymptotics for positive line bundles[END_REF]. Both these results concern line bundles. Here, we establish the estimates we need for the Bergman kernel in the case of a higher rank bundle using Hörmander-Tian peak sections (see sections 3.2 and 3.3 below).

The spectral function of the Laplacian

We consider the Riemannian setting of section 2.5. Let x ∈ M and let (x 1 , . . . , x n ) be normal coordinates centered at x. Let (y 1 , . . . , y n ) denote the same coordinates in a second copy of M , so that (x 1 , . . . , x n , y 1 , . . . , y n ) are normal coordinates around (x, x) ∈ M × M . We denote by ∂ xi (resp. ∂ yi ) the partial derivative with respect to x i (resp. y i ), and similarly ∂ xi,xj (resp. ∂ yi,yj ) denotes the second derivative with respect to x i and x j (resp. y i and y j ). Let

γ 0 = 1 (4π) n 2 Γ 1 + n 2 , γ 1 = 1 2(4π) n 2 Γ 2 + n 2 and γ 2 = 1 4(4π) n 2 Γ 3 + n 2 ,
where Γ is Euler's gamma function. Let us recall the main theorem of [START_REF] Bin | Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold[END_REF]. Theorem 3.1 (Bin). Let V λ be as in section 2.5 and let e λ denote its Schwartz kernel. The following asymptotics hold, uniformly in x ∈ M , as λ → +∞:

e λ (x, x) = γ 0 λ n 2 + O λ n-1 2 , (3.1) ∂ xi e λ (x, x) = O λ n 2 , (3.2) ∂ xi,x k e λ (x, x) =      -γ 1 λ n 2 +1 + O λ n+1 2 if i = k, O λ n+1 2 if i = k, (3.3)
∂ xi ∂ yj e λ (x, x) =      γ 1 λ n 2 +1 + O λ n+1 2 if i = j, O λ n+1 2 if i = j, (3.4)
∂ xi,x k ∂ yj e λ (x, x) = O λ n 2 +1 , (3.5) ∂ xi,x k ∂ yj ,y l e λ (x, x) =            3γ 2 λ n 2 +2 + O λ n+3 2 if i = j = k = l, γ 2 λ n 2 +2 + O λ n+3 2 if i = j = k = l or i = k = j = l, O λ n+3 2
otherwise.

(3.6)
Since e λ is symmetric, this also gives the asymptotics for ∂ yj e λ , ∂ yj ,y l e λ and ∂ xi ∂ yj ,y l e λ along the diagonal. This theorem, together with Lemma 2.6, gives the estimates we need for the kernel E λ of (V λ ) r .

We will need the following relations:

γ 0 γ 1 = n + 2 and γ 1 γ 2 = n + 4. (3.7)

Hörmander-Tian peak sections

We now recall the construction of Hörmander-Tian peak sections in the framework of section 2.6. Let X be a complex projective manifold. Let E be a rank r holomorphic Hermitian vector bundle and L be an ample holomorphic Hermitian line bundle, both defined over X . We assume that X , E and L are endowed with compatible real structures, and that the Kähler metric g C on X is induced by the curvature ω of L.

The goal of this subsection is to build, for every d large enough, a family of real sections of E ⊗ L d with prescribed 2-jets at some fixed point x ∈ RX . Moreover, we want this family to be orthonormal, up to an error that goes to 0 as d goes to infinity. Using these sections, we will compute (in section 3.3 below) the asymptotics we need for the Bergman kernel.

Let x ∈ RX and (x 1 , . . . , x n ) be real holomorphic coordinates centered at x and such that ( ∂ ∂x1 , . . . , ∂ ∂xn ) is orthonormal at x. The next lemma is established in [15, lemma 3.3], up to a factor π coming from different normalizations of the metric. Lemma 3.2. There exists a real holomorphic frame ζ 0 for L, defined over some neighborhood of x, whose potentialln(h L (ζ 0 , ζ 0 )) vanishes at x, where it reaches a local minimum with Hessian g C .

We choose such a frame ζ 0 . Let (ζ 1 , . . . , ζ r ) be a real holomorphic frame for E over a neighborhood of x, which is orthonormal at x. Since X is compact, we can find ρ > 0, not depending on x, such that local coordinates and frames as above are defined at least on the geodesic ball of radius ρ centered at x. The following results are proved in [16,

) ∈ N n , m > p 1 + • • • + p n and q ∈ {1, . .

. , r}.

There exists d 0 ∈ N such that, for any d d 0 , there exist C d,p > 0 and s ∈ RH 0 (X , E ⊗ L d ) such that s = 1 and

s(x 1 , . . . , x n ) = C d,p x p1 1 • • • x pn n + O (x 1 , . . . , x n ) 2m 1 + O d -2m ζ q ⊗ ζ d 0 in some neighborhood of x, where the estimate O(d -2m ) is uniform in x ∈ RX .
Moreover, d 0 depends on m but does not depend on x, p, q or our choices of local coordinates and frames. Finally, C d,p is given by:

(C d,p ) -2 = (x1,...,xn) ln(d) √ d |x p1 1 • • • x pn n | 2 h d L ζ d 0 , ζ d 0 dV X .
The following definitions use Proposition 3.3 with m = 3 and the corresponding d 0 .

Definitions 3.4. For any d d 0 and q ∈ {1, . . . , r} we denote the sections of RH 0 (X , E ⊗L d ) given by Proposition 3.3 by:

• s d,q 0 for p 1 = • • • = p n = 0, • s d,q i for p i = 1 and ∀ k = i, p k = 0, • s d,q i,i for p i = 2 and ∀ k = i, p k = 0, • s d,q i,j for p i = p j = 1 and ∀ k / ∈ {i, j}, p k = 0, when i < j.
Computing the values of the corresponding C d,p (see [14, lemma 2.5]), we get the following asymptotics as d goes to infinity. Once again, O(d -1 ) is uniform in x.

Lemma 3.5. For every q ∈ {1, . . . , r}, we have:

s d,q 0 = d n π n 1 + O (x 1 , . . . , x n ) 6 1 + O d -1 ζ q ⊗ ζ d 0 , (3.8) ∀i ∈ {1, . . . , n}, s d,q i = d n+1 π n x i + O (x 1 , . . . , x n ) 6 1 + O d -1 ζ q ⊗ ζ d 0 , (3.9) ∀i ∈ {1, . . . , n}, s d,q i,i = d n+2 π n x 2 i √ 2 + O (x 1 , . . . , x n ) 6 1 + O d -1 ζ q ⊗ ζ d 0 , (3.10)
and finally, ∀i, j ∈ {1, . . . , n} such that i < j,

s d,q i,j = d n+2 π n x i x j + O (x 1 , . . . , x n ) 6 1 + O d -1 ζ q ⊗ ζ d 0 . (3.11)
Remark 3.6. These values differ from the one given in [16, lemma 2.3.5] by a factor X dV X 1 2 and some power of √ π because we do not use the same normalization for the volume form. For the same reason they also differ from [32, lemma 2.3] by a factor π n 2 . The sections defined in Definition 3.4 are linearly independent, at least for d large enough. In fact, they are asymptotically orthonormal in the following sense. Let H d 2,x ⊂ RH 0 (X , E ⊗ L d ) denote the subspace of sections that vanish up to order 2 at x. Lemma 3.7. The sections (s d,q i ) 1 q r 0 i n and (s d,q i,j ) 1 q r 1 i j n defined in Definition 3.4 have L 2norm equal to 1 and their pairwise scalar product is dominated by a O(d -1 ) independent of x. Moreover, their scalar product with any unit element of H d 2,x is dominated by some O(d -1 ) not depending on x.

The Bergman kernel

In this section we compute some asymptotics for the Bergman kernel and its derivatives. Let x ∈ RX and let (x 1 , . . . , x n ) be real holomorphic coordinates around x such that ( ∂ ∂x1 , . . . , ∂ ∂xn ) is orthonormal at x. We denote by (y 1 , . . . , y n ) the same coordinates as (x 1 , . . . , x n ) in a second copy of X . Let ζ 0 be a real holomorphic frame for L given by Lemma 3.2 and (ζ 1 , . . . , ζ r ) be a real holomorphic frame for E that is orthonormal at x. For simplicity, we set

ζ d p = ζ p (x) ⊗ ζ d 0 (x) for every p ∈ {1, . . . , r} and d ∈ N, so that (ζ d 1 , . . . , ζ d r ) is an orthonormal basis of R(E ⊗ L d ) x .
Let ∇ d be any real connection on E ⊗ L d such that, for every p ∈ {1, . . . , r},

∇ d (ζ p ⊗ ζ d 0 )
vanishes in a neighborhood of x. In this neighborhood, we have for every function f :

∇ d (f ζ p ⊗ ζ d 0 ) = df ⊗ ζ p ⊗ ζ d 0 and ∇ 2,d (f ζ p ⊗ ζ d 0 ) = ∇ 2 f ⊗ ζ p ⊗ ζ d 0 , (3.12)
where ∇ 2,d stands for the associated second covariant derivative, as in section 2.6. This choice of connection may seem restrictive but the quantity we want to compute do not depend on a choice of connection so that we can choose one that suits us.

As usual, ∇ d induces a connection on (E ⊗ L d ) ⊠ (E ⊗ L d ). We denote by ∂ d xi and ∂ d yi the partial covariant derivatives with respect to x i and y i respectively. We also denote by

∂ d xi,xj (resp. ∂ d yi,yj
) the second derivative with respect to x i and x j (resp. y i and y j ). Proposition 3.8. The following asymptotics hold as d → +∞. They are independent of x ∈ RX and of the choice of the holomorphic frame (ζ 1 , . . . , ζ r ).

E d (x, x) , ζ d p ⊗ ζ d p ′ =    d n π n + O d n-1 if p = p ′ , O d n-1 otherwise, (3.13)
∂ d xi E d (x, x) , ζ d p ⊗ ζ d p ′ = O d n-1 2 , (3.14) ∂ d xi,x k E d (x, x) , ζ d p ⊗ ζ d p ′ = O(d n ) , (3.15) ∂ d xi ∂ d yj E d (x, x) , ζ d p ⊗ ζ d p ′ =    d n+1 π n + O(d n ) if p = p ′ and i = j, O(d n ) otherwise, (3.16)
∂ d xi,x k ∂ d yj E d (x, x) , ζ d p ⊗ ζ d p ′ = O d n+ 1 2 , (3.17) (3.18) ∂ d xi,x k ∂ d yj ,y l E d (x, x) , ζ d p ⊗ ζ d p ′ =                  2 d n+2 π n + O d n+1 if p = p ′ and i = j = k = l, d n+2 π n + O d n+1 if p = p ′ and i = j = k = l, or if p = p ′ and i = l = k = j, O d n+1 otherwise.
Remark 3.9. Since RH 0 (X , E ⊗ L d ) is not a product space, the terms with p = p ′ are usually not zero. However they are zero when E is trivial, for example.

Corollary 3.10. For every d large enough, RH 0 (X , E ⊗ L d ) is 0-ample.

Proof of Corollary 3.10. Let x ∈ RX . By (3.13), the matrix of

E d (x, x) in any orthonormal basis of R(E ⊗ L) x is d π n I r 1 + O d -1
, where I r stands for the identity matrix of size r.

Then E d (x, x) is positive-definite for d larger than some d 0 independent of x. By Lemma 2.10,

RH 0 (X , E ⊗ L d ) is 0-ample for d d 0 .
Proof of Proposition 3.8. First we build an orthonormal basis of RH 0 (X , E ⊗L d ) by applying the Gram-Schmidt process to the family of peak sections. Then we use formula (2.20) and the asymptotics of Lemma 3.5 to prove the proposition. We order the sections of Definition 3.4 as follows:

( and assume that for any k i and any j > i we have

b k , bj = O d -1 . Note that this is the case for i = r 2 (n + 1)(n + 2). Then, (3.20) bi = b i - j>i b i , bj bj -π i b i - j>i b i , bj bj -π i
where π i stands for the projection of b i onto H d 2,x . By Lemma 3.7,

π i 2 = b i , π i = O d -1 .
Then by our hypothesis b i -j>i b i , bj bj -

π i 2 = 1 + O d -1
. Using Lemma 3.7 and the above hypothesis once again, we get:

b k , bi = 1 + O d -1   b k , b i - j>i b k , bj b i , bj -b k , π i   = O d -1 ,
for any k < i. By induction, for any 1 i < j

r(n+1)(n+2) 2 , b i , bj = O d -1 . Then, using (3.20), for any i ∈ 1, . . . , r(n+1)(n+2) 2 , bi =   b i + O d -1 j>i bj -π i   1 + O d -1 .
Another induction gives:

(3.21) bi =   b i + O(d -1 ) j>i b j + πi   1 + O d -1 , for any i, where πi ∈ H d 2,x is such that πi 2 = O d -1
. Moreover, all the estimates are independent of x and i ∈ 1, . . . , r(n+1)(n+2)

2

.

