Expected number and distribution of critical points of real Lefschetz pencils
Michele Ancona
math.AG math.DG math.PR

[
    Feng Tian
    December 30, 2023
=====================





§ ABSTRACT


We give an asymptotic probabilistic real Riemann-Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere.
More generally,  we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety.



§ INTRODUCTION

The Riemann-Hurwitz formula says that the total ramification index  of a degree d branched covering  f:Σ→Σ'  between two compact Riemann surfaces equals d·χ(Σ')-χ(Σ). In particular, if Σ'=ℂℙ^1, the total ramification index is  2d+2g-2, where g is the genus of Σ.

More generally, if u:Xℂℙ^1 is a Lefschetz pencil on a complex manifold X of dimension n, then 

    (-1)^n#crit(u)=χ(X)-2χ(F)+χ(Y)
 
where F is a smooth fiber of u and Y is the base locus of u.

The questions that motivate this paper are the following: how do  these critical points distribute on the variety? When u is defined over ℝ, what about the number  of real critical points?

We answer these questions by computing the asymptotic expected number of real critical points of real Lefschetz pencils and also the asymptotic distribution of such points.

The chosen random setting has already been considered by  Shiffman and Zelditch in <cit.> to study the integration current over the  zero locus of a  random global section of a line bundle over a complex projective manifold.

In the real case, Kac <cit.>, Kostlan <cit.> and Shub and Smale <cit.> computed the expected number of real roots of a random real polynomial.
In higher dimensions, Podkorytov <cit.> and Bürgisser <cit.> computed the expected Euler characteristic of  random real algebraic submanifolds and Letendre  <cit.> the expected volume (see <cit.>  for the expected length of a random lemniscate). In <cit.> Gayet and Welschinger estimated from above and below the Betti numbers of the real locus of real algebraic submanifolds  (see also <cit.>).
 For intersection of real quadrics, a precise asymptotic of the total Betti number has been given by Lerario and Lundberg in <cit.>.
In <cit.> Nicolaescu computed the expected number of critical of a random smooth function on a Riemannian manifold have and how they distruibute.



 §.§ Statements of the results


Let X be a smooth real projective manifold of dimension n, that is a complex projective manifold equipped with an anti-holomorphic involution c_X, called the real structure. We denote by ℝ X=(c_X)  its real locus. Let ℒ be a positive real line bundle over X.
For large d, for almost all pairs  (α,β)∈  H^0(X;ℒ^d)^2 
(resp. ℝ H^0(X;ℒ^d)^2) of (real) global section, the map u_αβ:Xℂℙ^1 defined by x↦ [α(x):β(x)] is a (real) Lefschetz pencil, see Proposition <ref>. Recall that a real Lefschetz pencil is a Lefschetz pencil u:Xℂℙ^1 such that conj∘ u= u∘ c_X.
 We denote the set of critical points of u_αβ by crit(u_αβ) and by ℝcrit(u_αβ)=crit(u_αβ)∩ℝ X the set of real critical points. 
 
The number  of real critical points of a real Lefschetz pencil depends on the pair (α, β). The main theorem of this paper is the computation of the expected value of this number. Recall that, by definition, the expected value of #ℝcrit(u_αβ) equals

    𝔼[#ℝcrit(u_αβ)]=∫_(α,β)∈ℝ H^0(X,ℒ^d)^2(#ℝcrit(u_αβ))dμ(α,β).


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


    lim_d→ +∞1/√(d)^n𝔼[#ℝcrit(u_αβ)]={[ n!!/(n-1)!!e_ℝ(n)π/2Vol_h(ℝ X)                    if n is odd;    n!!/(n-1)!!e_ℝ(n)Vol_h(ℝ X)                  if n is even. ].



In this theorem, Vol_h(ℝ X) is the volume of ℝ X with respect to the Riemannian volume form  _h induced by the positive curvature  of the  metric h. The  probability measure we consider is a natural Gaussian probability on ℝ H^0(X;ℒ^d)^2 (see Section 2.1) and e_ℝ(n) is the expected value of (the absolute value) of the determinant of real symmetric matrices (for the explicit values of e_ℝ(n), see <cit.>).

