Damien Gayet

Jean-Yves Welschinger

Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Keywords: Mathematics subject classification 2010: 14P25, 32U40, 60F10, 60B20 Real projective manifold, ample line bundle, random matrix, random polynomial

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.

Introduction

How many real roots does a random real polynomial have? This question was answered by M. Kac in the 40's and, for a different measure, by E. Kostlan and M. Shub together with S. Smale in the 90's. In higher dimensions, this question may become: what is the topology of a random real hypersurface in a given smooth real projective manifold? The mean Euler characteristics of such random real hypersurfaces in RP n has been computed by S. S. Podkorytov [START_REF] Podkorytov | The mean value of the Euler characteristic of an algebraic hypersurface[END_REF] and P. Bürgisser [START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF], while the mean total Betti number has been recently estimated from above by the authors [START_REF]What is the total Betti number of a random real hypersurface?[END_REF] (see also [START_REF] Gayet | Exponential rarefaction of real curves with many components[END_REF]). In the case of spherical harmonics in dimension two, rather precise estimations have been obtained by F. Nazarov and M. Sodin [START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF].

Our aim is to improve our previous results [START_REF]What is the total Betti number of a random real hypersurface?[END_REF] by getting upper bounds for all individual Betti numbers of random real hypersurfaces. Let X be a smooth n-dimensional complex projective manifold defined over the reals and let RX be its real locus. Let L be a real ample line bundle over X. We equip L with a real Hermitian metric h of positive curvature ω and X with a normalized volume form dx. These induce an inner L 2 -product on all spaces of global holomorphic real sections RH 0 (X, L d ) for all tensor powers L d of L, d > 0, see §1.1. The latter spaces then inherit Gaussian probability measures µ R , with respect to which we are going to consider random sections, see §3.1.1 of [START_REF]What is the total Betti number of a random real hypersurface?[END_REF] for a discussion on this choice (previously considered in [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF], [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF], [START_REF] Podkorytov | The mean value of the Euler characteristic of an algebraic hypersurface[END_REF], [START_REF] Bürgisser | Average Euler characteristic of random real algebraic varieties[END_REF]) and on other possible ones (compare [START_REF] Shiffman | Equilibrium distribution of zeros of random polynomials[END_REF]).

For every generic section σ ∈ RH 0 (X, L d ), the real locus RC σ of its vanishing locus is a smooth hypersurface of RX, if non empty. For every i ∈ {0, • • • , n -1}, we denote by b i (RC σ ) the infinimum over all Morse functions f on RC σ of the number of critical points of index i of f . This fake Betti number bounds from above all ith Betti numbers of RC σ , whatever the coefficient rings are, as follows from Morse theory (see, e.g., [START_REF] Milnor | Morse theory[END_REF]). We then denote by E(b i ) the average value of this fake Betti number, namely E(b i ) = RH 0 (X,L d ) b i (RC σ )dµ R (σ). Our aim is to prove the following upper bound for this expectation (see Corollary 2):

Theorem 1 Let X be a smooth real projective manifold of dimension n > 0 and (L, h) be a real Hermitian line bundle of positive curvature ω over X. Then

lim sup d→∞ 1 √ d n E(b i ) ≤ 1 √ π e R (i, n -1 -i)V ol h (RX).
Moreover, when n = 1, the lim sup is a limit and the inequality an equality, so that

E(b 0 ) ∼ d→∞ Length h (RX) √ π √ d.
In Theorem 1, V ol h (RX) denotes the Riemannian volume of RX for the Kähler metric induced by the curvature form ω of h. It turns out that Theorem 1, as well as Theorem 2 and Corollary 1, does not depend on the normalized volume form dx chosen on X to define the L 2 -inner product on RH 0 (X, L d ), compare [START_REF] Shiffman | Equilibrium distribution of zeros of random polynomials[END_REF].

The coefficient e R (i, n -1 -i) is itself a mathematical expectation, namely the average value of (the absolute value of) the determinant on symmetric matrices of signature (i, n -1 -i), see §1.1. More precisely, the space Sym(n -1, R) of square symmetric matrices of size (n -1) × (n -1) has a natural Gaussian measure that we also denote by µ R . Let Sym(i, n -1 -i, R) be the open subset of matrices of signature (i, n -1 -i). Then, for every

i ∈ {0, • • • , n -1} e R (i, n -1 -i) = Sym(i,n-1-i,R) | det A|dµ R (A) and e R (n -1) = Sym(n-1,R) | det A|dµ R (A).
Here by convention, e R (0) = e R (0, 0) = 1.

Note that when X = CP 1 , L = O CP 1 (1) and h is the Fubini-Study metric, V ol F S (RP 1 ) = √ π, so that Theorem 1 recovers asymptotically the results of Kostlan and Shub-Smale, up to which a random degree d real polynomial in one variable has √ d roots for our choice of the probability measure. The initial result by M. Kac was rather expecting 2 π log d real roots, but for a different probability measure, see §3.1.1 of [START_REF]What is the total Betti number of a random real hypersurface?[END_REF]. When X = CP 2 , P. Sarnak and I. Wigman informed us in 2011 that they were also able to bound E(b 0 ) from above by a O(d) term as in Theorem 1. This Theorem 1 improves our previous results of [START_REF]What is the total Betti number of a random real hypersurface?[END_REF], where the best upper bounds we could get were by O( √ d log d n ) in some cases. Theorem 1 turns out to be the consequence of a more precise equidistribution result. Namely, when n > 1, we equip RX with a fixed Morse function p : RX → R.

Then, for every generic section σ ∈ RH 0 (X, L d ), p restricts to a Morse function on RC σ and we denote by Crit i (p |RCσ ) the set of critical points of index i of this restriction. We set

ν i (RC σ ) = 1 √ d n x∈Crit i (p |RCσ ) δ x
the empirical measure on these critical points, where δ x denotes the Dirac measure at x and E(ν i ) = RH 0 (X,L d ) ν σ dµ R (σ). When n = 1, ν 0 denotes the empirical measure on RC σ . Then, we get (see Theorem 6) Theorem 2 Under the hypotheses of Theorem 1, the measure E(ν i ) weakly converges to 1 √ π e R (i, n -1 -i)dvol h on RX as d grows to infinity.

In Theorem 2, dvol h denotes the Lebesgue measure of RX induced by its Riemannian metric, which is itself induced by the Kähler metric of X defined by ω. Note that we also establish such an equidistribution result for critical points of complex hypersurfaces, where the Morse function p on RX is replaced by a Lefschetz pencil on X and RH 0 (X, L d ) by H 0 (X, L d ), see Theorem 5. Similar equidistribution results can be found in [START_REF] Michael | Critical points and supersymmetric vacua. I[END_REF], [START_REF]Critical points and supersymmetric vacua. II: Asymptotics and extremal metrics[END_REF], [START_REF] Macdonald | Density of complex critical points of a real random SO(m + 1) polynomial[END_REF], [START_REF]What is the total Betti number of a random real hypersurface?[END_REF] or also [START_REF] Auffinger | Random matrices and complexity of spin glasses[END_REF], [START_REF] Dedieu | On the number of minima of a random polynomial[END_REF].

In order to prove Theorem 2, we roughly follow the approach of [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF]. We introduce the incidence variety

Σ i = {(σ, x) ∈ RH 0 (X, L d )×RX | x ∈ Crit i (p |RCσ )}
and express E(ν σ ) as the push-forward onto RX of the Gaussian measure µ R of RH 0 (X, L d ) "pulled-back" on Σ i . This push-forward measure is then computed asymptotically thanks to the coarea formula and peak sections of Hörmander. The latter indeed make it possible to compute pointwise the measure in terms of the 2-jets of sections, see §1.

Now, what are the values of the expectations e R (n), n > 0, and how do these distribute between the different e R (i, n -i), 0 ≤ i ≤ n? We devote the second paragraph to this question and get (see Proposition 6, Proposition 7 and Corollary 3):

Theorem 3 When n is odd, e R (n) = 2 √ 2 π Γ( n+2 2 ), while when n is even, e R (n) = (-1) m n! m!2 n + (-1) m-1 4 √ 2n! √ πm!2 n m-1 k=0 (-1) k Γ(k + 3/2) k! . In both cases, e R (n) is equivalent to 2 √ 2 π Γ( n+2
2 ) as n grows to infinity. The odd case in Theorem 3 was known, see §26.5 of [START_REF] Madan | Random matrices[END_REF], but we could only find the even case in terms of hypergeometric functions in the literature, see [START_REF] Delannay | Distribution of the determinant of a random realsymmetric matrix from the Gaussian orthogonal ensemble[END_REF]. It turns out that e R (n) is transcendental for odd n and algebraic in Q( √ 2) for even n. We can now rewrite the bound deduced from Theorem 1 for the expected total Betti number E(b * ) = n-1 i=0 E(b i ) as follows (see Remark 2). Corollary 1 Under the hypotheses of Theorem 1, for every even n > 0, lim sup

d→∞ 1 √ d n E(b * ) ≤ 2 √ 2 π V ol h (RX) V ol F S (RP n ) .
For odd n, this inequality holds asymptotically in n.✷

In particular, for every even-dimensional projective space, the right-hand side in Corollary 1 turns out not to depend on the dimension of the space. Finally, we get the following exponential decay away from the mid-dimensional Betti numbers (see Proposition 4):

Theorem 4 For every α ∈ [0, 1/2[, there exists c α > 0, such that ⌊αn⌋ i=0 e R (i, n -i) ≤ exp(-c α n 2 ).
This concentration near matrices having as many positive as negative eigenvalues actually follows from the large deviations estimates near Wigner semi-circle law established in [START_REF] Ben Arous | Large deviations for Wigner's law and Voiculescu's non-commutative entropy[END_REF]. As a consequence of Theorem 4, for large values of n, the upper bound for the expected total Betti number of RC σ given by the right hand side of Theorem 1 distributes between the different Betti numbers in such a way that it gets concentrated around the mid-dimensional ones and exponentially decreases away from them.

The first paragraph of this paper is devoted to Theorems 1 and 2. A key role is played by Hörmander peak sections, see §1.3. Note that we also prove along the same lines the complex analogue of Theorem 2, which is of independent interest. The second paragraph is devoted to Theorems 3 and 4 and the study of determinants of random symmetric matrices.