Among the elements of B, only sd,1 0 , . . . , sd,r 0 do not vanish at x. Using formula (2.20), we get E d (x, x) = 1 q r sd,q 0 (x) ⊗ sd,q 0 (x). Then,

E d (x, x) , ζ d p ⊗ ζ d p ′ = r q=1 sd,q 0 (x) , ζ d p sd,q 0 (x) , ζ d p ′ .
Recall that b0 = sd,1 0 , . . . , br = sd,r 0 . Because of (3.21), for all q ∈ {1, . . . , r},

(3.22) sd,q 0 (x) , ζ d p = s d,q 0 (x) , ζ d p + O d -1   r q ′ =1 s d,q ′ 0 (x) , ζ d p   =      d π n 2 1 + O d -1 if p = q, O d n 2 -1 otherwise,
where the last equality comes from equation (3.8). This establishes (3.13).

Likewise,

∂ d xi E d (x, x) , ζ d p ⊗ ζ d p ′ = 1 q r ∂ d xi sd,q 0 (x) ⊗ sd,q 0 (x) , ζ d p ⊗ ζ d p ′ = 1 q r ∂ d xi sd,q 0 (x) , ζ d p sd,q 0 (x) , ζ d p ′ .
The description (3.21) shows that ∂ d xi sd,q 0 (x) does not necessarily vanish, but it equals:

O d -1 1 q ′ r 1 j n ∂ d xi s d,q ′ j (x).
By (3.9), one gets that

∂ d xi sd,q 0 (x) , ζ d p = O d n-1 2
, for all p and q. Besides by (3.22),

sd,q 0 (x) , ζ d p = O d n 2
for all p and q. This proves (3.14). The remaining estimates can be proved in the same way, using Lemma 3.5 and the fact that the estimates for corresponding elements of B and B are the same.

An integral formula for the Euler characteristic of a submanifold

The goal of this section is to derive an integral formula for the Euler characteristic of a submanifold defined as the zero set of some f : M → R r , in terms of f and its derivatives. This section is independent of the previous ones and the results it contains are only useful for computing expected Euler characteristics (Theorems 1.2 and 1.4). We start by recalling the formalism of double forms, which was already used in this context by Taylor and Adler, see [31, section 7.2]. The Riemann curvature tensor and the second fundamental form of a submanifold being naturally double forms, this provides a useful way to formulate the Chern-Gauss-Bonnet theorem and the Gauss equation. This is done in sections 4.2 and 4.3 respectively. Finally, we express the second fundamental form of a submanifold in terms of the derivatives of a defining function and prove the desired integral formula in section 4.4.

The algebra of double forms

We follow the exposition of [31, pp. 157-158]. Let V be a real vector space of dimension n. For p and q ∈ {0, . . . , n} we denote by p,q (V * ) the space p (V * ) ⊗ q (V * ) of (p + q)-linear forms on V that are skew-symmetric in the first p and in the last q variables. The space of double forms on V is:

(4.1) • (V * ) ⊗ • (V * ) = 0 p,q n p,q (V * ).
Elements of p,q (V * ) are called (p, q)-double forms, or double forms of type (p, q). We set:

(4.2) •,• (V * ) = n p=0 p,p (V * ).
Note that 1,1 V * is the space of bilinear forms on V . On

• (V * ) ⊗ • (V * )
we can define a double wedge product. It extends the usual wedge product on

• (V * ) ≃ n p=0
p,0 (V * ), so we simply denote it by ∧. For pure tensors α ⊗ β

and α ′ ⊗ β ′ ∈ • (V * ) ⊗ • (V * ), we set: (4.3) (α ⊗ β) ∧ (α ′ ⊗ β ′ ) = (α ∧ α ′ ) ⊗ (β ∧ β ′ )
and we extend ∧ to all double forms by bilinearity. This makes

• (V * ) ⊗ • (V * ) into an algebra, of which •,• (V *
) is a commutative subalgebra. We denote by γ ∧k the double wedge product of a double form γ ∈

•,• (V * ) with itself k times.

Lemma 4.1. Let α be a symmetric (1, 1)-double form on V , then for every x, y, z and w ∈ V , α ∧2 ((x, y), (z, w)) = 2 (α(x, z)α(y, w)α(x, w)α(y, z)) .

Proof. Let (e 1 , . . . , e n ) be a basis of V and (e * 1 , . . . , e * n ) its dual basis. We have:

α = 1 i,k n α ik e * i ⊗ e * k and then α ∧2 = 1 i,j,k,l n α ik α jl (e * i ∧ e * j ) ⊗ (e * k ∧ e * l ).
Note that we do not restrict ourselves to indices satisfying i < j and k < l as is usually the case with skew-symmetric forms. By multilinearity, it is sufficient to check the result on elements of the basis. Let i, j, k and l ∈ {1, . . . , n}, then

α ∧2 ((e i , e j ), (e k , e l )) = α ik α jl -α jk α il -α il α jk + α jl α ik = 2(α ik α jl -α il α jk ) (since α is symmetric) = 2(
α(e i , e k )α(e j , e l )α(e i , e l )α(e j , e k )).

We can consider random vectors in spaces of double forms. The following technical result will be useful in the proofs of Theorems 1.2 and 1.4. See [31, lemma 12.3.1] for a proof. Lemma 4.2. Let V be a vector space of finite dimension n. Let α be a Gaussian vector in 1,1 V * . If α is centered, then for any p n 2 ,

E α ∧2p = (2p)! 2 p p! E α ∧2 ∧p .
Assume now that V is endowed with an inner product. It induces a natural inner product on

• (V * ) such that, if (e 1 , . . . , e n ) is an orthonormal basis of V ,

e * i1 ∧ • • • ∧ e * ip 1 p n and 1 i 1 < i 2 < • • • < i p n
is an orthonormal basis of • (V * ). We define the trace operator Tr on

•,• (V * ) in the following way. If α ⊗ β ∈ •,• (V *
) is a pure tensor, then:

(4.4) Tr(α ⊗ β) = α , β
and we extend Tr to •,• (V * ) by linearity. Let M be a smooth manifold of dimension n. Applying the previous construction pointwise to T x M , we define the vector bundle

• (T * M ) ⊗ • (T * M ) on M .
Sections of this bundle are called differential double forms on M , and we can take the double wedge product of two such sections. Finally, if M is equipped with a Riemannian metric, we have a trace operator Tr which is defined pointwise by (4.4). This operator is C ∞ (M )-linear and takes sections of the subbundle •,• (T * M ) = n p=0 p,p (T * M ) to smooth functions.

The Chern-Gauss-Bonnet theorem

Let (M, g) be a closed smooth Riemannian manifold of dimension n. We denote by ∇ M the Levi-Civita connection of M , and by κ its curvature operator. That is κ is the 2-form on M with values in the bundle End(T M ) = T M ⊗ T * M defined by:

κ(X, Y )Z = ∇ M X ∇ M Y Z -∇ M Y ∇ M X Z -∇ M [X,Y ] Z
for any vector fields X, Y and Z. Here [X, Y ] is the Lie bracket of X and Y . We denote by R the Riemann curvature tensor of M , defined by: R(X, Y, Z, W ) = g(κ(X, Y )W, Z),

for any vector fields X, Y , Z and W on M . This defines a four times covariant tensor on M which is skew-symmetric in the first two and in the last two variables, hence R can naturally be seen as a (2, 2)-double form. All this is standard material, except for the very last point, see for example [21, section 3.3].

We now state the Chern-Gauss-Bonnet theorem in terms of double forms. Recall that |dV M | denotes the Riemannian measure on M (see (2.1)).

Theorem 4.3 (Chern-Gauss-Bonnet). Let M be a closed Riemannian manifold of even dimension n. Let R denote its Riemann curvature tensor and χ(M ) denote its Euler characteristic. We have:

χ (M ) = 1 (2π) n 2 n 2 ! M Tr R ∧ n 2 |dV M | .
If M is orientable, this can be deduced from Atiyah-Singer's index theorem. The general case is treated in [START_REF] Palais | A topological Gauss-Bonnet theorem[END_REF]. The above formula in terms of double forms can be found in [31, thm. 12.6.18], up to a sign coming from different sign conventions in the definition of R.

The Gauss equation

Let (M, g) be a smooth Riemannian manifold of dimension n and M be a smooth submanifold of M of codimension r ∈ {1, . . . , n-1}. We denote by ∇ M and ∇ the Levi-Civita connections on M and M respectively. Likewise, we denote by R and R their Riemann curvature tensor. We wish to relate R and R. This is done by the Gauss equation, see Proposition 4.5 below.

We denote by II the second fundamental form of M ⊂ M which is defined as the section of T ⊥ M ⊗ T * M ⊗ T * M satisfying: Proposition 4.5 (Gauss equation). Let X, Y , Z and W be vector fields on M , then:

(4.5) II(X, Y ) = -∇ M X Y -∇X Y = -∇ M X Y ⊥ for
R(X, Y, Z, W ) = R(X, Y, Z, W ) + II(X, Z) , II(Y, W ) -II(X, W ) , II(Y, Z) .
We want to write this Gauss equation in terms of double forms. Let x ∈ M and X, Y , Z and W ∈ T x M . Let U ∼ N (0, Id) in T x M , by Lemma A.8:

II(X, Z) , II(Y, W ) -II(X, W ) , II(Y, Z) = E[ II(X, Z) , U II(Y, W ) , U -II(X, W ) , U II(Y, Z) , U ] .
Then we apply Lemma 4.1 to the symmetric (1, 1)-double form II , U for fixed U . This gives:

II(X, Z) , II(Y, W ) -II(X, W ) , II(Y, Z) = 1 2 E II , U ∧2 ((X, Y ), (Z, W )) .
We proved the following version of the Gauss equation.

∀x ∈ M , R(x) = R(x) + 1 2 E II(x) , U ∧2 ,
where U ∼ N (0, Id) with values in T x M , and R(x) is implicitly restricted to T x M .

An expression for the second fundamental form

Let us now express the second fundamental form II of a submanifold M of M defined as the zero set of a smooth map f : M → R r . For this we need some further definitions. Let V and V ′ be two Euclidean spaces of dimension n and r respectively. Let L : V → V ′ be a linear surjection, then the adjoint operator L * is injective and its image is ker(L) ⊥ , so that LL * is invertible. Definition 4.7. Let L : V → V ′ be a surjection, the pseudo-inverse (or Moore-Penrose inverse) of L is defined as

L † = L * (LL * ) -1 from V ′ to V .
The map L † is the inverse of the restriction of L to ker(L) ⊥ . It is characterized by the fact that LL † is the identity map of V ′ and L † L is the orthogonal projection onto ker(L) ⊥ .

Let f : M → R r be a smooth submersion and assume that M = f -1 (0). Recall that ∇ 2 f = ∇ M df . Lemma 4.8. Let M be a Riemannian manifold and let M ⊂ M be a submanifold of M defined as the zero set of the smooth submersion f : M → R r . Let II denote the second fundamental form of M ⊂ M . Then,

∀x ∈ M , II(x) = (d x f ) † • ∇ 2
x f, where ∇ 2

x f is implicitly restricted to T x M . Proof. Let x ∈ M , since II(x) and (d x f ) † take values in T x M ⊥ = ker(d x f ) ⊥ , we only need to prove that d x f • II(x) = ∇ 2

x f . Let X and Y be two vector fields on M . The map df • Y vanishes uniformly on M , hence:

d x (df • Y ) • X = (∇ M X df ) x • Y + d x f • (∇ M X Y ) = 0.
Then, using equation (4.5) and ker(d

x f ) = T x M , (d x f • II(x))(X, Y ) = -d x f • (∇ M X Y ) ⊥ = -d x f • (∇ M X Y ) = (∇ M X df ) x • Y = ∇ 2
x f (X, Y ). Proposition 4.9. Let (M, g) be a closed Riemannian manifold of dimension n and R its Riemann curvature. Let f : M → R r be a smooth submersion and Z f = f -1 (0). We denote by |dV f | the Riemannian measure on Z f and by R f its Riemann curvature.

If nr is even, the Euler characteristic of Z f is:

(4.6) χ(Z f ) = 1 (2π) m m! x∈Z f Tr (R f (x) ∧m ) |dV f | , where m = n-r 2 . Furthermore, for all x ∈ Z f , (4.7
) R f (x) = R(x) + 1 2 E ∇ 2 x f , (d x f ) † * (U ) ∧2 ,
where U is a standard Gaussian vector in T x M .

Proof. First, we apply the Chern-Gauss-Bonnet theorem 4.3 to Z f , which gives (4.6). Then let x ∈ Z f and U ∼ N (0, Id) in T x M , by Proposition 4.6,

R f (x) = R(x) + 1 2 E II f (x) , U ∧2 ,
where II f is the second fundamental form of Z f ⊂ M . We conclude by Lemma 4.8.