We recall that e_ℝ(1)=√(2/π), then we have:
 Let (Σ,c_Σ) be a real Riemann surface and (ℒ,h) be a real Hermitian line bundle of degree 1. Then, for every pair (α,β)∈ℝ H^0(X;ℒ^d)^2 without common zeros, the map u_αβ is a degree d branched covering between Σ and ℂℙ^1 and  the expected real total ramification index of u_αβ  is equivalent to 
    √(π/2)Vol_h(ℝΣ)√(d)
 as d  tends to +∞.



Theorem <ref> is a consequence of a more precise equidistribution   result. In order to introduce it, let us define a natural empirical measure associated with the real  critical points of a Lefschetz pencil as follows.
For any pair (α,β)∈ℝ H^0(X;ℒ^d)^2 of real global sections of ℒ^d, we define 
    ℝν_αβ=∑_x∈ℝcrit(u_αβ)δ_x.
 


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


    lim_d→ +∞1/√(d)^n𝔼[ℝν_αβ]={[ n!!/(n-1)!!e_ℝ(n)π/2_h            if n is odd;    n!!/(n-1)!!e_ℝ(n)_h          if n is even. ].

weakly in the sense of distributions. Here, _h is the Riemannian volume form induced by the curvature ω.


Theorem <ref> says that, for any continuous function  φ∈ C^0(ℝ X), we have 

    lim_d→ +∞1/√(d)^n𝔼[ℝν_αβ](φ)={[ n!!/(n-1)!!e_ℝ(n)π/2∫_ℝ Xφ_h                  if n is odd;    n!!/(n-1)!!e_ℝ(n)∫_ℝ Xφ_h                if n is even. ].

where the expected value is defined by 
    𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈ℝcrit(u_αβ)φ(x)dμ(α,β).


In the complex case, we obtain a similar  equidistribution theorem, whose proof   follows along the same lines. For any pair (α,β)∈  H^0(X;ℒ^d)^2 
 of  global sections of ℒ^d, we define 
    ν_αβ=∑_x∈crit(u_αβ)δ_x
 to be the empirical measure associated with the   critical points of the pencil u_αβ.


Let X be a smooth complex projective manifold of dimension n and (ℒ,h) be  a Hermitian line bundle over X with positive curvature ω. Then 


    lim_d→ +∞1/d^n𝔼[ν_αβ]=(n+1)ω^n

weakly in the sens of distribution.

As before, Theorem <ref> says that, for any continuous function  φ on X, we have


    lim_d→ +∞1/d^n𝔼[ν_αβ](φ)=(n+1)·∫_Xφω^n.



  §.§.§ Organisation of the paper

In Section <ref>, we introduce the Gaussian measure on H^0(X;ℒ^d)^2 associated with a Hermitian  line  bundle (ℒ,h) over a complex manifold X. We also give the same construction for the real case. We follow the approach of  <cit.>. In Section <ref>, we present some classical results about Lefschetz pencils on complex manifolds.
In Sections <ref> and <ref> we introduce our main tools, namely  the Hörmander peak sections (see also <cit.>, <cit.>) and the incidence manifold (see <cit.>).
Section <ref>  is completely devoted to the proofs of the Theorems  <ref>, <ref> and <ref>. In Sections <ref> and <ref>, we prove the equidistribution of critical points of a (real) Lefschetz pencil over a (real) algebraic variety X. This will be done using coarea formula and peak sections. These ideas are taken from <cit.>. In Section <ref> we will compute the universal constant by direct computation.






  §.§.§ Acknowledgments

I am very grateful to my advisor Jean-Yves Welschinger for all the time he devoted to me and for all the fruitful discussions we had. 

 This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070)
of Université de Lyon, within the program "Investissements d'Avenir"
(ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).