Aknowledgements. Let X be a smooth complex projective manifold of positive dimension n. When X is defined over R, we denote by c X : X → X the associated antiholomorphic involution, called real structure, and by RX ⊂ X the real locus of the manifold, that is the fixed point set of c X . Likewise, let L be an ample holomorphic line bundle over X equipped with a Hermitian metric h of positive curvature. We denote by ω the curvature form of h, so that for every local non-vanishing holomorphic section e of L defined over an open subset U of X,

ω |U = 1 2iπ ∂ ∂ log h(e, e).
We denote by g = ω(., J.) the induced Kähler metric on X, where J denotes the complex structure of T X. When X and L are defined over R, we denote by c L the associated real structure of L and assume that h is real, so that c * L h = h. The restriction of g to RX is a Riemannian metric and we denote by V ol h (RX) the total volume of RX for the associated Lebesgue measure dvol h . Note that the volume of X is independent of the metric h and equals

V ol(X) = X ω n n! = 1 n! X c 1 (L) n .
We denote by dx = 1 X ω n ω n the normalized volume form of X, or any volume form on X with total volume one.

For every d > 0, we denote by L d the d-th tensor power of L and by h d the induced Hermitian metric on L d . We denote by H 0 (X, L d ) its complex vector space of global holomorphic sections and by N d the dimension of H 0 (X, L d ). We denote then by ., . the L 2 -Hermitian product on this vector space, defined by the relation

∀σ, τ ∈ H 0 (X, L d ), σ, τ = X h d (σ, τ )dx.
The associated Gaussian measure is denoted by µ C . It is defined, for every open subset U of H 0 (X, L d ), by

µ C (U) = 1 π N d U e -||σ|| 2 dσ,
where dσ denotes the Lebesgue measure of H 0 (X, L d ). When L is defined over R, we denote by RH 0 (X, L d ) the real vector space of real sections of

L d , made of sections σ ∈ H 0 (X, L d ) satisfying c L • σ • c X = σ. Its dimension equals N d .
The Hermitian L 2 -product ., . restricts to a scalar product on RH 0 (X, L d ), which we also denote by ., . . The associated Gaussian measure is denoted by µ R and defined for every open subset RU of RH 0 (X, L d ) by

µ R (RU) = 1 √ π N d RU e -||σ|| 2 dσ.
For every d > 0, we denote by ∆ d (resp. R∆ d ) the discriminant hypersurface of H 0 (X, L d ) (resp. RH 0 (X, L d )), that is the set of sections σ ∈ H 0 (X, L d ) (resp. σ ∈ RH 0 (X, L d )) which do not vanish transversally. For every σ ∈ H 0 (X, L d ) \ {0}, we denote by C σ (resp. RC σ ) the vanishing locus of σ in X (resp. its real locus when σ ∈ RH 0 (X, L d )). For every σ ∈ H 0 (X, L d ) \ ∆ d , C σ is then a smooth hypersurface of X. When σ is real, RC σ is of dimension n -1 when non empty and we denote, for i ∈ {0, • • • , n -1}, by b i (RC σ ) the minimum number of critical points of a Morse function on RC σ . From Morse theory we know that b i (RC σ ) is bigger than any of its i-th Betti number, whatever the coefficient ring is. When X is real (resp. complex) and n > 1, we equip its real locus with a Morse function p : RX → R (resp. with a Lefschetz pencil p : X CP 1 ). We then denote, for every d > 0, by

R∆ d p (resp. ∆ d p ) the locus of sections σ ∈ RH 0 (X, L d ) (resp. σ ∈ H 0 (X, L d )) such that σ ∈ R∆ d (resp. ∆ d )
ν i (RC σ ) = 1 √ d n x∈Crit i (p |RCσ ) δ x .
When n = 1, we set likewise ν(C σ ) = 1

d x∈Cσ δ x and when σ ∈ RH 0 (X, L d ) \ R∆ d , ν 0 (RC σ ) = 1 √ d x∈RCσ δ x .
For every n ∈ N * , denote by Sym(n, R) (resp. by Sym(n, C)) the real (resp. complex) vector space of real (resp. complex) symmetric matrices of size n × n. These vector spaces are of dimension n(n+1) 2 and we equip them with the basis B given by the vectors

E ii = √ 2E ii and E ij = E ij + E ji , 1 ≤ i < j ≤ n
, where for every 1 ≤ k, l ≤ n, E kl denotes the elementary matrix whose entry at the i-th row and j-th column equals 1 if (i, j) = (k, l) and 0 otherwise. We equip then Sym(n, R) (resp. Sym(n, C) ) with the scalar (resp. Hermitian) product turning B into an orthonormal basis and we denote by ||.|| the associated norm. We then denote by µ R (resp. µ C ) the associated Gaussian probability measure, so that for every open subset

U of Sym(n, R) (resp. V of Sym(n, C)), µ R (U) = 1 √ π n(n+1) 2 U e -||A|| 2 dA and µ C (V ) = 1 π n(n+1) 2 V e -||A|| 2 dA,
where dA denotes the Lebesgue measure. For every p, q ∈ N, we denote by Sym(p, q, R) the open subset of Sym(p+q, R) made of non-degenerated matrices of signature (p, q).

We then set, for every integers p, q, n,

e C (n) = E C (| det | 2 ) = Sym(n,C) | det A| 2 dµ C (A), e R (n) = E R (| det |) = Sym(n,R) | det A|dµ R (A) and e R (p, q) = Sym(p,q,R) | det A|dµ R (A), so that p,q∈N p+q=n e R (p, q) = e R (n). By convention, e R (0) = e C (0) = e R (0, 0) = 1.

Statement of the results

Using the notations of §1.1, we set, for every d > 0,

E(ν) = H 0 (X,L d )\∆ d p ν(C σ )dµ C (σ)
the average of the measure ν.

Theorem 5 Let X be a smooth complex projective manifold of dimension n > 0 and L be an ample holomorphic line bundle on X equipped with a Hermitian metric of positive curvature ω. Let p : X CP 1 be a Lefschetz pencil. Then, the measure E(ν) weakly converges to e C (n -1) ω n n! = ω n as d grows to infinity. The form ω n n! in Theorem 5 is the Kähler volume form defined by ω. The equality e C (n -1)( ω n n! ) = ω n follows from Proposition 5, see §2.2.1. By "weak convergence" in Theorem 5 we mean that for every continuous function χ : X → R, E(ν), χ converges to X χω n as d grows to infinity, where

E(ν), χ = 1 d n H 0 (X,L d )\∆ d p x∈Crit(p |Cσ ) χ(x) dµ C (x).
Note that Theorem 5 slightly improves Theorem 3 of [START_REF]What is the total Betti number of a random real hypersurface?[END_REF]. Note also that this result, as well as the following Theorem 6 and Corollary 2, does not depend on the normalized volume form dx chosen on X in order to define the L 2 -inner product on H 0 (X, L d ), see §1.1. Likewise, when X and L are real, we set for every

d > 0 and i ∈ {0, • • • , n -1}, E(ν i ) = RH 0 (X,L d )\R∆ d p ν i (RC σ )dµ R (σ)
the average of the measure ν i .

Theorem 6 Let X be a smooth real projective manifold of dimension n > 0 and L be a real ample holomorphic line bundle over X equipped with a real Hermitian metric of positive curvature ω. Let p : RX → R be a Morse function. Then, for every

i ∈ {0, • • • , n -1}, the measure E(ν i ) weakly converges to 1 √ π e R (i, n -1 -i)dvol h as d grows to infinity.
Recall that dvol h denotes the Lebesgue measure associated to the Riemannian metric on RX induced by the Kähler metric g defined by ω. Under the hypotheses of Theorem 6 and using the notations of §1.1, we denote by

E(b i ) = RH 0 (X,L d )\R∆ d p b i (RC σ )dµ R (σ)
the average value of the i-th Betti number.

Corollary 2 Under the hypotheses of Theorem 6,

lim sup d→∞ 1 √ d n E(b i ) ≤ 1 √ π e R (i, n -1 -i)V ol h (RX).
Moreover, when n = 1, the lim sup is a limit and the inequality an equality, so that

E(b 0 ) ∼ d→∞ Length h (RX) √ π √ d.
Note that this Corollary 2 substantially improves Theorem 4 of [START_REF]What is the total Betti number of a random real hypersurface?[END_REF].

Proof. By definition, for every σ ∈ RH 0 (X, L d ) \ R∆ d p , 1 √ d n b i (RC σ ) ≤ RX ν i (RC σ )
with equality when n = 1 (and i = 0). By integration over RH 0 (X,

L d ) \ R∆ d p , we deduce that 1 √ d n E(b i ) ≤ RX E(ν i ),
with equality when n = 1. When n = 1, the result follows from Theorem 6 and from the definition of V ol h (RX) = RX dvol h . In general, we know from Theorem 6 that for every ǫ > 0, there exists d 0 > 0 such that

∀d > d 0 , RX E(ν i ) ≤ ǫ + 1 √ π e R (i, n -1 -i)V ol h (RX).
We get the result by taking the limsup on the left and having ǫ converge to zero. ✷ Remark 1 When X is the Riemann sphere CP 1 , L = O CP 1 (1) and h is the Fubini-Study metric, X equipped with its Kählerian metric is isometric to the round sphere of radius 1 2 √ π in the Euclidian three-space, so that its volume equals 1. It follows that V ol F S (RX) = √ π, and Corollary 2 then writes

E(b 0 ) ∼ d→∞ √ d, which is consistent
with Kostlan and Shub-Smale's results, see [START_REF] Kostlan | On the distribution of roots of random polynomials[END_REF] and [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF].

Remark 2 When X = CP n , L = O CP n (1)
and h is the Fubini-Study metric, the geodesics RP 1 of RP n have length √ π, so that RP n is isometric to the quotient of the sphere of radius 1 √ π by the antipodal map. Hence,

V ol F S (RP n ) = 1 2 √ π n V ol(S n ) = √ π Γ( n+1 2 )
, where S n denotes the unit sphere in R n+1 .

Hörmander peak sections

With the notations of §1.1, let x be a point of X (resp. RX). In a neighborhood of x in X there exists a local holomorphic (resp. real holomorphic) trivialization e of L whose associated potential Φ = -log h(e, e) vanishes at x, where it reaches a local minimum with Hessian of type [START_REF] Anderson | An introduction to random matrices[END_REF][START_REF] Anderson | An introduction to random matrices[END_REF].

Let (x 1 , • • • , x n ) be holomorphic (resp. real holomorphic) coordinates in the neighbourhood of x = (0, • • • , 0) in X, such that ( ∂ ∂x 1 , • • • , ∂ ∂xn
) be orthonormal at x for the Kähler metric g. In these coordinates, the Taylor expansion of Φ writes:

Φ(y) = - 1 2i ∂ ∂Φ(y, iy) + o(||y|| 2 ) = π||y|| 2 + o(||y|| 2 ),
where the norm is induced by the Kähler metric g at the point x.