Proposition 4.9 is also true for zero sets of sections. Let s be a section of some rank r vector bundle over M that vanishes transversally and Z s be its zero set. Let |dV s | denote the Riemannian measure on Z s and R s denote its Riemann tensor. As above, we can apply Theorem 4.3, so that:

(4.8) χ(Z s ) = 1 (2π) m m! x∈Zs Tr (R s (x) ∧m ) |dV s | .
The result of Proposition 4.6 is still valid for Z s . Besides, the same proof as in Lemma 4.8 shows that, for any connection ∇ d , the second fundamental form II s of Z s satisfies:

∀x ∈ Z s , II s (x) = (∇ d x s) † • ∇ 2,d
x s. Remark 4.10. This is not surprising since the terms of this equality do not depend on a choice of connection and the result in a trivialization is given by Lemma 4.8.

Finally, for every connection ∇ d and every x ∈ Z s , we get:

(4.9) R s (x) = R(x) + 1 2 E ∇ 2,d x s , (∇ d x s) † * (U ) ∧2 ,
where U is a standard Gaussian vector in T x M , as in (4.7).

Proofs of the main theorems

We now set to prove the main theorems. The proofs will be detailed in the Riemannian case but only sketched in the real algebraic one, since they are essentially the same.

The Kac-Rice formula

First, we state the celebrated Kac-Rice formula, which is one of the key ingredients in our proofs. This formula is proved in [5, thm. 4.2], see also [1, chap. 6]. For the reader's convenience, we include a proof in Appendix C.

Definition 5.1. Let L : V → V ′ be a linear map between Euclidean vector spaces. We denote by det ⊥ (L) the orthogonal determinant of L:

det ⊥ (L) = det(LL * ),
where L * : V ′ → V is the adjoint operator of L.

Remark 5.2. If L is not onto then det ⊥ (L) = 0. Else, let A be the matrix of the restriction of L to ker(L) ⊥ in any orthonormal basis of ker(L) ⊥ and V ′ , then we have det

⊥ (L) = |det(A)|.
As in section 2, we consider a closed Riemannian manifold M of dimension n and a subspace V ⊂ C ∞ (M, R r ) of dimension N (recall that 1 r n). We assume that V is 0-ample, in the sense of section 2.2, so that

Σ = {(f, x) ∈ V × M | f (x) = 0}
is a submanifold of codimension r of V ×M . Let f be a standard Gaussian vector in V . Then Z f is almost surely a smooth submanifold of codimension r of M (see section 2.2). Recall that E denotes both the Schwartz kernel of V and the covariance function of (f (x)) x∈M .

Theorem 5.3 (Kac-Rice formula). Let φ : Σ → R be a Borel measurable function, then

E x∈Z f φ(f, x) |dV f | = 1 (2π) r 2 x∈M 1 det E(x, x) E φ(f, x) det ⊥ (d x f ) f (x) = 0 |dV M | ,
whenever one of these integrals is well-defined.

The expectation on the right-hand side is to be understood as the conditional expectation of φ(f, x) det ⊥ (d x f ) given f (x) = 0. By det(E(x, x)), we mean the determinant of the matrix of the bilinear form E(x, x) in any orthonormal basis of R r .

Proof of Theorem 1.1

We start with the expectation of the volume, which is the toy-model for this kind of computations. In this case, the proof is closely related to [START_REF] Bleher | Universality and scaling of correlations between zeros on complex manifolds[END_REF][START_REF]Universality and scaling of zeros on symplectic manifolds, Random matrix models and their applications[END_REF], in a slightly different setting. The first step is to apply Kac-Rice formula above with φ : (f, x) → 1. We get:

(5.1)

E[Vol (Z f )] = 1 (2π) r 2 x∈M 1 det(E(x, x)) E det ⊥ (d x f ) f (x) = 0 |dV M | . Let x ∈ M , then j 1 x (f ) = (f (x), d x f ) is a Gaussian vector in R r ⊗ (R ⊕ T * x M
) whose distribution only depends on the values of E and its derivatives at (x, x), see Lemma 2.4. Thus E[Vol (Z f )] will only depend on the values of E and its derivatives along the diagonal as was expected from [4, thm. 2.2]. The next step is to compute pointwise asymptotics for the integrand on the right-hand side of (5.1). We will use Lemma 2.4, which describes the distribution of j 1

x (f ), and the estimates of section 3. Both in the Riemannian and the algebraic settings, the pointwise asymptotic turns out to be universal: it does not depend on x or even on the ambient manifold. This is because the distribution of j 1

x (f ) is determined by the asymptotics of section 3 which are universal.

We now specify to the case of Riemannian random waves (see section 2.5), that is V = (V λ ) r for some non-negative λ. Recall that (V λ ) r is 0-ample (Lemma 2.7) so that equation (5.1) is valid in this case. Let x ∈ M , by Lemma 2.6 and (3.1) we have:

(5.2) (E λ (x, x)) = (e λ (x, x)) r = γ 0 λ n 2 r 1 + O λ -1 2 .
Then we want to estimate the conditional expectation in (5.1). Before going further, the asymptotics of section 3.1 suggest to consider the scaled variables:

(5.3) (t λ , L λ ) = 1 √ γ 0 λ n 4 f (x), 1 √ γ 1 λ n+2 4 d x f instead of j 1 x (f )
. This is a centered Gaussian vector whose variance is determined by (5.3). Besides, by Definition 5.1, the orthogonal determinant is homogeneous of degree r for linear maps taking values in R r , so that:

(5.4) E det ⊥ (d x f ) f (x) = 0 = (γ 1 ) r 2 λ r(n+2) 4 E det ⊥ (L λ ) t λ = 0 .
Lemma 5.4. For every x ∈ M , we have:

E det ⊥ (L λ ) t λ = 0 = (2π) r 2 Vol (S n-r ) Vol (S n ) 1 + O λ -1 2 ,
where the error term does not depend on the point x ∈ M .

We postpone the proof of this lemma for now and conclude the proof of Theorem 1.1. By (5.2), (5.4) and Lemma 5.4, for every x ∈ M , 1 (2π)

r 2 det(E(x, x)) E det ⊥ (d x f ) f (x) = 0 = γ 1 γ 0 λ r 2 Vol (S n-r ) Vol (S n ) 1 + O λ -1 2 ,
where the error term does not depend on the point x ∈ M . By (3.7) this equals:

λ n + 2 r 2 Vol (S n-r ) Vol (S n ) 1 + O λ -1 2 .
Plugging this into equation (5.1) gives Theorem 1.1.

Remark 5.5. The same proof shows that, for any continuous function φ : M → R, we have:

E Z f φ |dV f | = λ n + 2 r 2 M φ |dV M | Vol (S n-r ) Vol (S n ) + O(λ r-1 2 ). Hence, n + 2 λ r 2 E[|dV f |] -----→ λ→+∞ Vol (S n-r ) Vol (S n ) |dV M |
in the sense of the weak convergence of measures.

We still have to prove Lemma 5.4. For this, we need to compute the variance of L λ given t λ = 0. Let (x 1 , . . . , x n ) be normal coordinates centered at x, and (ζ 1 , . . . , ζ r ) denote the canonical basis of R r . We equip R r ⊗ (R ⊕ T *

x M ) with the orthonormal basis:

(5.5) (ζ 1 , . . . , ζ r , ζ 1 ⊗ dx 1 , . . . , ζ 1 ⊗ dx n , ζ 2 ⊗ dx 1 , . . . , ζ 2 ⊗ dx n , . . . , ζ r ⊗ dx 1 , . . . , ζ r ⊗ dx n ).
Let Λ(λ) denote the matrix of Var(t λ , L λ ) in this basis. This matrix splits as

(5.6) Λ(λ) = Λ 00 (λ) Λ 01 (λ) Λ 10 (λ) Λ 11 (λ) ,
where Λ 00 (λ) and Λ 11 (λ) are the matrices of Var(t λ ) and Var(L λ ) and Λ 01 (λ) = t Λ 10 (λ) is the matrix of Cov(t λ , L λ ). We can decompose further Λ 10 and Λ 11 into blocks of size r × r:

Λ 10 (λ) = Λ i 10 (λ) 1 i n , Λ 11 (λ) = Λ i,j 11 (λ) 1 i,j n . (5.7)
By the Definition (5.3) of (t λ , L λ ), these blocks are obtained by scaling the corresponding blocks in the matrix of Var(j 1

x (f )). Lemmas 2.4 and 2.6 tell us that each one of these blocks is a scalar matrix. Then, using the estimates of section 3.1:

Λ 00 (λ) = e λ (x, x) γ 0 λ n 2 I r = I r + O λ -1 2 , (5.8) ∀i ∈ {1, . . . , n}, Λ i 10 (λ) = ∂ xi e λ (x, x) √ γ 0 γ 1 λ n+1 2 I r = O λ -1 2 , (5.9)
and ∀i, j ∈ {1, . . . , n}, Λ i,j 11 (λ) =

∂ xi ∂ yj e λ (x, x) γ 1 λ n 2 +1 I r =      I r + O λ -1 2 if i = j, O λ -1 2 otherwise, (5.10)
where I r stands for the identity matrix of size r. Thus Λ(λ) = I r(n+1) + O λ -1 2 and, by Corollary A.11, the distribution of L λ conditioned on t λ = 0 is a centered Gaussian with variance operator Λ(λ) = Id +O λ -1 2 . Note that these estimates do not depend on x or our choices of coordinates.

Proof of Lemma 5.4. Let Lλ ∼ N (0, Λ(λ)) in R r ⊗ T *

x M . For λ large enough, Λ(λ) is nonsingular, and we have:

(5.11) E det ⊥ (L λ ) t λ = 0 = E det ⊥ Lλ = 1 (2π) nr 2 det( Λ(λ)) det ⊥ (L) exp - 1 2 Λ(λ) -1 L , L dL,
where dL stands for the Lebesgue measure on R r ⊗ T * x M . Beware that we see L as a linear map in the term det ⊥ (L) but as a vector in Λ(λ) -1 L. The latter is not a composition.

Then Λ(λ) = Id +O λ -1 2 , so that Λ(λ) -1 -Id is bounded by C √ λ for some positive C. Hence, for all L ∈ R r ⊗ T * x M , Λ(λ) -1 -Id L , L C √ λ L 2 ,
and by the mean value theorem, exp -

1 2 Λ(λ) -1 -Id L , L -1 C 2 √ λ L 2 exp C 2 √ λ L 2 .
Then,

(5.12) det ⊥ (L) exp -

1 2 Λ(λ) -1 L , L -exp - L 2 2 dL C 2 √ λ det ⊥ (L) L 2 exp - L 2 2 1 - C √ λ dL.
The integral on the right-hand side of (5.12) converges to some finite limit as λ → +∞ by Lebesgue's dominated convergence theorem, so that:

(5.13) det ⊥ (L) exp - 1 2 Λ(λ) -1 L , L dL = det ⊥ (L) e -1 2 L 2 dL + O λ -1 2 . Since det Λ(λ) = 1 + O λ -1 2
, by (5.11), (5.13) we have:

E det ⊥ (L λ ) t λ = 0 = E det ⊥ (L) + O λ -1 2 ,
where L is a standard Gaussian vector in R r ⊗ T * x M . The result of the lemma is given by Lemma A.14.

Proof of Theorem 1.3

We now consider the real algebraic setting described in section 2.6. The proof goes along the same lines as above. Recall that X is a complex projective manifold of dimension n, equipped with a rank r holomorphic vector bundle E and an ample holomorphic line bundle L, and that X , E and L are endowed with compatible real structures. We are interested in the volume of the real zero set Z s of a standard Gaussian section s in RH 0 (X , E ⊗ L d ).

By Corollary 3.10, RH 0 (X , E ⊗ L d ) is 0-ample for d large enough, so that we can apply Kac-Rice formula (Theorem 5.3) with φ : (s, x) → 1, as in the Riemannian case. Note that we have to use the incidence manifold Σ d defined by (2.17) here. As in (5.1), we get:

(5.14) E[Vol (Z s )] = 1 (2π) r 2 x∈RX 1 det(E d (x, x)) E det ⊥ ∇ d x s s(x) = 0 |dV RX | , where ∇ d is any real connection on E ⊗ L d . Let x ∈ RX and (x 1 , . . . , x n ) be real holomorphic coordinates around x such that ( ∂ ∂x1 , . . . , ∂ ∂xn ) is orthonormal at x. Let (ζ d 1 , . . . , ζ d r ) be an orthonormal basis of R(E ⊗ L d )
x . This yields an orthonormal basis of J 2

x (E ⊗ L d ) similar to (5.5). The value of E det ⊥ ∇ d x s s(x) = 0 does not depend on the choice of ∇ d , since ∇ d x s does not depend on ∇ d when s(x) = 0. We choose a connection that satisfies the conditions of section 3.3, in order to compute the pointwise asymptotic of this quantity.

The estimates of Proposition 3.8 suggest to consider the scaled variables:

(5.15) (t d , L d ) = π n d n s(x), π n d n+1 ∇ d x s .
Then, (t d , L d ) is a centered Gaussian vector in R(E ⊗ L d ) x , and the matrix of Var(t d , L d ) in the basis described above is I r(n+1) +O(d -1 ). This is proved by the same kind of computation as in the Riemannian case, using the estimates of Proposition 3.8. The distribution of L d given t d = 0 is then a centered Gaussian with variance operator Λ(d) = Id +O(d -1 ). As in the previous section (cf. Lemma 5.4), for every x ∈ RX , (5.16)

E det ⊥ ∇ d x s s(x) = 0 = d n+1 π n r 2 E det ⊥ (L d ) t d = 0 = d n+1 π n r 2 (2π) r 2 Vol (S n-r ) Vol (S n ) 1 + O d -1 .
Besides, the estimate (3.13) shows that:

(5.17

) det (E d (x, x)) = d π rn 1 + O d -1 ,
Finally, by (5.14), (5.16) and (5.17), we have proved Theorem 1.3.