§ DEFINITIONS AND MAIN TOOLS



 §.§ Notations


Let X be a complex manifold of dimension n.
Let ℒ→ X be a holomorphic line bundle equipped with a Hermitian metric h of positive curvature ω∈Ω^(1,1)(X,ℝ). The curvature form induces a Kähler metric   and a normalized volume form  dx=ω^n/∫_Xω^n on X.

The Hermitian metric h induces a Hermitian metric h^d on ℒ^d for any integer d>0 and also a L^2-Hermitian product ⟨·,·⟩_L^2 on the space H^0(X;ℒ^d) of global holomorphic sections of ℒ^d. It is defined by 

    ⟨α,β⟩_L^2=∫_Xh^d(α,β)dx

for any α,β in H^0(X;ℒ^d).

This L^2-Hermitian product induces a Gaussian measure  on H^0(X;ℒ^d)^2
defined by


    μ(A)=1/π^2N_d∫_Ae^-α_L^2^2-β_L^2^2dαdβ

for any open subset A⊂ H^0(X;ℒ^d)^2 where dαdβ is the Lebesgue measure associated with ⟨·,·⟩_L^2 and N_d=_ℂH^0(X;ℒ^d).

Finally, a Lefschetz pencil on X is a rational map u:Xℂℙ^1 having only non degenerated critical points and defined by two global sections of a holomorphic line bundle with smooth and transverse vanishing loci.
 

All these definitions have a real counterpart. 


  * Let X be a real algebraic variety of dimension n, that is a complex manifold equipped with an anti-holomorphic involution c_X. We denote by
ℝX=(c_X) its real locus. 

  * A real holomorphic line bundle u:ℒ→ X  is a line bundle  equipped with an anti-holomorphic involution c_ℒ such that p∘ c_X=c_ℒ∘ p and c_ℒ is complex-antilinear in the fibers.

  * We denote by ℝ H^0(X;ℒ) the real vector space of real global section of ℒ, i.e. sections s∈ H^0(X;ℒ) such that s ∘ c_X=c_ℒ∘ s.

  * A real Hermitian metric on ℒ is a Hermitian metric h such that c_ℒ^*h=h̅. If (ℒ,h) is a   line bundle over X with positive curvature ω, then ω(.,i.) is a Hermitian metric over X and its real part defines a Riemannian metric over ℝ X. We denote  the Riemannian volume form induced by this metric by _h.

  * The L^2-Hermitian product ⟨·,·⟩_L^2 on H^0(X;ℒ^d) restricts to a L^2-scalar product on ℝ H^0(X;ℒ^d), also denoted  by ⟨·,·⟩_L^2. Then, as in the complex case, also in the real case we have a natural Gaussian measure on ℝ H^0(X;ℒ^d)^2 defined by


    μ(A)=1/π^N_d∫_Ae^-α_L^2^2-β_L^2^2dαdβ

for any open subset A⊂ℝ H^0(X;ℒ^d)^2 where dαdβ is the Lebesgue measure associated with ⟨·,·⟩_L^2 and N_d=_ℂH^0(X;ℒ^d)=_ℝℝ H^0(X;ℒ^d).


  * A real Lefschetz pencil u:Xℂℙ^1 is a Lefschetz pencil  such that p∘ c_X=conjconj∘ p.

We conclude this section by introducing some notation on symmetric matrices.
  For any n∈ℕ^*, we denote by (n,ℝ) the real vector space of real symmetric matrices of size n× n. It is a vector space of dimension n(n+1)/2. We equip it with the basis ℬ given by Ẽ_jj and Ẽ_ij=E_ij+E_ji for 1⩽ i<j⩽ n, where, for any k,l with 1⩽ k,l⩽ n, we denote by E_kl  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 (n,ℝ) with the scalar product turning ℬ into an orthonormal basis. Let μ_ℝ  the associated Gaussian probability measure. We then set 

    e_ℝ(n)=∫_A∈(n,ℝ)| A| dμ_ℝ(A).