The L 2 -estimates of Hörmander make it possible, for every d > 0, and after a small perturbation of e d in L 2 -norm, to extend e d into a global holomorphic (resp. real holomorphic) section of L d . The latter is called a Hörmander peak section. Moreover, G. Tian (Lemma 1.2 in [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF]) showed that this procedure can be controlled up to every order, as long as d be large enough. We recall this result in the following Lemma 1 where for every r > 0, B(x, r) denotes the ball centered at x and of radius r in X.

Lemma 1 (See [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], Lemma 1.2) Let (L, h) be a holomorphic Hermitian line bundle of positive curvature ω over a smooth complex projective manifold X.

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

X ω n ω n and

X\B(x, log d √ d ) h d (σ, σ)dx = O( 1 d 2p ′ ). ( 1
)
Moreover, if (x 1 , • • • , x n ) are local holomorphic coordinates in the neighborhood of x, we can assume that in a neighborhood of x, σ(x 1 , • • • , x n ) = λ x p 1 1 • • • x pn n + O(|x| 2p ′ ) e d 1 + O( 1 d 2p ′ ) , where λ -2 = B(x, log d √ d ) |x p 1 1 • • • x pn n | 2 h d (e d , e d )dx
and e is a local trivialization of L whose potential Φ = -log h(e, e) reaches a local minimum at x with Hessian πω(., i.).

This Lemma 1 admits a real counterpart Lemma 2 which is obtained by averaging the peak sections with the real structure:

Lemma 2 Let (L, h) be a real holomorphic Hermitian line bundle of positive curvature ω over a smooth real projective manifold X.

Let x ∈ RX, (p 1 , • • • , p n ) ∈ N n and p ′ > p 1 + • • • + p n . There exists d 0 ∈ N such that for every d > d 0 , the bundle L d has a global real holomorphic section σ satisfying X h d (σ, σ)dx = 1 for the volume form dx = 1 X ω n ω n and X\B(x, log d √ d ) h d (σ, σ)dx = O( 1 d 2p ′ ). Moreover, if (x 1 , • • • , x n ) are local real holomorphic coordinates in the neighborhood of x in X, we can assume that in a neighborhood of x in X, σ(x 1 , • • • , x n ) = λ x p 1 1 • • • x pn n + O(|x| 2p ′ ) e d 1 + O( 1 d 2p ′ ) , where λ -2 = B(x, log d √ d ) |x p 1 1 • • • x pn n | 2 h d (e d , e d )dx
and e is a local real trivialization of L whose potential Φ = -log h(e, e) reaches a local minimum at x with Hessian πω(., i.).

Let σ 0 be a section given by Lemma 1 in coordinates (

x 1 , • • • , x n ) with p ′ = 3 and p 1 = • • • = p n = 0.
Likewise, for every j ∈ {1, • • • , n}, let σ j be a section given by Lemma 1 with p ′ = 3, p j = 1 and p k = 0 for k ∈ {1, • • • , n} \ {j}. Finally, for every 1 ≤ k ≤ l ≤ n, let σ k,l be a section given by Lemma 1 with p ′ = 3, p j = 0 for every j ∈ {1, • • • , n} \ {k, l} and p k = p l = 1 if k = l, while p k = 2 otherwise. All these sections have their norms concentrated in the neighbourhood of x being close to 0 outside of a ball of radius log d √ d (from the mean value inequality, see Theorem 4.2.13 of [START_REF] Hörmander | Notions of convexity[END_REF] for instance). Likewise, by Lemma 1, the Taylor expansions of these sections write:

σ 0 (y) = λ 0 + O(||y|| 6 ) e d (y) 1 + O( 1 d 6 ) ∀j ∈ {1, • • • , n}, σ j (y) = λ 1 y j + O(||y|| 6 ) e d (y) 1 + O( 1 d 6 ) and ∀k, l ∈ {1, • • • , n}, k = l, σ k,l (y) = λ (1,1) y k y l + O(||y|| 6 ) e d (y) 1 + O( 1 d 6 ) , whereas ∀k ∈ {1, • • • , n}, σ k,k (y) = λ (2,0) y 2 k + O(||y|| 6 ) e d (y) 1 + O( 1 d 6 ) .
The asymptotic values of the constants λ 0 , λ 1 , λ (1,1) and λ (2,0) are given in Lemma 3 (compare Lemma 2.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF]):

Lemma 3 Under the hypotheses of Lemma 1, let δ L = X c 1 (L) n be the degree of L. Then

lim d→∞ 1 √ d n λ 0 = δ L ; lim d→∞ 1 √ d n+1 λ 1 = √ π δ L lim d→∞ 1 √ d n+2 λ (1,1) = π δ L and lim d→∞ 1 √ d n+2 λ (2,0) = π √ 2 δ L ,
for the inner L 2 -product induced by the volume form dx = 1 X ω n ω n .

These values differ from the ones given in Lemma 2.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF] by a constant π n since our choice of the Kähler metric slightly differs from [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF].

Proof. From Lemma 1, λ -2 0 is equivalent, as d grows to infinity, to 1

X ω n C n e -dπ||y|| 2 dvol(y) = 1 X d n ω n π n C n e -||z|| 2 dvol(z) = 1 X d n ω n , so that λ 0 ∼ d→∞ X c 1 (L) n d n . Likewise, λ -2 1 is equivalent to 1 X ω n C n |y 1 | 2 e -dπ||y|| 2 dvol(y) = 1 dπ X d n ω n π n C n |z 1 | 2 e -||z|| 2 dvol(z) = 1 dπ X d n ω n , so that λ 1 ∼ d→∞ √ dπλ 0 .
We obtain in the same way

λ (1,1) ∼ d→∞ dπλ 0 , whereas λ -1 (2,0) is equivalent to 1 X ω n C n |y 1 | 4 e -dπ||y|| 2 dvol(y) = 1 (dπ) 2 X d n ω n π n C n |z 1 | 4 e -||z|| 2 = 2 (dπ) 2 X d n ω n . Hence, λ (2,0) ∼ d→∞ dπ √ 2 λ 0 . ✷ Let ∇ X be
a torsion-free connection on T X and ∇ L be a connection (resp. real) on L. The connections ∇ X and ∇ L induce a connection denoted by ∇ X,L on T * X ⊗L. We then set

∇ 2 σ = ∇ X,L (∇ L σ) ∈ End(T X, T * X ⊗ L d ).
Now, the sections (σ i ) 0≤i≤n and (σ k,l ) 1≤k≤l≤n define a basis of a complement of the subspace of sections of H 0 (X, L d ) (resp. RH 0 (X, L d )) whose 2-jets at x vanish, which is denoted by

H 3x = {σ ∈ H 0 (X, L d ) | σ(x) = 0, ∇ L σ |x = 0 and ∇ 2 σ |x = 0} (2) (resp. RH 3x = {σ ∈ RH 0 (X, L d ) | σ(x) = 0, ∇ L σ |x = 0 and ∇ 2 σ |x = 0}). ( 3
)
This basis is not orthonormal and its spanned subspace is not orthogonal to H 3x . However, from Lemma 3.1 of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], we know that it becomes closer and closer to being orthonormal as d grows to infinity, see Lemma 4, as long as the chosen volume form is dx = 1 X ω n ω n , see Remark 3. Lemma 4 (See [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], Lemma 3.1) The sections (σ i ) 0≤i≤n and (σ k,l ) 1≤k≤l≤n have L 2 -norm equal to one and their pairwise scalar product are O( 1 d ). Likewise, their scalar products with every unitary element of

H 3x are O( 1 d 3/2 ). Remark 3 If the L 2 -scalar product is induced by a volume form different from dx = 1 X ω n ω n , say dx = f (x) 1
X ω n ω n , then Lemma 4 remains unchanged except that the L 2 -norms of the sections, instead of being one, would converge to f (x) from (1).

Incidence varieties and evaluation maps

Using the notations of §1.1, let us denote by Crit(p) (resp. Crit i (p)) the finite set of critical points of p (resp. of index i) and by Base(p) its base locus. Under the hypotheses of Theorem 5 (resp. Theorem 6) and following [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF], we set

Σ = {(σ, x) ∈ (H 0 (X, L d ) \ ∆ d p ) × (X \ (Crit(p) ∪ Base(p))| x ∈ Crit(p |Cσ )}, (resp. Σ i = {(σ, x) ∈ (RH 0 (X, L d ) \ R∆ d p ) × (RX \ Crit(p))| x ∈ Crit i (p |RCσ )}),
and

π 1 : (σ, x) ∈ Σ → σ ∈ H 0 (X, L d ) and π 2 : (σ, x) ∈ Σ → x ∈ X (resp. π 1 : (σ, x) ∈ Σ i → σ ∈ RH 0 (X, L d ) and π 2 : (σ, x) ∈ Σ i → x ∈ RX)
the associated projections on these incidence varieties.

For every (σ 0 , x 0 ) ∈ Σ (resp. (σ 0 , x 0 ) ∈ Σ i ), there exists a neighbourhood U (resp. RU) of σ 0 in H 0 (X, L d ) (resp. RH 0 (X, L d )) and a neighbourhood V (resp. RV ) of x 0 in X (resp. RX) such that for every σ ∈ U (resp. σ ∈ RU), the function p |Cσ (resp. p |RCσ ) has a unique critical point (resp. critical point of index i) in V (resp. RV ). We deduce from this an evaluation map at the critical point

ev (σ 0 ,x 0 ) : σ ∈ U → x ∈ Crit(p |Cσ ) ∩ V (resp. ev (σ 0 ,x 0 ) : σ ∈ RU → x ∈ Crit(p |RCσ ) ∩ RV ), so that Σ ∩ (U × V ) (resp. Σ i ∩ (RU × RV ))
is the graph of ev (σ 0 ,x 0 ) . This evaluation map is constant on π 1 (π -1 2 (x 0 )) ∩ U, so that its differential d |σ 0 ev (σ 0 ,x 0 ) vanishes on

T σ 0 π 1 (π -1 2 (x 0 )) ≃ π 1 (π -1 2 (x 0 )). When n = 1, we agree that π 1 (π -1 2 (x)) = {σ ∈ H 0 (X, L d )| σ(x) = 0}. We denote by d |σ 0 ev ⊥ (σ 0 ,x 0 ) the restriction of d |σ 0 ev (σ 0 ,x 0 ) to the orthogonal complement of π 1 (π -1 2 (x 0 )) in H 0 (X, L d ) (resp. RH 0 (X, L d )).
Proposition 1 Under the hypotheses of Theorem 5 (resp. Theorem 6),

E(ν) = 1 d n (π 2 ) * (π * 1 dµ C ) (resp. E(ν i ) = 1 √ d n (π 2 ) * (π * 1 dµ R )).
Moreover, at every point

x of X \ (Crit(p) ∪ Base(p)) (resp. RX \ Crit(p)), (π 2 ) * (π * 1 dµ C ) = 1 π n π 1 (π -1 2 (x)) | det d |σ ev ⊥ (σ,x) | -2 dµ C (σ) ω n n! resp. (π 2 ) * (π * 1 dµ R ) = 1 √ π n π 1 (π -1 2 (x)) | det d |σ ev ⊥ (σ,x) | -1 dµ R (σ) dvol h .
Note that π 1 is a map between manifolds of the same dimension, while µ R and µ C are absolute values of volume forms, so that the pull-backs π * 1 dµ R and π * 1 dµ C are well defined.