As for Riemannian random waves, the same proof shows that for any continuous function φ : RX → R, we have:

E Zs φ |dV s | = d r 2 RX φ |dV RX | Vol (S n-r ) Vol (S n ) + O d r 2 -1 .
Hence,

d -r 2 E[|dV s |] -----→ λ→+∞ Vol (S n-r ) Vol (S n ) |dV M |
in the sense of the weak convergence of measures.

Proof of Theorem 1.2

In this section we compute the expected Euler characteristic of our random submanifolds. The proof is basically the same as in the volume case: apply Kac-Rice formula then compute a pointwise asymptotic for the conditional expectation that appears in Theorem 5.3. Only, this time, we apply Kac-Rice formula to a quantity φ(f, x) that really depends on the couple (f, x) ∈ Σ. This makes the computations a bit more complex. Luckily, φ(f, x) only depends on the 2-jet of f at x, so we can still make pointwise computations. Consider first the general setting of sections 2.1 to 2.4: f is a standard Gaussian vector in the finite-dimensional subspace V ⊂ C ∞ (M, R r ). We assume that V is 0-ample, so that for almost every f ∈ V , Z f is a closed submanifold of dimension nr.

If nr is odd, then χ(Z f ) = 0 almost surely (see Remark 4.4). From now on, we assume that nr is even and set m = n-r 2 . If n = r, then Z f is almost surely a finite set, and χ(Z f ) = Vol (Z f ) is just the cardinal of Z f . In this case Theorems 1.2 and 1.1 coincide, so we need only consider the case r < n in the sequel.

We denote by R the Riemann curvature of the ambient manifold M and, for any f ∈ V , we denote by R f the Riemann curvature of Z f . By Proposition 4.9 and Kac-Rice formula (Theorem 5.3), (5.18)

E[χ(Z f )] = E 1 (2π) m m! Z f Tr ((R f ) ∧m ) |dV f | = 1 m!(2π) n 2 x∈M 1 det(E(x, x)) E det ⊥ (d x f ) Tr(R f (x) ∧m ) f (x) = 0 |dV M | .
Moreover, for all x ∈ M , let U ∼ N (0, Id) in T x M be independent of f . Then, Proposition 4.9 gives:

(5.19) R f (x) = R(x) + 1 2 E U ∇ 2 x f , d x f † * (U ) ∧2 ,
where the notation E U [ • ] means that we only take the expectation with respect to the variable U . Here and in everything that follows, R(x) and ∇ 2

x f are implicitly restricted to ker(

d x f ) = T x Z f .
As in the volume case, the next step is to compute the pointwise asymptotic for the integrand in the last term of (5.18). By (5.19), it only depends on R(x) and the distribution of j 2 x (f ) which is characterized by Lemma 2.4. This shows that the expected Euler characteristic only depends on R and the values of E and its derivatives (up to order 2 in each variable) along the diagonal in M × M . It turns out that, both in the Riemannian and the algebraic cases, the pointwise asymptotic is universal and no longer depends on R.

Focusing on random waves, V = (V λ ) r and we already know from Lemma 2.7 that

(V λ ) r is 0-ample. Let x ∈ M , recall that det(E λ (x, x)) = γ 0 λ n 2 r 1 + O λ -1 2 (see (5.2)).
Then, the main task is to estimate the conditional expectation in (5.18).

We consider the scaled variables:

(5.20

) (t λ , L λ , S λ ) = 1 √ γ 0 λ n 4 f (x), 1 √ γ 1 λ n+2 4 d x f, 1 √ γ 2 λ n+4 4 ∇ 2 x f in R r ⊗ (R ⊕ T * x M ⊕ Sym(T * x M ))
. By Lemma 2.4, (t λ , L λ , S λ ) is a centered Gaussian vector. We denote by ( Lλ , Sλ ) a random variable in R r ⊗(T *

x M ⊕Sym(T * x M )) distributed as (L λ , S λ ) given t λ = 0. Again, ( Lλ , Sλ ) is a centered Gaussian, see Corollary A.11.

Let U ∼ N (0, Id) in T x M be independent of f (hence of j 2 x (f )) and ( Lλ , Sλ ). Then, by (5.19), (5.20) and (3.7):

R f (x) = R(x) + 1 2 E U ∇ 2 x f , d x f † * (U ) ∧2 = R(x) + 1 2 E U   √ γ 2 λ n+4 4 S λ , L † * λ (U ) √ γ 1 λ n+2 4 ∧2   = R(x) + λ 2(n + 4) E U S λ , L † * λ (U ) ∧2 . Besides, det ⊥ (d x f ) = (γ 1 ) r 2 λ r(n+2) 4 det ⊥ (L λ ) , so that, (5.21) E det ⊥ (d x f ) Tr(R f (x) ∧m ) f (x) = 0 = (γ 1 ) r 2 λ r(n+2) 4 E det ⊥ (L λ ) Tr R(x) + λ 2(n + 4) E U S λ , L † * λ (U ) ∧2 ∧m t λ = 0 = (γ 1 ) r 2 λ r(n+2) 4 E det ⊥ Lλ Tr R(x) + λ 2(n + 4) E U Sλ , L † * λ (U ) ∧2 ∧m
.

To conclude the proof, we will use the following lemmas.

Lemma 5.6. The random vectors Lλ , Sλ converge in distribution to a Gaussian vector (L, S) as λ → +∞. Let U ∼ N (0, Id) in T x M be independent of Lλ , Sλ and (L, S), then:

E det ⊥ Lλ Tr R(x) + λ 2(n + 4) E U Sλ , L † * λ (U ) ∧2 ∧m = λ n + 4 m m! (2m)! E det ⊥ (L) Tr S , L † * (U ) ∧2m 1 + O λ -1 2 ,
where the error term is uniform in x ∈ M .

Lemma 5.7. Let (L, S) be distributed as the limit of Lλ , Sλ and U ∼ N (0, Id) in T x M be independent of (L, S). We have:

E det ⊥ (L) Tr S , L † * (U ) ∧2m = - n + 4 n + 2 m (2π) n 2 (2m)! Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1
) .

Assuming these lemmas and recalling (5.2) and (5.21), the integrand in the last term of (5.18) equals:

(-1) m m!(2π) n 2 γ 1 γ 0 λ r 2 λ n + 2 m Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1 ) 1 + O λ -1 2 ,
where the error term is uniform in x ∈ M . Since r + 2m = n and γ0 γ1 = n + 2, we finally get:

E[χ(Z f )] = (-1) m λ n + 2 n 2 Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1 ) x∈M 1 + O λ -1 2 |dV M | ,
and this is Theorem 1.2. We now have to prove Lemmas 5.6 and 5.7. For this we will need the following technical result which is a reformulation of [9, prop. 3.12]. The proof of Proposition 5.8 is mostly tedious computations and we postpone it until Appendix B. Proposition 5.8. Let V and V ′ be two Euclidean spaces of dimension n and r respectively, with 1 r n. Let L ∈ V ′ ⊗ V * and U ∈ V be independent standard Gaussian vectors. Then, L † is well-defined almost surely and det ⊥ (L) , (L † ) * U has the same distribution as

X n X n-1 • • • X n-r+1 , U ′ X n-r+1
,

where U ′ ∈ V ′ , X p ∈ R p for all p ∈ {nr + 1, . . . , n} and U ′ , X n , . . . , X n-r+1 are globally independent standard Gaussian vectors.

We start by computing the variance of (t λ , L λ , S λ ). As in section 5.2, we choose normal coordinates (x 1 , . . . , x n ) centered at x and denote by (ζ 1 , . . . , ζ r ) the canonical basis of R r . For any i and j such that 1 i < j n, we set dx ij = (dx i ⊗ dx j + dx j ⊗ dx i ) and dx ii = dx i ⊗ dx i . We complete the basis of J 1

x (R r ) given in (5.5) into an orthonormal basis of J 2

x (R r ) by adding the following elements (in this order) at the end of the list:

(5. The matrix of Var(t λ , L λ , S λ ) with respect to this basis is:

Λ(λ) =   Λ 00 (λ) Λ 01 (λ) Λ 02 (λ) Λ 10 (λ) Λ 11 (λ) Λ 12 (λ) Λ 20 (λ) Λ 21 (λ) Λ 22 (λ)   ,
where Λ 00 (λ), Λ 10 (λ) and Λ 11 (λ) are as in (5.6) and, similarly, Λ 22 (λ) is the matrix of Var(S λ ), Λ 02 (λ) = t Λ 20 (λ) is the matrix of Cov(t λ , S λ ) and Λ 12 (λ) = t Λ 21 (λ) is the matrix of Cov(L λ , S λ ). As in (5.7), we can decompose further each of these matrices in blocks of size r × r. That is, Λ 10 (λ) and Λ 11 (λ) satisfy (5.7) and, (

Λ 20 (λ) = Λ ik 20 (λ) 1 i k n , Λ 21 (λ) = Λ ik,j 21 (λ) 1 i k n 1 j n and Λ 22 (λ) = Λ ik,jl 22 (λ) 1 i k n 1 j l n 5.23)
.

By definition of (t λ , L λ , S λ ), these blocks are obtained by scaling the corresponding blocks in the matrix of Var(j 2

x (f )). By Lemmas 2.4 and 2.6 the matrices in (5.23) are scalar matrices. Recalling (3.7), we set: γ = -

γ 2 1 γ0γ2 = -n+4
n+2 . Then, by (5.20) and Theorem 3.1 we have:

Λ ik 20 (λ) = ∂ xi,x k e λ (x, x) √ γ 0 γ 2 λ n+2 2 I r =      γI r +O λ -1 2 if i = k, O λ -1 2 if i = k, (5.24)
Λ ik,j

= ∂ xi,x k ∂ yj e λ (x, x) √ γ 1 γ 2 λ n+3 2 I r = O λ -1 2 , 21 (λ)
Λ ik,jl 22 (λ) = ∂ xi,x k ∂ yj ,y l e λ (x, x) γ 2 λ n+4 2 I r =                  3I r +O λ -1 2 if i = j = k = l, I r +O λ -1 2 if i = j = k = l or i = k = j = l, O λ -1 2 otherwise, (5.25)
where I r denotes the identity matrix of size r. Similar estimates for Λ 00 (λ), Λ 10 (λ) and Λ 11 (λ) are given by (5.8), (5.9) and (5.10) respectively. Then Λ(λ) writes by blocks:

Λ(λ) =              I r γI r γI r • • • γI r I nr γI r 3I r I r • • • I r γI r I r 3I r . . . . . . . . . . . . . . . . . . I r γI r I r • • • I r 3I r I rn(n-1) 2              + O λ -1 2 ,
where the empty blocks are zeros. The distribution of ( Lλ , Sλ ), that is the distribution of (L λ , S λ ) given t λ = 0, is a centered Gaussian whose variance matrix is:

(5.27) Λ(λ) =           I nr β 0 I r βI r • • • βI r βI r β 0 I r . . . . . . . . . . . . . . . βI r βI r • • • βI r β 0 I r I rn(n-1) 2           + O λ -1 2 , where β = 1 -γ 2 = - 2 n + 2 and β 0 = 3 -γ 2 = 2n + 2 n + 2 (see Corollary A.11).
Let Λ denote the leading term in (5.27). Equation (5.27) shows that the random vectors Lλ , Sλ converge in distribution to a random vector (L, S) ∼ N (0, Λ) (see Lemma A.12).

We have:

det (Λ) = (β 0 + (n -1)β) r (β 0 -β) r(n-1) = 4(2n + 4) n-1 (n + 2) n r
, so that Λ is non-singular, and Λ(λ) is non-singular for λ large enough.

Proof of Lemma 5.6. We have already seen that Lλ , Sλ converges in distribution, as λ goes to infinity, to (L, S) ∼ N (0, Λ). We still have to prove the estimate in the lemma. We have:

R(x) + λ 2(n + 4) E U Sλ , L † * λ (U ) ∧2 ∧m = m q=0 m q λ 2(n + 4) q R(x) ∧(m-q) ∧ E U Sλ , L † * λ (U ) ∧2 ∧q
.