 §.§ Lefschetz pencils


In this section, we compute the asymptotic value of the number of critical points of a Lefschetz pencil (see also <cit.>).

Recall that a Lefschetz fibration is a map X→ℂℙ^1 with only non degenerate critical points. The following proposition is a kind of Riemann-Hurwitz formula for Lefschetz pencils, for a proof see <cit.>.

 Let X be a smooth complex projective manifold of positive dimension n equipped with a Lefschetz fibration p:X→ℂℙ^1 and let F be a regular fiber of p. Then we have the following equality:

    χ(X)=2χ(F)+(-1)^n#crit(p).


Remark that if u:Xℂℙ^1 a Lefschetz pencil and we blow-up the base locus Base(u)≑ Y, then we obtain a Lefschetz fibration ũ:X̃≑ Bl_YX→ℂℙ^1. By additivity of the Euler characteristic, we have that χ(X̃)=χ(X)+χ(Y), then by Proposition <ref> we have 
    χ(X)=2χ(F)-χ(Y)+(-1)^n#crit(u).




Let ℒ be an ample line bundle over a complex manifold X of dimension n. For almost all global sections (α, β)^2∈ H^0(X;ℒ^d)^2, the map u_αβ defined by x↦ [α(x):β(x)] is a Lefschetz pencil (see Prop. <ref>).
Then, as d goes to infinity, we have


    #crit(u_αβ)=(n+1)(∫_Xc_1(ℒ)^n)d^n+O(d^n-1).



  We will follow the lines of Lemma 2, Lemma 3 and Proposition 4 of <cit.>.

We have χ(F)=∫_F c_n-1(F) and χ(Y)=∫_Y c_n-2(Y).
We remark that the base locus is the intersection of the zero locus of α and β, that is Y=Z_α∩ Z_β. 
A regular fiber F over [a,b]∈ℂℙ^1 is the zero locus of the section bα-aβ∈ H^0(X;ℒ^d), thus the normal bundle N_X/F is ℒ^d_| F 
To compute χ(F) we will use the adjunction formula.
We have 

    0→ TF→ TX_| F→ N_X/F→ 0
 
then we have c(X)_| F=c(F)∧ c(ℒ^d)_| F, that is 
    (1+c_1(X)+…+c_n(X))_| F=(1+c_1(F)+…+c_n-1(F))∧(1+dc_1(ℒ)).

If we develop this we have c_1(X)=c_1(F)+dc_1(ℒ)  and, for j∈{2,…,n-1}, we have c_j(X)_| F=c_j(F)+dc_1(ℒ)_| F∧ c_j-1(F).
Then, summing up the term,


    c_j(F)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| F∧ c_j-k(X)_| F.

In particular, for j=n-1 we have


    c_n-1(F)=∑_k=0^n-1(-1)^kd^kc_1(ℒ)^k_| F∧ c_n-k-1(X)_| F.

Then χ(F) is equal to ∫_F∑_k=0^n-1(-1)^kd^kc_1(ℒ)^k)_| F∧ c_n-k-1(X)_| F.
But, for α∈ H_dR^2n-2(X), we have that 


    ∫_Fα_| F=∫_Xα∧ c_1(ℒ^d)

so, 
    χ(F)=∑_k=0^n-1∫_X(-1)^kd^k+1c_1(ℒ)^k+1∧ c_n-k-1(X)

and asymptotically we get 
    χ(F)∼(-1)^n-1(∫_Xc_1(ℒ)^n)d^n.

For Y=Z_α∩ Z_β, the same argument gives us


    c_j(Y)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| Y∧ c_j-k(Z_α)_| Y.