Remark 4

The pointwise expression of (π 2 ) * (π * 1 dµ C ) (resp. (π 2 ) * (π * 1 dµ R )) is invariant under dilation of the L 2 -inner product ., . on H 0 (X, L d ) (resp. on RH 0 (X, L d )). Indeed, for every λ ∈ C (resp. λ ∈ R), (σ 0 , x 0 ) in Σ (resp. in Σ i ) and σ in a neighborhood of σ 0 , ev (σ 0 ,x 0 ) (σ) = ev (λσ 0 ,x 0 ) (λσ) so that d |σ 0 ev (σ 0 ,x 0 ) = λd |λσ 0 ev (λσ 0 ,x 0 ) . We deduce that det

d |σ 0 ev ⊥ (σ 0 ,x 0 ) = λ n det d |λσ 0 ev ⊥ (λσ 0 ,x 0 )
if both determinants are computed in the orthonormal basis for the same inner product ., . at the source, but det d |σ 0 ev ⊥ (σ 0 ,x 0 ) becomes equal to det d |λσ 0 ev ⊥ (λσ 0 ,x 0 ) when the latter is computed in an orthonormal basis for the inner product dilated by λ 2 . Since under such a dilation the associated Gaussian measures are just push-forwards one with respect to the other by the corresponding homothety, the invariance follows.

Proof. Let χ : X → R be a continuous function. By definition,

E(ν), χ = 1 d n H 0 (X,L d )\∆ d p x∈Crit(p |Cσ ) χ(x) dµ C (σ) = 1 d n Σ (π * 2 χ)(π * 1 dµ C ) = 1 d n X χ (π 2 ) * (π * 1 dµ C ).
But from the coarea formula (see Theorem 3.2.3 of [START_REF] Federer | Geometric measure theory[END_REF] or Theorem 1 of [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF]), for every x ∈ X \ (Crit(p) ∪ Base(p)),

(π 2 ) * (π * 1 dµ C ) |x = 1 π N d π 1 (π -1 2 (x)) | det d |σ ev ⊥ (σ,x) | -2 e -||σ|| 2 dσ ω n n! ,
since the Jacobian of d |σ ev ⊥ (σ,x) , which is C-linear and computed with respect to the volume forms dσ at the source and ω n n! at the target, equals

| det d |σ ev ⊥ (σ,x) | 2 . We deduce that (π 2 ) * (π * 1 dµ C ) |x = 1 π n π 1 (π -1 2 (x)) | det d |σ ev ⊥ (σ,x) | -2 dµ C (σ) ω n n! |x .
Under the hypotheses of Theorem 6, if χ denotes now a continuous function χ : RX → R, we obtain likewise

E(ν i ), χ = 1 √ d n Σ i (π * 2 χ)(π * 1 dµ R ) = 1 √ d n RX χ (π 2 ) * (π * 1 dµ R ).
The coarea formula implies now for every x ∈ RX \ Crit(p) the relation

(π 2 ) * (π * 1 dµ R ) |x = 1 √ π n π 1 (π -1 2 (x)) | det d |σ ev ⊥ (σ,x) | -1 dµ R (σ) dvol h|x .

✷

We are going to compute the Jacobian | det d |σ ev ⊥ (σ,x) | appearing in Proposition 1. For every x ∈ X \ (Crit(p) ∪ Base(p)) (resp. x ∈ RX \ Crit(p)), we denote by

K x = ker d x p ⊂ T x X (resp. RK x = ker d x p ⊂ T x RX)
the kernel of d x p and set

H x = {σ ∈ H 0 (X, L d )| σ(x) = 0} (resp. RH x = {σ ∈ RH 0 (X, L d )| σ(x) = 0}.)
We now assume that the torsion-free connection ∇ X preserves the distribution K on X \ (Crit(p) ∪ Base(p)). This means that for every local vector field v of X taking value in K, we assume that ∇ X v also gets values in K. For every (σ, x) ∈ Σ, we set

λ ′ (σ,x) = ∇ L σ . σ . σ ∈ End T x X/K x , H 0 (X, L d )/H x ,
where .

σ denotes any non-trivial element of H 0 (X, L d )/H x . We consider ∇ 2 σ as a bilinear form on K x , that is

∇ 2 σ ∈ End(K x , K * x ⊗ L d x ). Hence, det(∇ 2 σ) ∈ End ∧ n-1 K x , ∧ n-1 K * x ⊗ L d(n-1)
x .

Define also the bilinear form

∇ L : (v, . σ) ∈ K x × H x /π 1 (π -1 2 (x)) → ∇ L v .
σ∈ L d x which we consider as an element of End K x , H x /π 1 (π -1 2 (x)) * ⊗ L d x and denote abusively by ∇ L . It follows that

det(∇ L ) ∈ End ∧ n-1 K x , ∧ n-1 H x /π 1 (π -1 2 (x)) * ⊗ L d(n-1)
x and we set

λ ′′ (σ,x) = det ∇ 2 σ det(∇ L ) ∈ End ∧ n-1 K x , ∧ n-1 H x /π 1 (π -1 2 (x)) .
Finally, we set

λ (σ,x) = λ ′ (σ,x) ∧ λ ′′ (σ,x) ∈ End ∧ n T x X, ∧ n H 0 (X, L d )/π 1 (π -1 2 (x))
when n > 1 and λ (σ,x) = λ ′ (σ,x) when n = 1. In the real case, ∇ X denotes a torsion-free connection on T RX |RX\Crit(p) which preserves the distribution RK, while ∇ L is real. For every (σ, x)

∈ Σ i , λ ′ (σ,x) belongs then to End T x RX/RK x , RH 0 (X, L d )/RH x and ∇ 2 σ to End RK x , RK * x ⊗ RL d x , so that det(∇ 2 ) ∈ End ∧ n-1 RK x , ∧ n-1 RK * x ⊗ RL d(n-1)
x .

The bilinear form

(v, . σ) ∈ RK x × RH x /π 1 (π -1 2 (x)) → ∇ L v . σ∈ RL d x is considered as an element of End RK x , RH x /π 1 (π -1 2 (x)) * ⊗ RL d x , so that det(∇ L ) ∈ End ∧ n-1 RK x , ∧ n-1 RH x /π 1 (π -1 2 (x)) * ⊗ RL d(n-1)
x .

Finally,

λ ′′ (σ,x) ∈ End ∧ n-1 RK x , ∧ n-1 RH x /π 1 (π -1 2 (x)) while λ (σ,x) ∈ End ∧ n T x RX, ∧ n RH 0 (X, L d )/π 1 (π -1 2 (x)) when n > 1 and λ (σ,x) = λ ′ (σ,x) when n = 1.
Proposition 2 Under the hypotheses of Theorem 5 (resp. Theorem 6), let

(σ 0 , x 0 ) ∈ Σ (resp. (σ 0 , x 0 ) ∈ Σ i ). Then, det(d |σ 0 ev ⊥ (σ 0 ,x 0 ) ) -1 = (-1) n λ (σ 0 ,x 0 ) .
Proof. Consider neighbourhoods U and V of σ 0 and x 0 respectively, such that the evaluation map ev (σ 0 ,x 0 ) : U → V is well defined. Under the hypotheses of Theorem 5, Σ ∩ (U × V ) is the vanishing locus of the map

F : (σ, y) ∈ U × V → (σ(y), ∇ L σ |y ) ∈ L d y × (K * y ⊗ L d y )
. It follows that for every σ ∈ U, F (σ, ev (σ 0 ,x 0 ) (σ)) = 0. By hypothesis the connection ∇ X restricts to a connection on the subbundle K * . Hence, the connection ∇ X,L restricts to a connection on K * ⊗ L d , denoted below by D X,L 2 . The latter makes it possible to differentiate F with respect to the second variable. After differentiation we deduce that

d 1 F |(σ 0 ,x 0 ) + D X,L 2 F |(σ 0 ,x 0 )
• d |σ 0 ev (σ 0 ,x 0 ) = 0, where d 1 F and D X,L 2 denote the partial derivatives of F with respect to the first and second variables respectively. Hence the relation

d |σ 0 ev (σ 0 ,x 0 ) = -(D X,L 2 F ) -1 |(σ 0 ,x 0 ) • d 1 F |(σ 0 ,x 0 ) . But the matrix of D X,L 2 F ∈ End K ⊥ x 0 ⊕ K x 0 , L d x 0 ⊕ (K * x 0 ⊗ L d x 0 ) at the point (σ 0 , x 0 ) is trigonal of the form ∇ L σ 0 0 * ∇ 2 σ 0 , so that det D X,L 2 F |(σ 0 ,x 0 ) = ∇ L σ 0 ∧ det(∇ 2 σ 0 ). Likewise, let .
σ 0 be a Bergman section at x 0 , that is a unitary vector in the orthogonal complement of H x 0 in H 0 (X, L d ). The restriction of

d 1 F ∈ End < . σ 0 > ⊕H x 0 /π 1 (π -1 2 (x 0 )), L d x 0 ⊕ (K * x 0 ⊗ L d x 0 ) at the point (σ 0 , x 0 ) to the orthogonal complement of π 1 (π -1 2 (x 0 )) in H 0 (X, L d ) has the matrix . σ 0 (x 0 ) 0 * ∇ L , so that det d 1 F |(σ 0 ,x 0 ) = . σ 0 (x 0 ) det(∇ L ).
Taking the quotient, we deduce the result under the hypotheses of Theorem 5. The proof goes along the same lines under the hypotheses of Theorem 6. ✷

Proofs of Theorems 5 and 6

Lemma 5 Under the hypotheses of Theorem 6, let (σ, x)

∈ Σ i . Let φ x : RL x → R be an isomorphism such that φ x • ∇ L |x σ = -dp x . Then, φ x • ∇ 2 σ |Kx = ∇ 2 (p |RCσ ) |x , so that the quadratic form φ x • ∇ 2 σ |Kx is non-degenerated of index i.
Proof. Let v, w be vector fields on RC σ at the neighbourhood of x. By definition, 0