We can apply Lemma 4.2 to each term in this sum. This yields:

(5.28)

E det ⊥ Lλ Tr R(x) + λ 2(n + 4) E U Sλ , L † * λ (U ) ∧2 ∧m = m q=0 m q q! (2q)! λ 2(n + 4) q E det ⊥ Lλ Tr R(x) ∧(m-q) ∧ Sλ , L † * λ (U ) ∧2q .
Then it is sufficient to show that, for all q ∈ {0, . . . , m},

(5.29) E det ⊥ Lλ Tr R(x) ∧(m-q) ∧ Sλ , L † * λ (U ) ∧2q = E det ⊥ (L) Tr R(x) ∧(m-q) ∧ S , L † * (U ) ∧2q + O λ -1 2 ,
and that these terms are finite. Then (5.28) and (5.29) yield the estimate in the lemma. Let q ∈ {0, . . . , m}, then we first show that the principal part on the right-hand side of (5.29) is finite. Let ζ ∈ R r , then (5.30)

E S Tr R(x) ∧(m-q) ∧ S , ζ ∧2q
is finite since it is the expectation of some polynomial in the coefficients of S. Thus, (5.30) only depends on ζ, and it is an homogeneous polynomial in ζ of degree 2q.

We assumed U to be independent of (L, S), and the expression (5.27) of Λ shows that L and S are independent. Let U ′ ∼ N (0, Id) in R r and X n-r+1 , . . . , X n be standard Gaussian vectors, with X p ∈ R p , such that U ′ , S, X n-r+1 , . . . , X n are globally independent. Applying Proposition 5.8, we have:

(5.31) E det ⊥ (L) Tr R(x) ∧(m-q) ∧ S , L † * (U ) ∧2q = E X n • • • X n-r+2 X n-r+1 2q-1 Tr R(x) ∧(m-q) ∧ S , U ′ ∧2q = E 1 X n-r+1 2q-1 E Tr R(x) ∧(m-q) ∧ S , U ′ ∧2q n p=n-r+2 E[ X p ] . Since 2q -1 2m -1 n -r -1 and X n-r+1 is a standard Gaussian in R n-r+1 , E 1
Xn-r+1 2q-1 < +∞ (see Lemma A.13). The other factors on the right-hand side of (5.31) are expectations of polynomials is some standard Gaussian variables, so they are finite.

Then, Λ(λ

) = Λ + O λ -1 2
, and the same kind of computations as in the proof of Lemma 5.4 gives (5.29), which concludes the proof of Lemma 5.6. Note that, since M is compact, R(x) is bounded, independently of x. We need this fact to ensure that the error term in (5.29) is independent of x.

Proof of Lemma 5.7.

Let (L, S) ∼ N (0, Λ) in R r ⊗ (T * x M ⊕ Sym(T * x M
)) and U ∼ N (0, Id) in T x M be independent of (L, S). By (5.31):

(5.32)

E det ⊥ (L) Tr S , L † * (U ) ∧2m = E X n-r+2 . . . X n X n-r+1 2m-1 E Tr S , U ′ ∧2m ,
where U ′ ∼ N (0, Id) in R r , X p ∼ N (0, Id) in R p for all p, and U ′ , S, X n-r+1 , . . . , X n are globally independent.

Recall that we are only interested in the restriction of S to ker(L) ⊂ T x M . But L and S are independent, as one can see on the expression of Λ (5.27), and the distribution of S is invariant under orthogonal transformations of T x M . Thus, we can consider S restricted to any 2m-dimensional subspace of T x M in our computations. For simplicity, we restrict S to V , the span of ∂ ∂x1 , . . . , ∂ ∂x2m . We now compute the term E Tr S /V , U ′ ∧2m . By Lemma 4.2,

E S S /V , U ′ ∧2m = (2m)! 2 m m! E S S /V , U ′ ∧2 ∧m . Assuming that S /V = 1 i,k 2m
S ik dx i ⊗ dx k , with S ik ∈ R r , we have:

S /V , U ′ = 1 i,k 2m S ik , U ′ dx i ⊗ dx k .
By Lemma A.8,

E S S /V , U ′ ∧2 = 1 i,j,k,l 2m E S [ S ik , U ′ S jl , U ′ ] (dx i ∧ dx j ) ⊗ (dx k ∧ dx l ) = 1 i,j,k,l 2m U ′ , Λ ik,jl U ′ (dx i ∧ dx j ) ⊗ (dx k ∧ dx l ),
where we denoted by Λ ik,jl the covariance operator of S ik and S jl . Then,

E S S , U ′ ∧2m = (2m)! 2 m m! 1 i1,...,im 2m 1 j1,...,jm 2m 1 k1,...,km 2m 1 l1,...,lm 2m m p=1 U ′ , Λ ipkp,jplp U ′ × (dx i1 ∧ dx j1 ∧ • • • ∧ dx im ∧ dx jm ) ⊗ (dx k1 ∧ dx l1 ∧ • • • ∧ dx km ∧ dx lm ) = (2m)! 2 m m! σ,σ ′ ∈S2m ε(σ)ε(σ ′ ) m p=1 U ′ , Λ σ(2p-1)σ ′ (2p-1),σ(2p)σ ′ (2p) U ′ (dx ⊗ dx),
where S 2m is the set of permutations of {1, . . . , 2m}, ε : S 2m → {-1, 1} denotes the signature morphism and dx = dx 1 ∧• • • ∧dx 2m . We get the last line by setting σ(2p-1) = i p , σ(2p) = j p , σ ′ (2p -1) = k p and σ ′ (2p) = l p and reordering the wedge products.

Since our local coordinates are such that ∂ ∂x1 , . . . , ∂ ∂x2m is orthonormal at x, we have Tr(dx ⊗ dx) = dx 2 = 1. Thus, (5.33)

Tr E S S , U ′ ∧2m = (2m)! 2 m m! σ,σ ′ ∈S2m ε(σσ ′ ) m p=1
U ′ , Λ σ(2p-1)σ ′ (2p-1),σ(2p)σ ′ (2p) U ′ .

By equation (5.27), for any σ, σ ′ ∈ S 2m and for any p ∈ {1, . . . , m},

Λ σ(2p-1)σ ′ (2p-1),σ(2p)σ ′ (2p) = K(p, σ, σ ′ ) Id,
where

(5.34) K(p, σ, σ ′ ) =      β if σ(2p -1) = σ ′ (2p -1) and σ(2p) = σ ′ (2p), 1 if σ(2p -1) = σ ′ (2p) and σ(2p) = σ ′ (2p -1), 0 otherwise. Note that K(p, σ, σ ′ ) = K(p, id, σ -1 • σ ′ )
, where id stands for the identity permutation. Then, setting

τ = σ -1 • σ ′ , (5.35) σ,σ ′ ∈S2m ε(σσ ′ ) m p=1 U ′ , Λσ(2p-1)σ ′ (2p-1),σ(2p)σ ′ (2p) U ′ = σ,σ ′ ∈S2m ε(σσ ′ ) U ′ 2m m p=1 K(p, σ, σ ′ ) = (2m)! U ′ 2m τ ∈S2m ε(τ ) m p=1 K(p, id, τ ).
From the Definition (5.34) of K(p, id, τ ), we get that m p=1 K(p, id, τ ) = 0 if and only if τ is a product of transpositions of the type ((2p -1) (2p)). Now, if I ⊂ {1, . . . , m} and τ = p∈I ((2p -1) (2p)), we have: (5.37) E Tr S , U ′ ∧2m = (-1)

m n + 4 n + 2 m ((2m)!) 2 2 m m! E U ′ 2m .
Then, by (5.32), (5.37) and Lemma A.13,

E det ⊥ (L) Tr S , L † * (U ) ∧2m = (-1) m n + 4 n + 2 m ((2m)!) 2 2 m m! (2π) r 2
Vol (S n-r ) Vol (S 1 )

Vol S n-r+1 Vol (S n ) Vol S r-1
Vol (S n-1 ) .

We conclude the proof of Lemma 5.7 by computing:

(2π) r 2 (2m)! 2 m m! Vol (S n-r ) 2 = (2π) r 2
2 m Γ m + 1 2 √ π π m+ 1 2 Γ m + 1 2 = (2π) n 2 .

Proof of Theorem 1.4

Let us now adapt the proof of the previous section to the case of real algebraic submanifolds.

Once again, we need only consider the case where r < n and nr is even. The framework is the same as in sections 2.6 and 5.3. We already know that RH 0 (X , E ⊗ L d ) is 0-ample by Corollary 3.10. We also have the estimate (5.17) for det(E d ) along the diagonal, where E d is the Bergman kernel of E ⊗ L d . As in the Riemannian case, we use (4.8) and Kac-Rice formula:

(5.38)

E[χ(Z s )] = 1 m!(2π) n 2 x∈RX E det ⊥ ∇ d x s Tr (R s (x) ∧m ) s(x) = 0 det(E d (x, x)) |dV RX | ,
where ∇ d is any real connection on (E ⊗ L d ) and R s denotes the Riemann tensor of Z s . We need to compute the conditional expectation in (5.38) for some fixed x ∈ RX . Since it does not depend on our choice of connection, we will use one that is adapted to x as in sections 3.3 and 5.3. Let x ∈ RX , then we consider the scaled variables: x (E ⊗ L d ) similar to the one defined by (5.5) and (5.22).

We compute the matrix Λ(d) of Var(t d , L d , S d ) in this basis. For this we use the estimates for the blocks of Λ(d) given by Proposition 3.8 and (5.39). Namely, for all i, j, k and l ∈ {1, . . . , n}, with i k and j l,

Λ ik 20 (d) = O d -1 , Λ ik,j 21 (d) = O d -1 , (5.40) Λ ik,jl 22 (d) =      2I r +O d -1 if i = j = k = l, I r +O d -1 if i = j = k = l, O d -1
otherwise.

(

2   + O d -1 ,
where the empty blocks are zeros. Let Λ denote the leading term in equation (5.42) and let (L, S) ∼ N (0, Λ) in R(E ⊗ L d ) x ⊗ (T * x RX ⊕ Sym(T * x RX )). By Lemma A.12, ( Ld , Sd ) converges in distribution to (L, S).

Let U ∼ N (0, Id) in T x RX be independent of all the other variables. By (4.9),

(5.43)

R s (x) = R(x) + 1 2 E ∇ 2,d x s , (∇ d x s) † * (U ) ∧2 .
As in the case of Riemannian random waves (5.21), we have:

(5.44) E det ⊥ ∇ d x s Tr(R s (x) ∧m ) s(x) = 0 = d n+1 π n r 2 E det ⊥ Ld Tr R(x) + d E U Sd , L † * d (U ) ∧2 ∧m
.

The proof of Lemma 5.6 adapts immediately to this setting, so that:

(5.45

) E det ⊥ Ld Tr R(x) + d E U Sd , L † * d (U ) ∧2 ∧m = d m m! (2m)! E det ⊥ (L) Tr S , L † * (U ) ∧2m 1 + O d -1 ,
where the error term is uniform in x ∈ M .

Lemma 5.9. Let (L, S) ∼ N (0, Λ) R(E ⊗L d ) x ⊗(T * x RX ⊕Sym(T * x RX )) and U ∼ N (0, Id) in T x M be independent of (L, S). We have:

E det ⊥ (L) Tr S , L † * (U ) ∧2m = (-1) m (2π) n 2 (2m)!
Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1 ) .

Once this lemma is proved, we get Theorem 1.4 immediately by (5.17), (5.38), (5.44), (5.45) and Lemma 5.9. We sketch the proof of Lemma 5.9 which is, unsurprisingly, adapted from the proof of Lemma 5.7.

Proof. The only difference between Lemmas 5.7 and 5.9 comes from the definition of Λ which is not the same in the algebraic case. The proof is exactly the same as the proof of Lemma 5.7 until the definition of K (5.34). The Λ ik,jl are now given by (5.42), hence we have to change the definition of K. In this setting,

K(p, σ, σ ′ ) = 1 if σ(2p -1) = σ ′ (2p) and σ(2p) = σ ′ (2p -1), 0 otherwise, so that m p=1 K(p, id, τ ) is 0, unless τ = τ 0 = m p=1
((2p -1)(2p)). Then (5.36) becomes:

τ ∈S2m ε(τ ) m p=1 K(p, id, τ ) = ε(τ 0 ) m p=1 K(p, id, τ 0 ) = (-1) m .
This explains why the factor -n+4 n+2 m becomes (-1) m in the algebraic case. What remains of the proof is as in Lemma 5.7.

Two special cases

In some special cases, the covariance kernel is known explicitly. It is then possible to prove more precise results. In this section, we sketch what happens on the flat torus and in the real projective space. In these cases, we get the expectation of the volume and the Euler characteristic of our random submanifolds for fixed λ (resp. fixed d).

The flat torus

Let T n = R n (2πZ) n denote the torus of dimension n that we equip with the quotient of the Euclidean metric on R n . We have Vol (T n ) = (2π) n . We identify functions on T n and (2πZ) n -periodic functions on R n . Then, the Laplacian is ∆ = -

n i=1 ∂ 2 ∂x 2 i
, and it is known that its eigenvalues are the integers of the form p 2 = (p i ) 2 , with p = (p 1 . . . , p n ) ∈ N n . The eigenspace associated to 0 is spanned by the constant function x → (2π) -n 2 . For λ > 0, the eigenspace associated to λ is spanned by the normalized functions of the form:

x → √ 2 (2π) n 2 sin( p , x ) and x → √ 2 (2π) n 2 cos( p , x ),
where p ∈ Z n is such that p 2 = λ, and • , • is the canonical scalar product on R n .