But, as before, 
    c_j-k(Z_α)=∑_h=0^j-k(-1)^hd^hc_1(ℒ)^h∧ c_j-k-h(X).

and so,  replacing in the above equation 

    c_j(Y)=∑_k=0^j(-1)^kd^kc_1(ℒ)^k_| Y∧ (∑_h=0^j-k(-1)^hd^hc_1(ℒ)^h_| Y∧ c_j-k-h(X)_| Y)

For j=n-2 we have

    c_n-2(Y)=
    ∑_k=0^n-2(-1)^kd^kc_1(ℒ)^k_| Y∧ (∑_h=0^n-2-k(-1)^hd^hc_1(ℒ)^h_Y∧ c_n-2-k-h(X)_| Y)

and this is equivalent to 

    ∑_k=0^n-2(-1)^n-2d^n-2c_1(ℒ)^n-2_| Y
    =
    (-1)^n-2(n-1)d^n-2c_1(ℒ)^n-2_| Y

 as d→∞.
So we have, as d→∞,

    χ(Y)∼
    (-1)^n-2(n-1)d^n-2∫_Yc_1(ℒ)^n-2_| Y=


    =(-1)^n-2(n-1)d^n-1∫_Z_αc_1(ℒ)^n-2∧ c_1(ℒ)=


    =(-1)^n-2(n-1)(∫_Xc_1(ℒ)^n)d^n.

Combining this with χ(X)=2χ(F)-χ(Y)+(-1)^n#crit(u_αβ)  we obtain the result.








 §.§ Hörmander's peak sections

In this section we recall the theory of Hörmander's peak sections, an essential tool for our proofs of Theorems   <ref> and <ref> (see also <cit.>, <cit.>, <cit.>).
Let ℒ be a holomorphic line bundle over a smooth complex projective manifold equipped with a Hermitian metric h of positive curvature ω and let dx=ω^n/∫_Xω^n be the normalized volume form. Let x be a point of X.
There exists, in the neighborhood of x, a holomorphic trivialization e of ℒ such that the associated potential reaches a local minimum at x with Hessian of type (1,1).
The following result was proved in  <cit.> (see also <cit.>) .

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 ∫_Xh^d(σ,σ)dx=1 and 

    ∫_X∖ B(x,log d/√(d))h^d(σ,σ)dx=O(1/d^2p')


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_1^p_1⋯ x_n^p_n+O(x^2p'))e^d(1+O(1/d^2p'))

where 

    λ^-2=∫_B(x,log d/√(d))|x_1^p_1⋯ x_n^p_n|^2 h^d(e^d,e^d)dx

and e is a holomorphic trivialization of ℒ in the neighborhood of x  whose potential ϕ=-log h(e,e) reaches a local minimum at x with Hessian πω(.,i.).


This lemma is true also in the real setting, in the following sense:

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 holomorphic section σ satisfying ∫_Xh^d(σ,σ)dx=1 and 

    ∫_X∖ B(x,log d/√(d))h^d(σ,σ)_h=O(1/d^2p')

Moreover, if (x_1,…,x_n) are local real holomorphic coordinates in the neighborhood of x, we can assume that in a neighborhood of x,

    σ(x_1,…,x_n)=λ(x_1^p_1⋯ x_n^p_n+O(x^2p'))e^d(1+O(1/d^2p'))

where 

    λ^-2=∫_B(x,log d/√(d))| x_1^p_1⋯ x_n^p_n|^2 h^d(e^d,e^d)dx

and e is a real trivialization of ℒ in the neighborhood of x  whose potential ϕ=-log h(e,e) reaches a local minimum at x with Hessian πω(.,i.).


This real counterpart follows from Lemma <ref> by averaging the peak sections with the real structure.

Let σ_0 be  the section given by the Lemma <ref> with p'=3 
and p_i=0 for all i, σ_i the section given by Lemma <ref> with p'=3 and p_j=δ_ij, σ_ij the section given by the Lemma <ref> with p_i=p_j=1 and p_k=0 otherwise and σ_kk the section given by the Lemma <ref> with p_k=2 and p_l for l≠ k. 