= ∇ L v (∇ L w σ) = ∇ 2 v,w σ + ∇ L ∇ X v w σ, so that φ x • ∇ 2 v,w σ = -φ x • ∇ L ∇ X v w σ = dp |x (∇ X v w).
Applying the same equality to the function p, we get

d v (d w p) = ∇ X (dp)(v, w) + dp(∇ X v w) = dp(∇ X v w) by hypothesis on ∇ X , so that φ x • ∇ 2 v,w σ = d v (d w p).
Finally, applying this equality to the restriction p |RCσ , we get

d v (d w p) |x = ∇ 2 v,w p |RCσ|x + dp |RCσ (∇ Cσ v w) |x = ∇ 2 v,w (p |RCσ ) |x ,
where ∇ Cσ denotes any connexion on T RC σ . Hence the result. ✷ Proposition 3 Under the hypotheses of Theorem 5,

1 d n π 1 (π -1 2 (x)) λ (σ,x) ∧ λ (σ,x) dµ C (σ) = π n e C (n -1) + O( 1 √ d ) dvol h ,
whatever the normalized volume form dx chosen on of X to define dµ C is (see §1.1), and where O( 1 √ d ) ∈ L 1 (X, dvol h ) denotes a sequence of integrable functions having pole at Crit(p) (resp. Base(p)) of order at most 2n -2 (resp. 2). Likewise, under the hypotheses of Theorem 6,

1 √ d n π 1 (π -1 2 (x)) |λ (σ,x) |dµ R (σ) = √ π n-1 e R (i, n -1 -i) + O( 1 √ d ) dvol h ,
whatever the normalized volume form chosen on X is, and where O( 1 √ d ) ∈ L 1 (RX, dvol h ) denotes a sequence of integrable functions having poles at Crit(p) of order at most n -1.

In Proposition 3, a function f ∈ L 1 (X, dvol h ) is said to have a pole of order at most k along a submanifold Y if r k f is bounded near Y , where r denotes the distance function to Y . (3)). In the previous equality, σ ∈ π 1 (π -1 2 (x)) if and only if a j = 0 for every j ∈ {0, • • • , n} \ {1} and we assume that this holds true. Moreover, from Lemmas 1 and 3,

σ 0 = λ 0 e d (x)(1 + O( 1 d 6 )), ∀j ∈ {1, • • • , n}, ∇ L σ j|x = √ πdλ 0 e d (x) 1 + O( 1 d 6 ) dx j and ∇ 2 σ jj|x = πdλ 0 √ 2 e d (x) 1 + O( 1 d 6 ) (2dx j ⊗ dx j ), while ∀1 ≤ k < l ≤ n, ∇ L σ kl|x = 0 and ∇ 2 σ kl|x = πdλ 0 e d (x) 1 + O( 1 d 6 ) (dx k ⊗ dx l + dx l ⊗ dx k ).
These equations do not depend on the chosen connexions ∇ L , ∇ X . It follows that

∇ 2 σ 1|Kx = √ πdλ 0 e d (x)(1 + O( 1 d 6 ))∇ X (dx 1 ) (4)
since by hypothesis, the restriction of dx 1 to K x vanishes. Likewise,

1 πdλ 0 ∇ 2 σ |Kx e d (x) = n j=2 b jj πdλ 0 ∇ 2 σ jj|Kx e d (x) + 2≤k<l≤n b kl πdλ 0 ∇ 2 σ kl|Kx e d (x) + a 1 πdλ 0 ∇ 2 σ 1|Kx e d (x) , (5)
so that this restriction writes

n j=2 √ 2b jj dx j ⊗ dx j + 2≤k<l≤n b kl (dx k ⊗ dx l + dx l ⊗ dx k ) + O( 1 √ d ).
Using the notations of §1.1, let B be the matrix 2≤k≤l≤n b kl E kl , and let us first assume that dx = 1 X ω n ω n , so that from Lemma 4, the sections (σ j ) 0≤j≤n and (σ kl ) 0≤k≤l≤n are asymptotically orthonormal. We deduce that pointwise on X \ (Crit(p) ∪ Base(p)),

1 √ πd n-1 det(∇ 2 σ |Kx ) det(∇ L ∂ ∂x k σ l|x ) 2≤k≤l≤n = | det B||dx 2 ∧ • • • ∧ dx n | + O( 1 √ d ).
Moreover,

1 √ πd ∇ L σ σ 0 (x) = a 1 dx 1 + O( 1 d 6 ), so that 1 (πd) n λ (σ,x) ∧ λ (σ,x) = |a 1 | 2 | det B| 2 + O( 1 √ d ) dvol h|x .
Now, we decompose the space π 1 (π -1 2 (x)) as 2) and (3).

π 1 (π -1 2 (x)) = H 3x ∩ π 1 (π -1 2 (x)) ⊕ H 3x ∩ π 1 (π -1 2 (x)) ⊥ (resp. π 1 (π -1 2 (x)) = RH 3x ∩ π 1 (π -1 2 (x)) ⊕ RH 3x ∩ π 1 (π -1 2 (x)) ⊥ ), see (
Denote by H ′ the vector space spanned by σ 1 and σ k,l , 1 ≤ k ≤ l ≤ n and by

π ′ : H 3x ∩ π 1 (π -1 2 (x)) ⊥ → H ′
the projection onto H ′ directed by H 3x (resp. RH 3x ). We deduce that

1 d n π 1 (π -1 2 (x)) λ (σ,x) ∧ λ (σ,x) dµ C (σ) = 1 d n (H 3x ∩π 1 (π -1 2 (x))) ⊥ λ (σ,x) ∧ λ (σ,x) dµ C (σ) = π n H ′ |a 1 | 2 | det B| 2 (π ′ * dµ C )(a 1 , B) +O( 1 √ d )dvol h|x
From Lemma 4, the pushforward measure π ′ * µ C coincides with the Gaussian measure on the space with coordinates a 1 and (b kl ) 1≤k≤l≤n up to a O(

1 √ d ) term. Hence, 1 d n π 1 (π -1 2 (x)) λ (σ,x) ∧ λ (σ,x) dµ C (σ) = π n Sym(n-1,C) | det B| 2 dµ C (B) + O( 1 √ d ) dvol h|x = π n e C (n -1) + O( 1 √ d ) dvol h|x .
This result remains unchanged if a different normalized volume form dx is used on X to define the L 2 -scalar product, since from Remark 3, this asymptotically just has the effect of dilating the scalar product on the subspace (H 3x ∩ π 1 (π -1 2 (x))) ⊥ , while from Remark 4 and Proposition 2, such a dilation does not affect the integral

(H 3x ∩π 1 (π -1 2 (x))) ⊥ λ (σ,x) ∧ λ σ,x dµ C (σ).
Since ∇ X is not defined at the critical and base points of p, we have now to estimate the singularities of ∇ X (dx 1 ) near these loci. In the coordinates (x 1 , • • • , x n ) around x, let us write dp = n i=1 α i dx i , so that at the point x, α

2 (x) = • • • = α n (x) = 0 and |α 1 (x)| = ||dp x ||. Then, 0 = ∇ X (dp) |Kx = α 1 (∇ X dx 1 ) |Kx + n i=1 (dα i ⊗ dx i ) |Kx so that ||∇ X dx 1|Kx || = 1 ||dp |x || || n i=1 dα i ⊗ dx i|Kx ||
has a pole of order one at x, since by definition of a Lefschetz pencil, dp vanishes transversally at x. By developing the determinant of ∇ 2 σ and using (5), we thus get that

1 (πd) n λ (σ,x) ∧ λ (σ,x) = |a 1 | 2 | det B| 2 + O( 1 √ d ) dvol h ,
where for every d > 0, the singularities of the function O( 1 √ d ), which is polynomial in a 1 and b k,l for 2 ≤ k ≤ l ≤ n, are poles of order at most 2(n -1) near the critical points.

Near the base points, these are poles of order at most 2. Indeed, the normal form for p near a base point writes p : (y

1 , • • • , y n ) ∈ C n → y 1 /y 2 ∈ C, so that dp |(y 1 ,••• ,yn) = y 2 dy 1 -y 1 dy 2 y 2
is not well defined along y 2 = 0. Denote by β the numerator one-form y 2 dy 1 -y 1 dy 2 , which is well defined everywhere. When the point x lies in such a chart, there is no obstruction in finding local coordinates around x which are orthonormal at x and such that in these coordinates, β writes α 1 dx 1 + α 2 dx 2 where α 1 , α 2 only depend on x 1 , x 2 , α 2 (x) = 0 and α 1 (x) is a function of x having a simple zero along the base locus. Then, by hypothesis on

∇ X , 0 = ∇ X β |Kx = α 1 ∇ X (dx 1 ) |Kx + ∂α 2 ∂x 2 dx 2 ⊗ dx 2|Kx .
It follows that ∇ X (dx 1 ) |Kx has a simple pole along the base locus, and by ( 4) the matrix of

∇ 2 σ 1|Kx in the basis ( ∂ ∂x 2 , • • • , ∂ ∂xn
) is elementary, with only one diagonal coefficient having a simple pole along the base locus. After developing the determinant, we deduce that the O( 1 √ d ) function has a pole of order at most 2 near the base locus.

In the real case, finally we get likewise that

1 √ πd n |λ (σ,x) | = |a 1 || det B| + O( 1 √ d ) dvol h|x ,
where the singularities of the functions O( 1 √ d ) are poles of orders at most n -1 near the critical points. After integration, we deduce that

1 √ d n π 1 (π -1 2 (x)) |λ (σ,x) |dµ R (σ) = √ π n H ′ i |a 1 || det B|(π ′ * dµ R (a 1 , B)) + O( 1 √ d ) dvol h|x ,
where from Lemma 5,

H ′ i = {σ ∈ H ′ | φ x • ∇ 2 σ |Kx is of index i}. Again, we deduce from Lemma 4 that 1 √ d n π 1 (π -1 2 (x)) |λ (σ,x) |dµ R (σ) = √ π n R |a 1 |dµ R (a 1 ) Sym(i,n-1-i,R) | det B|dµ R (B) + O( 1 √ d ) |dvol h|x | = √ π n-1 e R (i, n -1 -i) + O( 1 √ d ) |dvol h|x |,
and this result remains unchanged if a different normalized volume form dx is used on X. ✷ Proof of Theorems 5 and 6.