We set

B λ = {p ∈ Z n | p 2 λ}.
After some computations, we get e λ , the spectral function of the Laplacian on T n :

(6.1) ∀λ 0, ∀x, y ∈ T n , e λ (x, y) = 1 (2π) n p∈B λ cos( p , x -y ).
Let λ 0 and r ∈ {1, . . . , n}, let V λ be spanned by the eigenfunctions of ∆ associated to eigenvalues smaller than λ, and let f ∼ N (0, Id) in (V λ ) r . For all x ∈ T n , j 2

x (f ) is a centered Gaussian variable whose variance is determined by Lemma 2.4 , Lemma 2.6 and the above formula (6.1). Note that e λ and its derivatives are constant along the diagonal.

We can compute explicitly the variance of j 2 x (f ) (which is independent of x) and follow the same steps as in sections 5.2 and 5.4. Only, this time, we can make exact computations with λ fixed, instead of deriving asymptotics. These computations are not difficult, and similar to what we already did, so we simply state the final results. Note that the scaling of the variables has to be adapted. Proposition 6.1. On the flat torus T n , let λ 0 and let f 1 , . . . , f r be independent standard Gaussian functions in V λ , with 1 r n. We have:

E[Vol (Z f )] =   1 |B λ | (p1,...,pn)∈B λ (p 1 ) 2   r 2 (2π) n Vol (S n-r ) Vol (S n ) ,
where

B λ = {p ∈ Z n | p 2 λ} and |B λ | denotes the cardinal of B λ .
This result was already known, see [START_REF] Rudnick | On the volume of nodal sets for eigenfunctions of the Laplacian on the torus[END_REF], where Rudnick and Wigman compute the variance of Vol (Z f ) when r = 1, and the references therein. Proposition 6.2. On the flat torus T n , let λ 0 and let f 1 , . . . , f r be independent standard Gaussian functions in V λ , with 1 r n. If nr is even, we have:

E[χ(Z f )] = (-1) n-r 2   1 |B λ | (p1,...,pn)∈B λ (p 1 ) 2   n 2 (2π) n Vol S n-r+1 Vol S r-1 π Vol (S n ) Vol (S n-1 ) , where B λ = {p ∈ Z n | p 2 λ} and |B λ | denotes the cardinal of B λ .
Remark 6.3. For this last result, one of the points that make the computations tractable is that the Riemann tensor of the ambient manifold is zero.

The projective space

We consider the algebraic case with X = CP n , E = C r × CP n with the standard Hermitian metric on each fiber, and L = O(1) the hyperplane bundle with its usual metric.

E d (x, y) = r q=1 ζ q ⊗ ζ q ⊗ e d (x, y).
In this case, RX = RP n and the elements of RH 0 (X , E ⊗ L d ) are r-tuples of real homogeneous polynomials of degree d in n + 1 variables. We denote by R hom d [X 0 , . . . , X n ] the space of real homogeneous polynomials of degree d in X 0 , . . . , X n . Let α = (α 0 , . . . , α n ) ∈ N n+1 , then we set [START_REF]Expected topology of random real algebraic submanifolds[END_REF]) is given by the sections:

X α = X α0 0 • • • X αn n , |α| = α 0 + • • • + α n and, if |α| = d, we also set d α = d! α0!•••αn! . It is well-known that an orthonormal basis of R hom d [X 0 , . . . , X n ] for the inner prod- uct (2.
(6.3) s α = (n + d)! π n d! d α X α , with |α| = d,
see [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] Bürgisser | Average volume, curvatures, and Euler characteristic of random real algebraic varieties[END_REF][START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF]. Then, formally, (6.4)

e d = (n + d)! π n d! |α|=d d α X α Y α = (n + d)! π n d! X , Y d .
More precisely, we consider the local coordinates (x 1 , . . . , x n ) → [1 : x 1 : • • • :

x n ], defined on a neighborhood of [1 : 0 : • • • : 0], and the real holomorphic frame s (d,0,...,0) for O(d) on this neighborhood. In these coordinates, (6.5) e d (x, y) = (1 + x , y ) d s (d,0,...,0) (x) ⊗ s (d,0,...,0) (y) , with x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ). Since everything is invariant under orthogonal transformations of RP n , this totally describes e d . In particular, the covariances that appear in the proofs of Theorems 1.3 and 1.4 are given, in this case, by the values of the function

(x, y) → (n + d)! π n d! (1 + x , y ) d
and its derivatives at the point (0, 0) in R n × R n . Let d ∈ N, r ∈ {1, . . . , n} and s ∼ N (0, Id) in (R hom d [X 0 , . . . , X n ]) r . For every x ∈ RP n , j 2,d x (s) is a centered Gaussian variable whose variance is determined by Lemma 2.13, (6.2) and (6.5). Once again, we can compute the variance of j 2,d x (s) explicitly, and it does not depend on x ∈ RP n . Then, we can follow the steps of the proof of Theorem 1.3, and make exact computations for fixed degree d. This yields Kostlan's result [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF]. Note that we have to adapt the scaling of the variables. Theorem 6.4 (Kostlan). In the real projective space RP n , let d ∈ N and let P 1 , . . . , P r be independent standard Gaussian polynomials in R hom d [X 0 , . . . , X n ], with 1 r n. Let Z s denote the common zero set of P 1 , . . . , P r . We have:

E[Vol (Z s )] = d r 2 Vol RP n-r .
Assuming that nr is even and equals 2m, we can also adapt the proof of Theorem 1.4 to get the expected Euler characteristic for fixed d. This computation is a bit more complex than on the torus since the Riemann curvature of RP n is not zero.

By Kac-Rice formula and equations (6.2) and (4.9),

(6.6) E[χ(Z s )] = 1 m!(2π) n 2 x∈RP n 1 (e d (x, x)) n 2 × • • • E det ⊥ ∇ d x s Tr R(x) + 1 2 E ∇ 2,d x s , (∇ d x s) † * (U ) ∧2 ∧m |dV RP n | ,
where R is the Riemann curvature of RP n and U ∼ N (0, Id) in T x RP n . We used the fact that s(x), ∇ d x s and ∇ 2,d x s are independent in this particular case. This is why the expectation is not conditioned on s(x) = 0.

Since everything is invariant under orthogonal transformations of RP n , we only need to compute the expectation in the integrand at the point x = [1 : 0 : • • • : 0]. We do this in the same chart as for (6.5) above. We get:

R(x) = 1 2 1 i,j n dx i ∧ dx j ⊗ dx i ∧ dx j .
Following the computations of section 5.4, we get (after a suitable rescaling) that the expectation in the integrand of (6.6) equals:

(6.7) d r 2 m q=0 (d -1) q 2 q m q E X n-r+2 . . . X n X n-r+1 2q-1 E Tr R(x) ∧(m-q) ∧ S , U ′ ∧2q ,
where U ′ ∈ R r , S ∈ (RO(d) x ) r ⊗ Sym(T * x RP n ) and X p ∈ R p (for nr + 1 p n) are globally independent standard Gaussian vectors. By the same kind of computations as in the proof of Lemma 5.7, we get:

(6.8) E S S , U ′ ∧2 = -U ′ 2 1 i,j n dx i ∧ dx j ⊗ dx i ∧ dx j .
Restricting R and S to the span of ∂ ∂x1 , . . . , ∂ ∂x2m , we have:

∀q ∈ {0, . . . , m}, Tr R(x) ∧m-q ∧ E S S , U ′ ∧2 ∧q = (-1) q 1 2 m-q (2m)! U ′ 2q .
As in the proof of Lemma 5.7, we compute the expectation of this term with respect to the variable U ′ by Lemma A.13. The same lemma allows us to compute the value of (6.7). Finally, we recover Bürgisser's result [START_REF] Bürgisser | Average volume, curvatures, and Euler characteristic of random real algebraic varieties[END_REF]. Theorem 6.5 (Bürgisser). In the real projective space RP n , let d ∈ N and let P 1 , . . . , P r be independent standard Gaussian polynomials in R hom d [X 0 , . . . , X n ], with 1 r n. Let Z s denote the common zero set of P 1 , . . . , P r . If nr is even, we have:

E[χ(Z s )] = d r 2 n-r 2 p=0 (1 -d) p Γ p + r 2 p! Γ r 2 .
These bilinear forms are associated, as above, to the following operators:

Λ 11 : V * 1 → V 1 , Λ 12 : V * 2 → V 1 , Λ 22 : V * 2 → V 2 and Λ 21 : V * 1 → V 2 .
Since Var(X) is symmetric, Cov(X 1 , X 2 )(ξ, η) = Cov(X 2 , X 1 )(η, ξ) for any ξ and η.

Definition A.6. We say that Cov(X 1 , X 2 ) is the covariance of X 1 and X 2 , and that Λ 12 is their covariance operator.

We choose orthonormal bases (e 1 , . . . , e n ) and (e ′ 1 , . . . , e ′ r ), of V and V ′ respectively, such that e r = θ, e ′ r = θ ′ and (e 1 , . . . , e r ) is a basis of ker(L) ⊥ . Then, ∀i ∈ {1, . . . , n}, Le i , θ ′ = e i , L * θ ′ = L * θ ′ e i , θ .

Thus the matrix of L in these bases has the form:

(B.3)   A * . . . * 0 0 ••• 0 L * θ ′ 0 ••• 0   , and det ⊥ (L) = |det(A)| L * θ ′ . Let π θ and π θ ′ denote the orthogonal projections along R • θ in V and along R • θ ′ in V ′ respectively. We define L ′ : V → (R • θ ′ ) ⊥ by L ′ = π θ ′ • L • π θ . Then |det(A)| = det ⊥ (L ′ ) ,
and L ′ does not depend on our choice of bases. Finally, we have:

(B.4) det ⊥ (L) , L † * U = det ⊥ (L ′ ) L * θ ′ , ρθ ′ L * θ ′ .
To prove Proposition 5.8, we will show that det ⊥ (L ′ ) , L * θ ′ , ρ and θ ′ are independent and identify their distributions.

If L and U are independent standard Gaussians, then almost surely L ∈ B and we can consider (L, U ) as a random element of B × V . Then (L, Ũ ) is a random element of F and its distribution is characterized by:

E φ(L, Ũ ) = L∈B U∈V φ(L, Ũ ) dν n (U ) dν nr (L) = L∈B Ũ∈ker(L) ⊥ φ(L, Ũ ) dν r ( Ũ ) dν nr (L),
for any bounded continuous function φ : F → R. Recall that dν N stands for the standard Gaussian measure in dimension N . We get the distribution of (L, θ, ρ) ∈ S × R + by a polar change of variables in the innermost integral: for any bounded continuous φ :

S × R + → R, E[φ(L, θ, ρ)] = L∈B θ∈S(ker(L) ⊥ ) +∞ ρ=0 φ(L, θ, ρ)ρ r-1 e -ρ 2 2 dρ (2π) r 2 dθ dν nr (L),
where dρ is the Lebesgue measure on R and dθ is the Euclidean measure on the sphere S(ker(L) ⊥ ).

This distribution is a product measure on S × R + , thus (L, θ) and ρ are independent variables. Since det ⊥ (L ′ ) , θ ′ , L * θ ′ only depends on (L, θ), this triple is independent of ρ. Besides, ρ is distributed as the norm of a standard Gaussian vector in R r since its density with respect to the Lebesgue measure is ρ → Vol S r-1 (2π) -r 2 ρ r-1 e -ρ 2 2 on R + and vanishes elsewhere. Finally, the distribution of (L, θ) satisfies:

(B.5) E[φ(L, θ)] = L∈B θ∈S(ker(L) ⊥ ) φ(L, θ) dθ Vol (S r-1 ) dν nr (L),
for any bounded and continuous φ : S → R.

We will now compute the distribution of (L, θ ′ ) in B × S(V ′ ).

Lemma B.1. For any bounded and continuous φ :

B × S(V ′ ) → R, (B.6) E[φ(L, θ ′ )] = θ ′ ∈S(V ′ ) L∈B φ(L, θ ′ ) det ⊥ (L ′ ) L * θ ′ r-1 e -L 2 2 dθ ′ Vol (S r-1 ) dL (2π) nr 2
, where dθ ′ is the Euclidean measure on S(V ′ ), dL is the Lebesgue measure on V ′ ⊗ V * and L ′ is defined as in (B.4).

Proof. Fixing some φ, we see from (B.5) and (B.1) that:

E[φ(L, θ ′ )] = L∈B θ∈S(ker(L) ⊥ ) φ L, (L † ) * θ (L † ) * θ e -L 2 2 dθ Vol (S r-1 ) dL (2π) nr 2 .
Then we make the change of variables θ

′ = ψ(θ) = (L † ) * θ (L † ) * θ in the innermost integral, with L fixed. Recalling (B.2), we have ψ -1 : θ ′ → L * θ ′ L * θ ′ from S(V ′ ) to S(ker(L) ⊥ ). Now, the differential of ψ -1 at θ ′ ∈ S(V ′ ) satisfies: ∀v ∈ (R • θ ′ ) ⊥ , d θ ′ (ψ -1 ) • v = 1 L * θ ′ L * v -L * v , L * θ ′ L * θ ′ L * θ ′ L * θ ′ = π θ (L * v) L * θ ′ .
As above, we choose an orthonormal basis e ′ 1 , . . . , e ′ r-1 of (R • θ ′ ) ⊥ and an orthonormal basis (e 1 , . . . , e r-1 ) of (R • θ ⊕ ker(L))

⊥ . In these coordinates we have:

det(d θ ′ (ψ -1 )) = det π θ • L * /(θ ′ ) ⊥ L * θ ′ r-1 = |det(A * )| L * θ ′ r-1 = |det(A)| L * θ ′ r-1 = det ⊥ (L ′ ) L * θ ′ r-1 ,
where A is as in (B.3) and L ′ as in (B.4). This proves (B.6).