These sections are called peak sections. Their Taylor expansions are:


    σ_0(y)=(λ_0+O( y^6))e^d(1+O(1/d^6));


    σ_i(y)=(λ_iy_i+O( y^6))e^d(1+O(1/d^6)) ∀ i;


    σ_ij(y)=(λ_ijy_iy_j+O( y^6))e^d(1+O(1/d^6))∀ i≠ j;


    σ_kk(y)=(λ_kky_k^2+O( y^6))e^d(1+O(1/d^6)) ∀ k.

The following lemma shows the asymptotic of the constants 
λ_0, λ_i, λ_ij et λ_kk.

[Lemma 2.5 of <cit.>]
Under the hypothesis of Lemma <ref> and <ref>, we have

    lim_d⟶∞1/√(d)^nλ_0=√(δ_ℒ)


    lim_d⟶∞1/√(d)^n+1λ_i=√(π)√(δ_ℒ)


    lim_d⟶∞1/√(d)^n+2λ_ij=π√(δ_ℒ)


    lim_d⟶∞1/√(d)^n+2λ_kk=π/√(2)√(δ_ℒ)

for the L^2-product induced by dx=ω^n/∫_Xω^n where  δ_ℒ=∫_X c_1(ℒ)^n is the degree of the line bundle ℒ.


Set
 
    H_2x={s∈ H^0(X;ℒ^d) | s(x)=0, ∇ s(x)=0,∇^2s(x)=0}


 
    (resp.  ℝ H_2x={s∈ℝ H^0(X;ℒ^d) | s(x)=0, ∇ s(x)=0,∇^2s(x)=0}).


This space is formed by  sections whose 2-jet vanishes at x. The sections (σ_i)_0≤ i≤ n (σ_ij)_1≤ i≤ j≤ n provide a basis of a complement of H_2x. This basis is not orthonormal and its spanned subspace is not orthogonal to H_2x. However, this basis is aymptotically an orthonormal basis and its spanned subspace is asymptotically orthonormal to H_2x, in the following sense:


The section (σ_i)_0≤ i≤ n and (σ_ij)_1≤ i≤ j≤ n have L^2-norm equal to 1 and  and their pairwise L^2-scalar product are O(1/d). Likewise, their scalar products with every unitary element of H_2x are O(1/d^3/2).








 §.§ Incidence manifolds

Following <cit.>, we define an incidence manifold associated with the complex (resp. real) manifold X and to the (real) positive line bundle ℒ. We will use this incidence manifold to prove  that,  for almost all pairs global sections (α, β)∈ H^0(X;ℒ^d)^2  (resp. (α, β)∈ℝ H^0(X;ℒ^d)^2),  the map u_αβx↦ [α(x):β(x)] defines   a Lefschetz pencil, see Proposition <ref>.

Let (ℒ,h) be a (real) Hermitian line bundle with positive curvature ω over a (real) algebraic variety X of dimension n.
 
 Let α, β∈ H^0(X;ℒ^d) (resp. ℝ H^0(X;ℒ^d)) be two  (real) global sections such that the map x↦ [α(x):β(x)] is a Lefschetz pencil. We define 


  * the base locus of a Lefschetz pencil as the points x such that α(x)=β(x)=0; 

  * the critical points as the points x∈ X∖Base(u_αβ) such that (α∇β-β∇α)(x)=0 (this expression does not depend on the choice of a connection ∇ on ℒ).
We denote by crit(u_αβ) the set of critical points of (u_αβ) and by ℝcrit(u_αβ)=crit(u_αβ)∩ℝ X the set of real critical points. 