From Proposition 3 follows that under the hypotheses of Theorem 5 (resp. of Theorem 6), the measure

1 d n π 1 (π -1 2 (x)) λ (σ,x) ∧ λ (σ,x) dµ C (σ) (resp. 1 √ d n |λ (σ,x) |dµ R (σ))
weakly converges to the measure

π n e C (n -1)dvol h (resp. √ π n-1 e R (i, n -1 -i)dvol h ).
Theorems 5 and 6 then follow from Propositions 1 and 2, whatever the normalized volume form dx is chosen on X to define the L 2 -scalar product ., . . ✷

Expected determinant of random symmetric matrices

In §2.1 we study the asymptotic distribution of e R (p, q) for large n = p + q. We then compute e C (n) in §2.2.1 and e R (n) in § §2.2.2 and 2.2.3. We also give in §2.2.4 the values of e R (p, q) for p + q ≤ 3.

2.1 Large random real symmetric matrices

2.1.1 The energy functional Let f : R 2 -→ R ∪ {∞} (x, y) -→ 1 2 (x 2 + y 2 ) -log |x -y| if x = y +∞ if x = y.
Let M + 1 (R) be the space of probability measures on R and H be the energy functional defined by the relation:

H : M + 1 (R) -→ R ∪ {∞} µ -→ 1 2 R 2 f (x, y)dµ(x)dµ(y) if R log(|x| + 1)dµ(x) < +∞ +∞ otherwise.
This functional is lower semicontinuous, strictly convex and reaches its unique minimum at the semi-circle law µ W of Wigner, see §2.6.1 of [START_REF] Anderson | An introduction to random matrices[END_REF]. Moreover, H(µ W ) =

1 4 ( 3 2 + log 2).
For every 0 ≤ α ≤ 1, define

M + α,1-α (R) = {µ ∈ M + 1 (R) | µ(R * -) = α and µ(R * + ) = 1 -α}.
Since the functional H is strictly convex and equals +∞ on atomic measures, its restriction to M + α,1-α reaches its minimum at a unique measure µ α ∈ M + α,1-α which has no atom. In particular, µ 1 2 = µ W . For every 0 ≤ α ≤ 1, we set

m α = min M + α,1-α (R) H = H(µ α ). Lemma 6 The function m : α ∈ [0, 1] → m α ∈ R + is strictly decreasing over [0, 1/2]
and strictly increasing over

[1/2, 1]. More precisely, for every α ∈ [0, 1] \ {1/2}, there exists c α > 0 such that ∀t ∈ [0, 1], m tα+(1-t) 1 2 ≤ m α + (t 2 -1)c α . Proof. Let α ∈ [0, 1] \ { 1 2 } and f α ∈ L 1 (R, dx
) be the density of µ α with respect to the Lebesgue measure dx. We decompose f α = f p +f i into odd and even functions, so that

f p = 1 2 (f α + f α • (-Id)) and f i = 1 2 (f α -f α • (-Id))
. Likewise, we set µ p = f p dx and µ i = f i dx, so that µ α = µ p + µ i . Then, for every t ∈ [-1, 1],

H(µ p + tµ i ) = 1 2 R 2 f (x, y) dµ p (x)dµ p (y) + tdµ p (x)dµ i (y) + tdµ i (x)dµ p (y) + t 2 2 R 2 f (x, y)dµ i (x)dµ i (y) = H(µ p ) + t 2 H(µ i ) from Fubini's theorem, since R 1 2 (x 2 + y 2 )dµ i (x) = R 1
2 (x 2 + y 2 )dµ i (y) = 0 while likewise R log |y -x|dµ p (x) and R log |y -x|dµ p (y) are even functions of y and x respectively. Since H is strictly convex, so does its restriction over the interval {µ p + tµ i , t ∈ [-1, 1]}, so that H(µ i ) has to be positive. However, for every t ∈ [0, 1], µ p +tµ i belongs to M + (tα+(1-t) 1 2 ,(1+t) 1 2 -tα) (R) and we deduce that m tα+(1-t)

1 2 ≤ H(µ p )+ t 2 H(µ i ) = m α + (t 2 -1)H(µ i ). Hence the result. ✷ Remark 5 It would be of interest to compute explicitely the function α ∈ [0, 1] → m α ∈ [m1 2
, ∞] and in particular its asymptotics near α = 1/2. Likewise, we saw in the proof of Lemma 6 that the functional H restricted to measures of type g(x)dx, where g is an odd function in L 1 (R, dx) with R + gdx = 1, is positive, but does its infinimum remain positive? Finally, what is the measure µ α ? (see [START_REF] Dean | Large deviations of extreme eigenvalues of random matrices[END_REF] for the case α = 0)

⌊αn⌋ i=0 e R (i, n -i) ≤ exp(-c α n 2 ) and µ R ⌊αn⌋ i=0 Sym(i, n -i, R) ≤ exp(-c α n 2 ).
Proof. The orthogonal group O n (R) acts by conjugation on real symmetric matrices and a fundamental domain for this action is given by diagonal matrices with stabilizer {±1} n . From the coarea formula (see Theorem 3.2.3 of [START_REF] Federer | Geometric measure theory[END_REF] or Theorem 1 of [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF]), we deduce that for every 0

≤ i ≤ n, e R (i, n -i) = V ol(O n (R)) 2 n √ π n(n-1) 2 λ 1 <•••<λ i <0 0<λ i+1 <•••<λn n i=1 √ 2λ i 1≤i<j≤n |λ j -λ i |dµ(λ),
where the volume of O n (R) is computed with respect to the right invariant metric for which the basis (E ij -E ji ) 1≤i<j≤n of its Lie algebra is orthonormal, see §2.1.3, and where dµ(λ) denotes the Gaussian measure on R n . As a consequence, for every 0

≤ i ≤ n, e R (i, n -i) = V ol(O n (R)) √ 2 n√ π n(n+1) 2 λ 1 <•••<λ i <0 0<λ i+1 <•••<λn exp - n j=1 λ 2 j + 1≤j<k≤n log |λ j -λ k | n j=1 (|λ j |dλ j ) = c n n! γ 1 <•••<γ i <0 0<γ i+1 <•••<γn exp 1≤j<k≤n log |γ j -γ k | - 1 2 1≤j<k≤n (γ 2 j + γ 2 k ) exp(- 1 2 n j=1 γ 2 j ) n j=1 (|γ j |dγ j )
where

c n = V ol(O n (R)) n! √ 2 n√ π n(n+1) 2 n 2 n(n-1) 4
+n , and where we wrote, for every 1 ≤ j ≤ n, λ j = n 2 γ j . We now proceed as in §3.1 of [3] (or §2.6.1 of [START_REF] Anderson | An introduction to random matrices[END_REF]). Define, for every

γ 1 < • • • < γ i < 0 < γ i+1 < • • • < γ n , µ n = 1 n n j=1 δ γ j ∈ M + i/n,1-i/n (R), so that 1≤j<k≤n |γ j -γ k | - 1 2 1≤j<k≤n (γ 2 j + γ 2 k ) = -n 2 x<y f (x, y)dµ n (x)dµ n (y). Let M > m 0 , f M = min(f, M) and ∀µ ∈ M + 1 (R), H M (µ) = 1 2 R 2 f M (x, y)dµ(x)dµ(y).
By Lemma 6, min

β≤α M + β,1-β (R) H M = min β≤α M + β,1-β (R)
H.

As a consequence, for every 0

≤ i ≤ ⌊αn⌋, x<y f (x, y)dµ n (x)dµ n (y) = H M (µ N ) - M 2n ≥ m α - M 2n .
Moreover, by Lemma 8 (see §2.1.3 below) and Stirling's formula, ln

(c n ) = n 2 m 1/2 + O(n).
Hence, there exists a constant D > 0 such that

⌊αn⌋ i=0 e R (i, n -i) ≤ exp -n 2 (m α -m 1/2 ) + Dn R |γ|e -γ 2 2 dγ n ,
and the first part of Proposition 4 follows. The proof of the second part goes along the same lines. ✷

Volume of the orthogonal group

Let us equip the vector space of real antisymmetric matrices with the scalar product turning the basis (E ij -E ji ) 1≤i<j≤n into an orthonormal one. This scalar product on the Lie algebra of O n (R) induces on O n (R) a Riemannian metric for which the multiplications on the right by elements produce isometries. We recall in the following Lemma 7 the value of the total volume of O n (R) for this metric.

Lemma 7 For every positive integer n,

V ol(O n (R)) = n! √ π n(n+1) 2 √ 2 n(n-1) 2 n j=1 Γ(1 + j/2)
.

Proof. The coarea formula gives as in the proof of Proposition 4

V ol(O n (R)) = 2 n √ π n(n-1) 2 λ 1 <•••<λn 1≤i<j≤n (λ j -λ i )dµ(λ) . But λ 1 <•••<λn 1≤i<j≤n (λ j -λ i )dµ(λ) = 1 n! √ 2 n(n+1) 2 √ π n R n 1≤i<j≤n (λ j -λ i ) e -||λ|| 2 2 dλ.
The latter integral can be computed using Selberg's formula, see Theorem 3.3.1 of [START_REF] Madan | Random matrices[END_REF], which writes

R n 1≤i<j≤n (λ j -λ i ) e -||λ|| 2 2 dλ = 2 n √ 2 n n j=1 Γ(1 + j/2)).
Hence the result. ✷

Lemma 8

The following asymptotic development holds :

ln V ol(O n (R)) √ 2 n√ π n(n-1) 2 = - n 2 ln n 4 + n 2 ( 3 8 + ln 2 2 ) + 1 4 n ln n + O(n).
Proof. From Lemma 7, when n = 2m is even,

V ol(O n (R)) √ 2 n√ π n(n-1) 2 = n! √ π n √ 2 n(n-3) 2 n/2 j=1 (j!Γ(j + 1/2))
.