We can now compute the joint distribution of det ⊥ (L ′ ) , θ ′ , L * θ ′ from the one of (L, θ ′ ). We fix θ ′ ∈ S(V ′ ) and an orthonormal basis (e ′ 1 , . . . e ′ r ) of V ′ such that e ′ r = θ ′ . The choice of L ∈ V ′ ⊗ V * is equivalent to the choice of the r independent standard Gaussian vectors L * e ′ 1 , . . . , L * e ′ r in V . For simplicity, we set L i = L * e ′ i . Note that if we choose a basis for V as well, these are the rows of the matrix of L. We can rewrite (B.6) as: , where dL i denotes the Lebesgue measure in the i-th copy of V . We set α r = Lr Lr and ρ r = L r . Here, L ′ = π θ ′ • L • π αr depends on α r and θ ′ but not on ρ r . Making a polar change of variables, the above integral equals:

(n+1)(r-1) 2

.

Then, ρ r = L * θ ′ is independent of (θ ′ , α r , L 1 , . . . , L r-1 ), hence of θ ′ , det ⊥ (L ′ ) . Moreover ρ r is distributed as the norm of a standard Gaussian in R n-r+1 , since it has the same density. Finally, (θ ′ , α r , L 1 , . . . , L r-1 ) has the density: (θ ′ , α r , L 1 , . . . , L r-1 ) → e det ⊥ (L ′ ) Vol (S r-1 ) Vol (S n-r ) with respect to dθ ′ ⊗ dα r ⊗ dL 1 ⊗ • • • ⊗ dL r-1 .

For i ∈ {1, . . . , r-1} we denote by L ⊥ i the orthogonal projection of L i onto the orthogonal of the subspace spanned by (α r , L 1 , . . . , L i-1 ).

Lemma B.2. For any

L ∈ B, det ⊥ (L ′ ) = L ⊥ 1 • • • L ⊥ r-1 .
Proof. If one of the L ⊥ i is zero, then the vectors α r , L 1 , . . . , L r-1 are linearly dependent and L is singular. Since we assumed L ∈ B this is not the case and

L ⊥ 1 L ⊥ 1 , . . . , L ⊥ r-1 L ⊥ r-1
is an orthonormal basis of ker(L ′ ) ⊥ . Writing the matrix of the restriction of L ′ to ker(L ′ ) ⊥ in this basis and e ′ 1 , . . . , e ′ r-1 , we see that it is lower triangular with diagonal coefficients L ⊥ 1 ,. . . , L ⊥ r-1 . This proves the lemma.

Let φ be a continuous bounded function from S(V ′ ) × R + to R. We have: dL ⊥ r-1 . . . dL ⊥ 1 dθ ′ dα r Vol (S r-1 ) Vol (S n-r ) .

E φ θ ′ , det ⊥ (L ′ ) = φ θ ′ , det ⊥ (L ′ ) e -1 2 r-1 i=1 Li 2 det ⊥ (L ′ ) (2π)
Then we make polar changes of variables: for each i we set

ρ i = L ⊥ i and α i = L ⊥ i L ⊥ i .
Note that, when L 1 , . . . , L i-1 are fixed, L ⊥ i is a vector in a space of dimension ni. We have: .

E φ θ ′ , det ⊥ (L ′ ) = φ θ ′ , r-1 i=1 ρ i e -1 2 r-1 i=1 ρ 2 i r-1 i=1 (ρ i ) n-i (2π)
This shows that θ ′ , ρ 1 , . . . , ρ r-1 are independent variables, that θ ′ is uniformly distributed in S(V ′ ) and that, for all i ∈ {1, . . . , r -1}, ρ i is distributed as the norm of a standard Gaussian vector in R n+1-i . Finally, this shows that det ⊥ (L ′ ) is distributed as

r-1 i=1 ρ i .
Putting all we have done so far together, we see that det ⊥ (L) , L † * U is distributed as

X n • • • X n-r+1 , ρθ ′ X n-r+1
, where X p is a standard Gaussian vector in R p for all p. Moreover, θ ′ is uniformly distributed in S(V ′ ), ρ is distributed as the norm of a standard Gaussian vector in R r , and all these variables are globally independent. Finally U ′ = ρθ ′ is a standard Gaussian in V ′ , independent of X n , . . . , X n-r+1 so we have proved Proposition 5.8. An immediate corollary of this is the following.

Corollary B.3. Let V and V ′ be two Euclidean spaces of dimension n and r respectively, with 1 r n. Let L ∼ N (0, Id) in V ′ ⊗ V * . Then det ⊥ (L) is distributed as X n-r+1 • • • X n , where for all p ∈ {nr + 1, . . . , n}, X p ∼ N (0, Id) in R p and these vectors are globally independent.

C Proof of the Kac-Rice formula

In this appendix, we give a proof of the Kac-Rice formula using Federer's coarea formula. This was already done by Bleher, Shiffman and Zelditch in [5, thm. 4.2]. See also [1, chap. 6].

C.1 The coarea formula

We start by stating the coarea formula in the case of a smooth map between smooth Riemannian manifolds. A proof in this special case can be found in [START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF]Appendix] (see [11, thm. 3.2.12] for the general case).

Let π : M → M be a smooth map between smooth Riemannian manifolds of respective dimensions m and n. We assume that m n. Note that the innermost integral on the right-hand side is only defined almost everywhere.

C.2 The double-fibration trick

We now describe the double-fibration trick, which consists in applying the coarea formula twice, for different fibrations. Let M 1 and M 2 be two smooth Riemannian manifolds of dimension n 1 and n 2 respectively. Let F : M 1 × M 2 → R r be a smooth submersion, and let Σ = F -1 (0). We equip Σ with the restriction of the product metric on We would like the integrand on the right-hand side to depend on F rather than on π 1 and π 2 . Let ∂ 1 F and ∂ 2 F denote the partial differentials of F with respect to the first and second variable respectively. For any x = (x 1 , x 2 ) ∈ Σ,

T x Σ = {(v 1 , v 2 ) ∈ T x1 M 1 × T x2 M 2 | ∂ 1 F (x) • v 1 + ∂ 2 F (x) • v 2 = 0} .

C.3 Proof of Theorem 5.3

Finally, we prove Theorem 5.3. Let M be a closed Riemannian manifold of dimension n and V be a subspace of C ∞ (M, R r ) of dimension N (recall that 1 r n). We assume that V is 0-ample, so that F : (f, x) → f (x) is a smooth submersion from V × M to R r and

Σ = F -1 (0) = {(f, x) ∈ V × M | f (x) = 0}
is a submanifold of codimension r of V × M . Let df denote the Lebesgue measure on V or on a subspace of V . Let φ : Σ → R be a Borel measurable function, by Proposition C.3,

E x∈Z f φ(f, x) |dV f | = 1 (2π) N 2 f ∈V x∈Z f φ(f, x)e -f 2 2 |dV f | df = 1 (2π) N 2 x∈M f ∈ker(j 0 x ) φ(f, x)e -f 2 2 det ⊥ (∂ 2 F (x)) |det ⊥ (∂ 1 F (x))| df |dV M | .
Recall that for all x ∈ M , j 0 x : f → f (x) is onto, since V is 0-ample. In particular, ker(j 0 x ) has codimension r and V splits as ker(j 0

x ) ⊕ ker(j 0 x ) ⊥ . We recognize the innermost integral to be a conditional expectation given f (x) = 0 (see Corollary A.11). Thus, Since f ∼ N (0, Id), equation (A.8) shows that j 0 x j 0 * x is the variance operator of f (x) = j 0 x (f ). Then, by (2.9), det j 0 x (j 0 x ) * = det(Var(f (x))) = det(E(x, x)). In particular, this quantity does not depend on f ∈ V . Equations (C.5), (C.6) and this last equality prove Theorem 5.3.

(C.5) E x∈Z f φ(f, x) |dV f | = 1 (2π)

1 2 . 2 2 . 3 2 . 4 2 . 5 2 . 6 10 3 16 4 21 5 34 6

 1222324252610162134 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The incidence manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The covariance kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Random jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Riemannian random waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The real algebraic setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimates for the covariance kernels 13 3.1 The spectral function of the Laplacian . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Hörmander-Tian peak sections . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The Bergman kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An integral formula for the Euler characteristic of a submanifold 18 4.1 The algebra of double forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 The Chern-Gauss-Bonnet theorem . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 The Gauss equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 An expression for the second fundamental form . . . . . . . . . . . . . . . . . Proofs of the main theorems 23 5.1 The Kac-Rice formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.3 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.4 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.5 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two special cases 36 6.1 The flat torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 The projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

  Let (M, g) be a smooth closed manifold of positive dimension n. We denote by |dV M | the Riemannian measure on M induced by g. That is, if x = (x 1 , . . . , x n ) are local coordinates in an open set U ⊂ M and φ is a smooth function with compact support in U , (2.1) M φ |dV M | = x∈U φ(x) det(g(x)) dx 1 . . . dx n .

Remark 4 . 4 .

 44 If M is a closed manifold of odd dimension then χ(M ) = 0, see[18, cor. 3.37].

Proposition 4 . 6 (

 46 Gauss equation). Let (M, g) be a Riemannian manifold and let M be a smooth submanifold of M , such that dim(M ) > dim( M )1. Let R and R denote the Riemann curvature of M and M respectively, and let II be the second fundamental form of M ⊂ M . Then, in the sense of double forms:

2 m.

 2 , id, τ ) = β m-|I| and ε(τ ) = (-1) |I| , where |I| stands for the cardinal of I. p β m-p = (β -1) m = (-1) m n + 4 n + Finally, by equations (5.33), (5.35) and (5.36),

( 5 .

 5 39) (t d , L d , S d ) = π n d n s(x), π n d n+1 ∇ d x s, π n d n+2 ∇ 2,d x s , and ( Ld , Sd ) ∈ R(E ⊗ L d ) x ⊗ (T * x RX ⊕ Sym(T * x RX )) distributed as (L d , S d ) given t d = 0. As in section 5.3, let (x 1 , . . . , x n ) be real holomorphic coordinates centered at x and such that ( ∂ ∂x1 , . . . , ∂ ∂xn ) is orthonormal at x. Let (ζ d 1 , . . . , ζ d r ) be an orthonormal basis of R(E ⊗ L d ) x . We get an orthonormal basis of J 2

E 1 2

 1 [φ(L, θ ′ )] = θ ′ ∈S(V ′ ) L1,...,Lr-1∈V Lr∈V φ(L, θ ′ ) det ⊥ (L ′ ) Li 2 dθ ′ Vol (S r-1 ) dL 1 • • • dL r (2π) nr 2

θ 2 ri=1

 2 ′ ∈S(V ′ ) αr ∈S(V ) L1,...,Lr-1∈V +∞ ρr=0 φ(L, θ ′ ) (ρ r ) n-r e -ρ Li 2 det ⊥ (L ′ ) Vol (S r-1 ) dθ ′ dα r dL 1 • • • dL r-1

2 dθ

 2 ′ dα r dL 1 . . . dL r-1 Vol (S r-1 ) Vol (S n-r ) = αr ,θ ′ L ⊥ 1

4 dρ 1  2 i 2 ( 2 

 41222 . . . dρ r-1 dα 1 . . . dα r dθ ′ Vol (S r-1 ) Vol (S n-r ) = ρ1,...,ρr-1,θ ′ φ θ ′ , Vol S n-i e -ρ ρ i ) n-i (2π) n+1-i dρ 1 . . . dρ r-1dθ ′ Vol (S r-1 )

  Let |dV M | (resp. |dV M |) denote the Riemannian measure on M (resp. M ) induced by its metric. By Sard's theorem, for almost every y ∈ M , π -1 (y) is a smooth submanifold of dimension (mn) of M . For such y ∈ M , we denote by |dV y | the Riemannian measure on π -1 (y) induced by the metric of M . When m = n, the dimension π -1 (y) is 0 and |dV y | is just x∈π -1 (y) δ x , where δ x is the Dirac measure at x. Theorem C.1 (Coarea formula, Federer). Let π : M → M be a smooth map between smooth Riemannian manifolds of dimension m and n respectively, with m n. Let φ : M → R be a Borel measurable function. Then:x∈ M φ(x) det ⊥ (d x π) |dV M | = y∈M x∈π -1 (y) φ(x) |dV y | |dV M | ,whenever one of these integrals is well-defined.