 
We denote by Δ (resp. by ℝΔ) the set of  (α, β,x)∈ H^0(X;ℒ^d)^2× X (resp. (α, β,x)∈ℝ H^0(X;ℒ^d)^2×ℝ X) such that α(x)=β(x)=0.
Set

    ℐ={(α,β,x)∈(H^0(X;ℒ^d)^2 × X)∖Δ| x∈crit(u_αβ) }


    (resp.  ℝℐ={(α,β,x)∈(ℝ H^0(X;ℒ^d)^2 ×ℝ X)∖ℝΔ| x∈crit(u_αβ) })


 Let ℒ be a (real) holomorphic line bundle over a smooth complex (resp. real) projective manifold X. If ℒ^d is 1-ample, that is if the 1-jet map 

    H^0(X;ℒ^d)× X→ J^1(ℒ^d)


    (s,x)↦ j^1_x(s)=(s(x),∇ s(x))
  is surjective, then ℐ (resp. ℝℐ) is a smooth manifold of complex  (resp. real) dimension 2N_d, where N_d= H^0(X;ℒ^d). 


We study the differential of the map 
    q:(H^0(X;ℒ^d)^2 × X)∖Δ→ T^*X⊗ℒ^2d
 defined by

    (α,β,x)↦(α∇β-β∇α)(x)∈ T_x^*X⊗ℒ_x^2d

defining ℐ. If we prove that 0 is a regular value, then, by Implicit Function Theorem, we have the result.
Now, for (α,β,x)∈ℐ we have 
    d_|(α,β,x)q·(α̇,β̇,ẋ)=(α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α
    +∇_ẋα∇β-∇_ẋβ∇α)(x).

For any η∈ T_x^*X⊗ℒ_x^2d we have to prove that there exists (α̇,β̇,ẋ) such that
d_|(α,β,x)q·(α̇,β̇,ẋ)=η.
As (α,β,x)∉Δ, we know that at least one between  α(x) and β(x)  is not zero.
Without loss of generality, suppose that α(x)≠ 0, then, as ℒ^d is 1-ample, there exists β̇ such that  β̇(x)=0 and   α(x)∇β̇(x)=η, then d_|(α,β,x)q·(0,β̇,0)=η.


If ℒ is ample, then, for large d, the line bundle ℒ^d is 1-ample. 
Then, for large d, ℐ (resp. ℝℐ ) is a smooth manifold, called the incidence manifold. 
The tangent space T_(α,β,x)ℐ of ℐ  at a point (α,β,x) (resp. T_(α,β,x)ℝℐ of ℝℐ) equals

    {
    (α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX|(α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)
    (x)=0}.


    (resp.  {(α̇
    ,β̇,ẋ)∈ℝ H^0(X;ℒ^d)^2× T_xℝ X|(α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}).




  * In the equation defining the tangent space there is also the term 

    (∇_ẋα∇β-∇_ẋβ∇α)
    (x).
 However, it equals zero (both in the complex and real case) because on ℐ and ℝℐ we have the condition (α∇β-β∇α)(x)=0
so that 
    (∇_ẋα∇β-∇_ẋβ∇α)(x)=((∇_ẋαβ/α-∇_ẋβ)∇α)
    (x)=0.


  * The incidence manifold comes equipped with two natural projections 


    π_H:ℐ→ H^0(X;ℒ^d)^2  and  π_X:ℐ→ X



    (resp.  π_ℝ H:ℝℐ→ℝ H^0(X;ℒ^d)^2  and  π_ℝ X:ℝ ℐ→ℝ X).






Let ℒ be an ample  holomorphic line bundle (resp. real holomorphic) over a smooth complex projective manifold X (resp. real projective).
For large d and for almost all pairs (α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2), the map 
    u_αβ:Xℂℙ^1


    x↦ [α(x):β(x)].

is a Lefschetz pencil (resp. real Lefschetz pencil).



The critical points of the projection π_H (resp. π_ℝ H) are exactly the triples (α,β,x) such that the Hessian (α∇^2β-β∇^2α)(x) is degenerate.
By Sard's theorem (π_H) has zero Lebesgue, and then Gaussian, measure.
Also, for large d, the set Γ composed by the pairs (α,β)∈ H^0(X;ℒ^d)×  H^0(X;ℒ^d)
 such that {x∈ X, α(x)=β(x)=0} is not smooth has zero Lebesgue and Gaussian measure (see for example <cit.>).
Then (Γ∪(π_H)) has zero measure and its complement is exactly the set of pairs of sections defining a Lefschetz pencil.