From Stirling's formula, n! is equivalent to n n e -n √ 2πn as n grows to infinity and Γ(j + 1/2) to (j - (j + 1) 2 ln(j + 1) -j 2 ln j -3j + 1 2 (j + 1) ln(j + 1) -j 2 ln j

+m ln m + O(n) = n 2 4 ln( n 2 ) - 3 8 n 2 + 3n 4 ln( n 2 ) + O(n).
Finally, we obtain

ln V ol(O n (R)) √ 2 n√ π n(n-1) 2 = n ln n + n 2 4 ln 2 + n 2 4 ln 2 - n 2 4 ln(n) + 3 8 n 2 - 3n 4 ln(n) + O(n)
and the result when n is even. When n = 2m + 1 is odd, we have

V ol(O n (R)) √ 2 n√ π n(n-1) 2 = √ 2 m+n π m 2 m 2 m-1 j=0 (j!Γ(j + 3/2)) . But ln m-1 j=0 (j!Γ(j + 3/2)) = 2 m-1 j=1 (j ln j -j + 3 4 ln j) + O(n) = (n -1) 2 4 ln( n -1 2 ) - 3 8 (n -1) 2 + n -1 4 ln( n -1 2 ) + O(n).
We deduce that

ln V ol(O n (R)) √ 2 n√ π n(n-1) 2 = n 2 4 ln (
(det A)(det A)dµ C (A) = σ∈Sn (-1) ǫ(σ) τ ∈Sn (-1) ǫ(τ ) Sym(n,C) √ 2 #Fix(σ)+#Fix(τ ) • • • • • • a 1σ(1) a 1τ (1) • • • a nσ(n) a nτ (n) dµ C (A),
since the diagonal entries of A have weight √ 2. Now, the integral C z α zβ dµ C (z) vanishes when α = β, so that for every σ ∈ S n , the only permutations τ ∈ S n which contribute to the integral are the ones for which {a 1σ [START_REF] Anderson | An introduction to random matrices[END_REF]

, • • • , a nσ(n) } = {a 1τ (1) , • • • , a nτ (n) }.
This implies that for every j ∈ {1, • • • , n}, {a jσ(j) , a σ -1 (j)j } = {a jτ (j) , a τ -1 (j)j }.

If j belongs to a cycle of length 1 or 2 of σ, we deduce that σ(j) = τ (j). More generally, we deduce that if σ is an element of Cycles(σ), then either σ or σ-1 is an element of Cycles(τ ). In particular, ǫ(σ) = ǫ(τ ). Conversely, every permutation τ which can be written as a product of the form σ∈Cycles(σ) σ±1 contributes to the integral. There are 2 #Cycles ≥3 (σ) such permutations, where Cycles ≥3 (σ) denotes the set of elements of Cycles(σ) having length ≥ 3. As a consequence,

e C (n) = σ∈Sn 2 #Cycles =2 (σ) Sym(n,C) n i=1 |a iσ(i) | 2 dµ C (A),
where Cycles =2 (σ) denotes the subset of elements in Cycles(σ) having length different from 2. Now, C |z| 2 dµ C (z) = 1 whereas C |z| 4 dµ C (z) = 2. Every transposition of Cycles(σ) produces an element of this second type whereas the other elements of Cycles(σ) give rise to products of the first type. Hence the result. ✷ Lemma 10 For every n ∈ N * , σ∈Sn 2 #Cycles(σ) = (n + 1)! Proof. When n = 1, this equality is satisfied. Assume that it is satisfied up to a rank n and let us prove it for the rank n + 1. Let σ ∈ S n+1 and σ = σ1 • • • σk be its decomposition into a product of cycles with disjoint supports. If we remove the element (n + 1) of the cycle which contains this element, we get a permutation τ of S n together with its decomposition into a product of cycles with disjoint supports. We deduce from this a (n + 1) to 1 forgetful map f n : σ ∈ S n+1 → τ ∈ S n such that #Cycles(σ) = #Cycles(f n (σ)) if (n + 1) is not fixed by σ and #Cycles(σ) = #Cycles(f n (σ)) + 1 otherwise. Hence, Proof. This Proposition is a consequence of Lemmas 9 and 10 when n > 0 and of our convention when n = 0. ✷

σ∈S n+1 2 #Cycles(σ) = τ ∈Sn σ∈f -1 n (τ ) 2 #Cycles(σ) = (n + 2)

Real symmetric matrices of odd size

We recall here the values of e R (n), n > 0, distinguishing between the cases n even and n odd, see §25.5 and §26.6 of [START_REF] Madan | Random matrices[END_REF]. The odd case turns out to be easier than the even one:

Proposition 6 (Formula 26.5.2 of [START_REF] Madan | Random matrices[END_REF]) For every odd integer n,

e R (n) = 2 √ 2 π Γ( n + 2 2 ).
Let us briefly recall the proof of this Proposition 6 as taken out from [START_REF] Madan | Random matrices[END_REF].

Proof. As in the proof of Proposition 4, it follows from the coarea formula (see Theorem 3.2.3 of [START_REF] Federer | Geometric measure theory[END_REF] or Theorem 1 of [START_REF] Shub | Complexity of Bezout's theorem. II. Volumes and probabilities[END_REF]) that

e R (n) = V ol(O n (R)) 2 n √ π n(n-1) 2 λ 1 <•••<λn n i=1 √ 2λ i 1≤i<j≤n |λ j -λ i |dµ(λ),
where as before the volume of O n (R) is computed with respect to the right invariant metric for which the basis (E ij -E ji ) 1≤i<j≤n of its Lie algebra is orthonormal, see §2.1.3, and where dµ(λ) denotes the Gaussian measure on R n . The integrand is a Vandermonde determinant. Integrating the odd lines of this determinant and then expanding by pairs of rows in the Laplace manner, we get the relation

e R (n) = V ol(O n (R)) √ 2 n√ π n(n-1) 2 det B.
Here, writing n = 2m + 1, B denotes a square matrix of size (m + 1) × (m + 1) with entries (b ij ) 0≤i<j≤m defined by

∀0 ≤ i ≤ m, ∀0 ≤ j < m, b ij = 2(ψ ij + η 2i η 2j+1 )
and b im = 2η 2i where

ψ ij = 0≤x<y<+∞ |xy|(x 2i y 2j+1 -y 2i x 2j+1 )dµ(x)dµ(y) (6)
and

η k = +∞ 0 x k+1 dµ(x) = 1 2 √ π Γ( k + 2 2 ).
Using linear combinations of rows and columns of B with the help of the relations

∀i, j ≥ 0, ψ i+1,j = (i + 1)ψ ij - 1 π2 i+j+7/2 Γ(i + j + 5/2) and η 2i+2 = (i + 1)η 2i , we get det B = 1 √ π n √ 2 m 2 m(m+1) det(Γ(i + j + 5/2)) 0≤i,j≤m-1 = 1 √ π n √ 2 m 2 m(m+1) m-1 j=0
(j!Γ(5/2 + j))(see formula A.18.7 in [START_REF] Madan | Random matrices[END_REF])).

When n = 2m + 1 we deduce from this that

e R (n) = V ol(O n (R)) √ π n(n+1) 2 √ 2 m+1 2 m(m+2) m-1 j=0 (j!Γ(5/2 + j)).
The result now follows from Lemma 7. ✷

The proof of Proposition 6 may also provide an alternative proof of Lemma 7 for odd n's, as suggested in [START_REF] Madan | Random matrices[END_REF].

Alternative proof of Lemma 7 in odd dimensions. Proceeding as in the proof of Proposition 6, we get

1 = V ol(O n (R)) 2 n √ π n(n-1) 2 λ 1 <•••<λn 1≤i<j≤n (λ j -λ i )dµ(λ) = V ol(O n (R)) 2 n √ π n(n-1) 2 det(B ′ ),
where B ′ denotes a square matrix of size (m + 1) × (m + 1) with entries (b ′ ij ) 0≤i,j≤m defined by

∀0 ≤ i ≤ m, ∀0 ≤ j < m, b ′ ij = 2(ψ ′ ij + η 2i-1 η 2j ) and b ′ im = 2η 2i-1 , whereas ψ ′ ij = 0≤x<y≤+∞ (x 2i y 2j+1 -y 2i x 2j+1 )dµ(x)dµ(y) = -ψ j(i-1) .
From linear combinations and the relation

∀i, j ≥ 0, ψ ′ i+1,j = ( 2i + 1 2 )ψ ′ ij - 1 π2 i+j+5/2 Γ(i + j + 3/2), we get det(B ′ ) = 1 π m √ 2 m 2 m 2 m-1 j=0 (j!Γ(3/2 + j)).
Finally,

V ol(O n (R)) = √ π m(n+2) √ 2 m 2 m 2 +n m-1 j=0 (j!Γ(3/2 + j)) (7
) √ mm! as n = 2m + 1 grows to infinity.

and e R (n) = 2 √ 2 π Γ( n+2 2 ), since Γ(1/2) = √ π. ✷ Remark

Real symmetric matrices of even size

When the dimension n = 2m is even, the value of e R (n) is given by the following Proposition 7.

Proposition 7 For every even positive integer

n = 2m, e R (n) = (-1) m n! m!2 n + (-1) m-1 4 √ 2n! √ πm!2 n m-1 k=0 (-1) k Γ(k + 3/2) k! .
The expression given by Proposition 7 can be rewritten as

e R (n) = (-1) m 4 √ 2n! √ πm!2 n +∞ 0 √ t(e -t - m-1 k=0 (-t) k k! )e -t dt.
The first term in the right-hand side of the expression given by Proposition 7 is alternated and negligible for large values of n with respect to the second one which is always non negative. The latter can be checked by pairing the terms of the sum, see the proof of Corollary 3.

Remark 7

The first values of e R (n) for even n's are:

e R (0) = 1, e R (2) = √ 2 -1/2, e R (4) = 3 4 ( √ 2
(n) = 2 √ 2 π Γ( n+2
2 ) from Proposition 6. When n = 2m is even, the first term in the right-hand side given by Proposition 7 is equivalent to

(-1) m m m e -m √ 2, that is (-1) m Γ( n+2 2 )
√ πm from Stirling's formula. Pairing the terms of the sum in the second one, we get

m-1 k=0 (-1) k Γ(k + 3/2) k! = - 1 2 m 2 -1 j=0 Γ(2j + 3/2) (2j + 1)! when m is even and m-1 k=0 (-1) k Γ(k + 3/2) k! = Γ(m + 1/2) (m -1)! - 1 2 m-1 2 -1 j=0 Γ(2j + 3/2) (2j + 1)!
when m is odd. In both cases, this sum gets equivalent to (-1) m-1 √ m 2 as n grows to infinity, hence the result. ✷ In order to prove Proposition 7, we first compute in the following Proposition 8 e R (n) in terms of a sequence (b m ) m∈N which we now introduce. Let (a j ) j=0 be the sequence defined by the relations a 0 = 8 √ 2-7 3

and

∀j > 0, a j = ( 4j + 2 2j + 3 )a j-1 + 1. Let b 1 = a 0 + 1 = 4 3 (2 √ 2 -1) and for every m > 1, b m = m-1 j=0 (-1) m-1-j m -1 j a j .
Proposition 8 For every even integer n , so that the entry of the i-th row and j-th column, 1 ≤ i ≤ m -1, 0 ≤ j ≤ m -1 equals 1/(j -i + 1)! if j -i + 1 ≥ 0 and 0 otherwise. Subtracting to the m -1 first lines multiples of the last one, we obtain zeros on the last column, whereas the entry of the penultimate column on the i-th line, 1 ≤ i ≤ m -2 equals m-1-i (m-i)! . Then, subtracting to the m -2 first lines multiples of the penultimate one, we get zeros on the penultimate column whereas the entry of the (m -2)-th column on the i-th line, 1 ≤ i ≤ m -3, equals (m-1-i)(m-2-i) (m-i)!