M 1 ×M 2

 12 and denote by |dV M1 |, |dV M2 | and |dV Σ | the Riemannian measures on the corresponding manifolds. Finally, let π 1 : Σ → M 1 and π 2 : Σ → M 2 be the projections from Σ to each factor. Assuming that r min(n 1 , n 2 ), we have dim(Σ) = n 1 + n 2r max(n 1 , n 2 ). Thus we can apply the coarea formula both to π 1 and π 2 .Let φ : Σ → R be a Borel measurable function, then:(C.1)y1∈M1 x∈π -1 1 (y1) φ(x) |dV y1 | |dV M1 | = x∈Σ φ(x) det ⊥ (d x π 1 ) |dV Σ | det ⊥ (d x π 1 ) |det ⊥ (d x π 2 )| |dV y2 | |dV M2 | ,whenever one of these integrals is well-defined. Note that if det ⊥ (d x π 2 ) vanishes then π 2 (x) is a critical value of π 2 , and the set of such critical values has measure 0 in M 2 .

Lemma C. 2 .

 2 Let x ∈ Σ, then det ⊥ (d x π 2 ) = 0 if and only if det ⊥ (∂ 1 F (x)) = 0. Moreover, (C.2) det ⊥ (d x π 1 ) det ⊥ (∂ 1 F (x)) = det ⊥ (d x π 2 ) det ⊥ (∂ 2 F (x)) .

r 2 x∈M

 2 E φ(f, x) det ⊥ (∂ 2 F (x)) |det ⊥ (∂ 1 F (x))| f (x) = 0 |dV M | . By equation (2.4), det ⊥ (∂ 2 F (x)) = det ⊥ (d x f ) and (C.6) det ⊥ (∂ 1 F (x)) = det ⊥ j 0 x = det (j 0 x (j 0 x ) * ).

  section 2.3]. See also [14, section 2.2] and the paper by Tian [32, lemmas 1.2 and 2.3], without the higher rank bundle E but with more details. Proposition 3.3. Let p = (p 1 , . . . , p n

  This family is linearly independent for d large enough and spans a space whose direct sum withH d 2,x is RH 0 (X , E ⊗ L d ).We complete it into a basis B of RH 0 (X , E ⊗ L d ) by adding an orthonormal basis of H d 2,x at the end of the previous list. We apply the Gram-Schmidt process to B, starting by the last elements and going backwards. Let B denote the resulting orthonormal basis, and sd,1

	3.19)
	s d,1 0 , . . . , s d,r 0 , s d,1 1 , . . . , s d,r 1 , . . . , s d,1 n , . . . , s d,r n , s d,1 1,1 , . . . , s d,r 1,1 , s d,1 2,2 , . . . , s d,r 2,2 , . . . , s d,1 n,n , . . . , s d,r n,n ,
	s d,1 1,2 , . . . , s d,r 1,2 , s d,1 1,3 , . . . , s d,r 1,3 , . . . , s d,1 1,n , . . . , s d,r 1,n , s d,1 2,3 , . . . , s d,r 2,3 , . . . , s d,1 n-1,n , . . . , s d,r n-1,n .
	0 , . . . , sd,r n , sd,1 1,1 , . . . , sd,r n-1,n
	denote its first elements. This way, sd,r n-1,n is a linear combination of s d,r n-1,n and elements
	of H d 2,x , and sd,1 0 (b i ) 1 i r(n+1)(n+2) 2 elements of B. Let i ∈ 1, . . . , r(n+1)(n+2) is a linear combination of (possibly) all elements of B. We denote by the first elements of B listed above (3.19) and by ( bi ) the corresponding 2

  any vector fields X and Y on M . Here, ∇ M X Y ⊥ stands for the orthogonal projection of ∇ M X Y on T ⊥ M . It is well-known that II is symmetric in X and Y , see [21, lemma 3.6.2]. The second fundamental form encodes the difference between R and R in the following sense, see [21, thm. 3.6.2].

  22) ζ 1 ⊗ dx 11 , . . . , ζ r ⊗ dx 11 , ζ 1 ⊗ dx 22 , . . . , ζ r ⊗ dx 22 , . . . , ζ 1 ⊗ dx nn , . . . , ζ r ⊗ dx nn , ζ 1 ⊗ dx 12 , . . . , ζ r ⊗ dx 12 , ζ 1 ⊗ dx 13 , . . . , ζ r ⊗ dx 13 , . . . , ζ 1 ⊗ dx 1n , . . . , ζ r ⊗ dx 1n ,

. . . , ζ 1 ⊗ dx (n-1)n , . . . , ζ r ⊗ dx (n-1)n .

  Then, ω is the standard Fubini-Study form. We consider the real structures induced by the standard conjugation in C. Let e d denote the Bergman kernel of L d and E d denote the Bergman kernel of E ⊗ L d . Since E is trivial we have a product situation, as in section 2.5, and E d and e d are related as in Lemma 2.6. Let (ζ 1 , . . . , ζ r ) be any orthonormal basis of C r , for all x, y ∈ CP n ,

	(6.2)

A Concerning Gaussian vectors

In this appendix we survey the few facts we need concerning random vectors, especially Gaussian ones. It is essentially borrowed from [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF]Appendix A]. We include it here for the reader's convenience.

A.1 Variance and covariance as tensors

Let V be a real vector space of finite dimension and X a random vector with values in V . For any ξ ∈ V * , ξ(X) is a real random variable. From now on, we assume that these variables are square integrable.

Definition A.1. The expectation (or mean) of X is the linear form on V * defined by:

If m X = 0, we say that X (resp. its distribution dP X ) is centered.

Under the canonical isomorphism V * * ≃ V , we have m X = V x dP X .

Definition A.2. The variance of X is the non-negative symmetric bilinear form on V * defined by:

Remark A.3. Traditionally, the term "variance" is only used when V has dimension 1 and one speaks of "covariance" when dim(V ) 2. We chose to use the term "covariance" for couples of distinct random vectors (see below) and "variance" otherwise. This is the convention of [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], for example.

As a bilinear form on V * × V * , Var(X) is naturally an element of V ⊗ V and we have the following lemma.

Lemma A.4. Let X be a random vector in V , then we have:

Proof. For any ξ and η ∈ V * , we have:

Definition A.5. The variance operator of X is the linear map Λ X : V * → V such that, for any ξ and η ∈ V * ,

By Lemma A.4 we have:

, with X i a random vector in V i , then m X = m X1 + m X2 and the variance form Var(X) splits accordingly into four parts:

Lemma A.7. Let X 1 and X 2 be random vectors in V 1 and V 2 respectively, then we have:

Moreover, for any

Let L : V → V ′ be a linear map between finite-dimensional vector spaces and X be a random vector in V . Then L(X) is a random vector in V ′ with dP L(X) = L * ( dP X ). An immediate consequence of (A.1), (A.2) and (A.4) is that:

where

If X is a random vector in a Euclidean space (V, • , • ), we can see Var(X) as a bilinear symmetric form on V , and Λ X as a self-adjoint operator on V . Then, by (A.3) and (A.5) :

where for any v ∈ V , we set

Lemma A.8. Let X be a random vector in a Euclidean space V , then we have:

Proof. Let v and w ∈ V , then we have:

A.2 Gaussian vectors

The following material can be found either in [1, section 1.2] or [31, section 1.2]. We present it in a coordinate-free fashion, in the spirit of [START_REF] Nicolaescu | Critical sets of random smooth functions on compact manifolds[END_REF]. Let m ∈ R and σ 0, then the Gaussian (or normal ) distribution on R with expectation m and variance σ 2 is the distribution whose characteristic function is ξ → exp imξ -1 2 σ 2 ξ 2 . If σ = 0, this is the Dirac measure centered at m, otherwise it has a density with respect to the Lebesgue measure, given by x → 1

Let V be a real vector space of dimension n, then a random vector X in V is said to be Gaussian, or normally distributed, if for any ξ ∈ V * , ξ(X) is a Gaussian variable in R. Recall that a Gaussian vector has finite moments of all orders and that its distribution is totally determined by its expectation and variance. We denote by N (m, Λ) the Gaussian distribution with expectation m and variance operator Λ and by X ∼ N (m, Λ) the fact that X is distributed according to N (m, Λ). From (A.6) and (A.8) we deduce the following.

Lemma A.9. Let L : V → V ′ be a linear map between finite-dimensional vector spaces and

with the notations of section A.1, Lemma A.9 shows that X 1 ∼ N (m X1 , Λ 11 ) and X 2 ∼ N (m X2 , Λ 22 ). Besides, X 1 and X 2 are independent if and only if Cov(X 1 , X 2 ) = 0, or equivalently Λ 12 = 0.

If Var(X 1 ) is non-degenerate then X 2 has the same distribution as

This is shown in [1, prop. 1.2]. From this, we deduce that the distribution of X 2 given X 1 = x 1 is Gaussian in V 2 with expectation m X2 + Λ 21 (Λ 11 ) -1 (x 1m X1 ) and variance operator Λ 22 -Λ 21 (Λ 11 ) -1 Λ 12 . We use this in the case where X is centered and x 1 = 0.

Corollary A.11. Let X = (X 1 , X 2 ) be a centered Gaussian vector in V 1 ⊕ V 2 and assume that Var(X 1 ) is non-degenerate. Then the distribution of X 2 given X 1 = 0 is a centered Gaussian in V 2 with variance operator

In what follows, we assume that V is a Euclidean space. Recall that in this case, we can see the variance operator of a random vector as an endomorphism of V . We will say that N (0, Id) is the standard normal distribution on V , where Id denotes the identity map on V .

If Var(X) is non-degenerate, then dP X has the following density with respect to the Lebesgue measure on V :

If Var(X) is singular, dP X is supported on ker(Λ X ) ⊥ .

Lemma A.12. Let (X k ) k∈N be a sequence of random vectors in V such that

Proof. Under the hypothesis of the lemma, the characteristic function of X k converges pointwise to the characteristic function of a Gaussian vector X ∼ N (m, Λ). Then, Lévy's continuity theorem gives the result.

We conclude this appendix by computing two Gaussian expectations.

Lemma A.13. Let X ∼ N (0, Id) with values in a Euclidean space of dimension n and let k ∈ Z such that k > -n. Then,

Vol S n-1 Vol (S n+k-1 ) .

Proof.

Assuming we proved Proposition 5.8 (more precisely its Corollary B.3), we have the following.

Lemma A.14. Let V and V ′ be two Euclidean spaces of dimension n and r respectively, with 1 r n. Let L ∼ N (0, Id) in V ′ ⊗ V * . Then:

Gaussian in R p for all p and X n-r+1 , . . . , X n independent. Then, using Lemma A.13,

Vol (S n-r ) Vol (S n ) .

B Proof of Proposition 5.8

This appendix is devoted to the proof of Proposition 5.8 which is a reformulation of [9, prop. 3.12]. Let V and V ′ be two Euclidean spaces of dimension n and r respectively, with 1 r n. The space V ′ ⊗ V * of linear maps from V to V ′ comes with a natural scalar product induced by those on V and V ′ . The set of linear maps of rank less than r is an algebraic submanifold of V ′ ⊗ V * of codimension at least 1, hence it has measure 0 for any non-singular Gaussian measure. Let L be a standard Gaussian vector in V ′ ⊗ V * , then the rank of L is r almost surely. Hence L † is well-defined almost surely (recall Definition 4.7).

We introduce some further notations. Let B ⊂ V ′ ⊗ V * denote the set of maps of rank r. We set

Here and in the sequel, S(•) stands for the unit sphere of the concerned space. Given L ∈ B and U ∈ V , we denote by Ũ the orthogonal projection of U onto ker(L) ⊥ . Then we set:

and finally:

The space T x Σ splits as the following orthogonal direct sum:

where G is the orthogonal complement of ker(d x π 1 ) ⊕ ker(d x π 2 ) in T x Σ. Then, det ⊥ (d x π 2 ) = 0 if and only if d x π 2 is not onto. Recalling that dim(M 2 ) = n 2 and dim(Σ) = n 1 + n 2r, this is equivalent to dim(ker(d x π 2 )) > n 1r. In the same way, det ⊥ (∂ 1 F (x)) = 0 if and only if dim(ker(∂ 1 F (x))) > n 1r. But the kernels of d x π 2 and ∂ 1 F (x) have the same dimension by (C.3), so that det ⊥ (d x π 2 ) = 0 if and only if det ⊥ (∂ 1 F (x)) = 0. A similar argument shows that det ⊥ (d x π 1 ) = 0 if and only if det ⊥ (∂ 2 F (x)) = 0. Thus the lemma is true if any of the four maps in (C.2) is singular.

From now on, we assume that these maps are all surjective. In this case, we have: An immediate consequence of (C.1) and (C.2) is the following.

Proposition C.3. Let M 1 and M 2 be two smooth Riemannian manifolds of dimension n 1 and n 2 respectively. Let F : M 1 × M 2 → R r be a smooth submersion, and let Σ = F -1 (0). Let φ : Σ → R be a Borel measurable function. Then:

whenever one of these integrals is well-defined.