= 2m > 0, e R (n) = n!Γ( n+3
. By recurrence, we get a lower triangular matrix and (-1) j 2 j m-1 k=m-j k j!Γ(j + 5/2) (4πψ 0j + Γ(j + 3/2)).

We deduce then from Lemma 7 the relation

e R (n) = n!Γ( n+3 2 ) m!(m -1)!2 n √ π 2 √ 2 m-1 j=0
(-1) m-1-j 2 j m-1 j Γ(j + 5/2) (4πψ 0j + Γ(j + 3/2)).

The result follows by setting, for every j ∈ {0, • • • , m -1}, a j = 2 j+1 √ 2 Γ(j + 5/2) (4πψ 0j + Γ(j + 3/2)) -1.

Indeed,

ψ 00 = 1 8 √ 2π ( √ 2 -1) (8)
so that a 0 = 8 √ 2-7 3

and the recurrence relation satisfied by (a j ) j≥0 follows from the relation ∀j > 0, ψ 0j = (j + 1/2)ψ 0j-1 + 1 π2 j+5/2 Γ(j + 3/2) (compare formula 26.4.13 of [START_REF] Madan | Random matrices[END_REF]). ✷ Remark 8 As in the alternative proof of Lemma 7, we may get that for every n = 2m > 0, x 2 (2x 2 -1) m-1 dx -4 1 0

x 2 (x 2 -1) m-1 dx.

From the relations ∀j > 0 1 0 x 2 (x 2 -1) j dx = -2j 2j + 3 Remark 9 An expression of e R (n) in terms of hypergeometric series can be extracted from [START_REF] Delannay | Distribution of the determinant of a random realsymmetric matrix from the Gaussian orthogonal ensemble[END_REF], whereas an equivalent in logarithmic scale can be extracted from [START_REF] Tao | A central limit theorem for the determinant of a Wigner matrix[END_REF].

2.2.4

Values of e(p, q) for p + q ≤ 3

We have not been able to compute the numbers e R (p, q) in general and only give here their values for p + q ≤ 3. Proof. The multiplication of symmetric matrices by -Id preserves the measure of Sym(n, R) as well as the absolute value of the determinant, so that for every p, q ∈ N, e R (p, q) = e R (q, p). As a consequence, e R (1, 0) = e R (0, 1) = 1 2 e R (1) = 1

Proof.

  Let x ∈ X \(Crit(p)∪Base(p)) (resp. x ∈ RX \Crit(p)) and (x 1 , • • • , x n ) be local holomorphic (resp. real holomorphic) coordinates in the neighbourhood of x = (0, • • • , 0) such that ( ∂ ∂x 1 , • • • , ∂ ∂xn )be orthonormal at x and ( ∂ ∂x 2 , • • • , ∂ ∂xn ) spans ker(dp) at the point x. With the notations of §1.3, every element σ ∈ H 0 (X, L d ) (resp. σ ∈ RH 0 (X, L d )) writes σ = n j=0 a j σ j + 1≤k≤l≤n b kl σ kl + τ, where a j , b kl ∈ C (resp. a j , b kl ∈ R) and τ ∈ H 3x (resp. τ ∈ RH 3x ), see (2) (resp.

2. 1 . 2

 12 Measure concentration around matrices of vanishing signature Proposition 4 For every α ∈ [0, 1/2[ , there exists c α > 0 such that

τ ∈Sn 2 #Proposition 5

 25 Cycles(τ ) = (n + 2)! by induction. ✷ For every n ∈ N, e C (n) = (n + 1)!

2 2 √ 1 = 2 j 1 ,

 22121 )b m m!(m -1)!2 n √ π .Proof. As in the proof of Proposition 6, we establish thate R (n) = V ol(O n (R)) √ 2 n√ π n(n-1) 2 det(C),where C denotes a square matrix of size m × m and entries (c ij ) 0≤i,j≤m-1 satisfying∀0 ≤ i, j ≤ m -1, c ij = 2(ψ ij + η 2i η 2j+1 ) with ψ ij = 0≤x<y≤+∞ |xy|(x 2i y 2j+1 -y 2i x 2j+1 )dµ(x)dµ(y)andη k = +∞ 0 x k+1 dµ(x) = 1 π Γ( k+22). Following §26.6 of[START_REF] Madan | Random matrices[END_REF],det(C) = -2 m -1 (η 2j+1 ) 0≤j≤m-1 (η 2i ) 0≤i≤m-1 (ψ ij ) 0≤i,j≤m-1 = -2 m -1 (η 2j+1 ) 0≤j≤m-1 η 0 (ψ 0j ) 0≤j≤m-1 0 (ψ ij -iψ i-1j = -Γ(i+j+3/2) π2 i+j+5/2 ) 1≤i≤m-1, 0≤j≤m-(4πψ 0j + Γ(j + 3/2)) 0≤j≤m-1 0 Γ(5/2) Γ(j + 5/2) 1≤j≤m-1 0 0 (Γ(i + j + 3/2) -(i + 1/2)Γ(i + j + 1/2) = jΓ(i + j + 1/2)) 2≤i≤m-1, 1≤j≤m-

1 1 .( 2 √ 2 -(- 1 ) 1 j=0(- 1 )

 1122111 and for every m > 1, b ′ m = m-1 j=0 (-1) m-1-j m-1 j a ′ j . This sequence (a ′ j ) j≥0 is defined by the relations a ′ 0 = 1 and ∀j > 0, a j = ( 4j 2j+1 )a j-1 + Proof of Proposition 7. When m = 1, Proposition 7 is a consequence of Proposition 8 since b 1 = 4 3 1). From Proposition 8, when m > 1, we have to compute the values of bm ∈ Q[ √ 2]. For every j ≥ 0, we deduce from the recurrence relation that m-1-j m -1 j2 √ 22 j -1 2j + 3 x 2j+3 ′ = mm-1-j m -1 j (2 √ 22 j -1)x 2j+2 = x 2 (2 √ 2(2x 2 -1) m-1 -(x 2 -1) m-1 ),

1 0x 2 (x 2 - 1 ) j-1 and 1 0 x 2 (2x 2 - 1 )

 12211221 7 follows from Proposition 8. ✷

  , • • • , 3}, e R (p, 3 -p)

2 .√ π 0<λ 1 <λ 2 <+∞ det |λ 1 | λ 2 1 |λ 2 | λ 2 2 dµ(λ 1 )

 21221 Proceeding as in the beginning of the proof of Proposition 6, we get thate R (2, 0) = V ol(O 2 (R))2 so that e R (2, 0) = e R (0

  The research leading to these results has received funding from the European Community's Seventh Framework Progamme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement n o [258204]. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Hörmander peak sections . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Incidence varieties and evaluation maps . . . . . . . . . . . . . . . . . 12 1.5 Proofs of Theorems 5 and 6 . . . . . . . . . . . . . . . . . . . . . . . 16 2 Expected determinant of random symmetric matrices 20 2.1 Large random real symmetric matrices . . . . . . . . . . . . . . . . . 20 2.1.1 The energy functional . . . . . . . . . . . . . . . . . . . . . . 20 2.1.2 Measure concentration around matrices of vanishing signature 21 2.1.3 Volume of the orthogonal group . . . . . . . . . . . . . . . . . 22 2.2 Determinants of random symmetric matrices . . . . . . . . . . . . . . 24 2.2.1 Complex symmetric matrices . . . . . . . . . . . . . . . . . . 24 2.2.2 Real symmetric matrices of odd size . . . . . . . . . . . . . . . 25

	Contents
	1 Expected Betti numbers of random real hypersurfaces	5
	1.1

2.2.3 Real symmetric matrices of even size . . . . . . . . . . . . . . 27 2.2.4 Values of e(p, q) for p + q ≤ 3 . . . . . . . . . . . . . . . . . . 31 1 Expected Betti numbers of random real hypersurfaces 1.1 Notations

  or the restriction of p to RC σ (resp. C σ ) is not Morse (resp. not a Lefschetz pencil). For every σ ∈ H 0 (X, L d ) \ ∆ d p , we denote by Crit(p |Cσ ) the set of critical points of the restriction of p to C σ and set where δ x denotes the Dirac measure of X at the point x. When σ is real, we denote similarly, for every i ∈ {0, • • • n -1}, by Crit i (p |RCσ ) the set of critical points of index i of p |RCσ , and set

	ν(C σ ) =	1 d n	x∈Crit(p |Cσ )	δ x ,

  For every n ∈ N * , denote by S n the group of permutations of {1, • • • , n} and for every σ ∈ S n , by Cycles(σ) the set of cycles appearing in the decomposition of σ into a product of cycles with disjoint supports. For instance, if σ denotes the permutation For every n ∈ N * , e C (n) = σ∈Sn 2 #Cycles(σ) .Proof. For every A ∈ Sym(n, C), denote by A = 1≤i≤j≤n a ij E ij and set a ji = a ij if i > j. By definition,

		2) -	n 2 4	ln(n) +	n 2	ln(n) +	n 2 4	ln 2 +	3 8	n 2 -	n 4	ln n + O(n)
	and the result. ✷
	2.2 Determinants of random symmetric matrices
	2.2.1 Complex symmetric matrices
	1 2 3 4 5 3 2 1 5 4	of {1, • • • , 5}, then Cycles(σ) = {(13), (2), (45)}.
	Lemma 9 e C (n) =
		Sym(n,C)

6

  The first values given by Proposition 6 are

	e R (1) =	2 π	, e R (3) =	3 √ 2π	and e R (5)	2	15 √ 2π	.
	√ Moreover, from Stirling's formula, e R (n) is equivalent to 2 π	2

  Whatever the parity of n is, e R (n) gets equivalent to2

	+ 1), 2] for even n and transcendental for odd values of 8 , e R (8) = 3×5×7 16 ( 13 8 √ 2 + 1). 2 -15 Note that e R (n) is algebraic in Q[ e R (6) = 165 32 √ √
	n.
	Corollary 3 √ π Γ( n+2 2 2 ) as n grows to infinity.
	Proof. For every odd n, e R

We get along the same lines that e R [START_REF] Ben Arous | Large deviations for Wigner's law and Voiculescu's non-commutative entropy[END_REF]0)

From the recurrence relation given in the proof of Proposition 6, we deduce that

Likewise,