§ PROOF OF  THE MAIN THEOREMS

In this section we prove  Theorems <ref>, <ref> and  <ref>.  Hörmander's peak sections and the coarea formula play an important role here.



 §.§ Coarea formula

In this section we  use the incidence manifold defined in Section <ref> and the coarea formula to write the expected distribution of critical points of a (real)  Lefschetz pencil as an integral over X (resp. ℝ X).


The normal jacobian Jac_Nf of a submersion f:M→ N between two Riemannian manifolds is the determinant of the differential of the map restricted to the orthogonal complement of its kernel, that is Jac_Nf=Jac(df_| (df)^⊥). Equivalently, if df_p is the differential of f at p, then the normal jacobian is equal to √((df_pdf_p^*)),
where df_p^* is the adjoint of df_p with respect to the scalar product on T_pM and T_f(p)N.


Let X be a smooth complex (resp. real) projective manifold of dimension n and (ℒ,h) be a (real) holomorphic line bundle with positive curvature ω.


We define a Dirac measure for (real) critical points of a (real) Lefschetz pencil u_αβ associated with a pair (α,β) ∈ H^0(X;ℒ^d)^2 (resp. (α,β)∈ℝ H^0(X;ℒ^d)^2) by  

    ν_αβ=∑_x∈crit(u_αβ)δ_x      (resp.  ℝν_αβ=∑_x∈ℝcrit(u_αβ)δ_x).
 

Let φ be a continuous function on ℝ X. Then, by definition, we have


    𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈crit(u_αβ)φ(x)dμ(α,β)

where dμ is the Gaussian measure on ℝ H^0(X;ℒ^d)^2 constructed in Section <ref>. 
Finally, recall that we denote by π_ℝ H and π_ℝ X the two natural projections from ℝℐ to ℝ H^0(X;ℒ^d)^2 and ℝ X. The projection π_ℝ H is (almost everywhere) a local isomorphism and, by a slight abuse of notation, we will denote by π_ℝ H^-1 any local inverse.
 Following the notation of Section <ref>, we have 
    𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))_h.

where the measure dμ_|π_ℝ H(π^-1_ℝ X(x)) is the following: first we restrict the scalar product ⟨·,·⟩_L^2 on ℝ H^0(X;ℒ^d)^2 to π_ℝ H(π^-1_ℝ X(x)), which is a codimension n submanifold, then  we  consider the Riemannian measure associated with this metric, and finally we multiply it by the  factor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^d).


We denote by  π_ℝ  H^*dμ  the pull-backed measure on ℝℐ, which is well defined since π_ℝ  H is (almost everywhere) a local isomorphism. By definition of the pull-backed measure, 
the integral 
    𝔼[ℝν_αβ](φ)=∫_ℝ H^0(X;ℒ^d)^2∑_x∈crit(u_αβ)φ(x)dμ(α,β)
  which defines the expected value equals the following integral over the incidence manifold ℝℐ

    ∫_ℝℐ(π_ℝ X^*φ)(α,β,x)(π_ℝ  H^*dμ)(α,β,x).
 
We use the coarea formula (see <cit.> or <cit.>) for the map π_ℝ X and we obtain
 
    𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)∫_π^-1_ℝ X(x)1/|Jac_N(π_ℝ X)|(π_ℝ  H^*dμ)_|π^-1_ℝ X(x)_h

where the measure (π_ℝ  H^*dμ)_|π^-1_X(x) is the following: first we restrict the (singular) metric π_H^*⟨·,·⟩_L^2 on ℝℐ to π^-1_ℝ X(x), that is a codimension n submanifold, then  we  consider the Riemannian measure associated with this metric, and finally we multiply it by the  factor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^d).
Another application of coarea formula for the map π_ℝ H gives us the result.
  
 
The space π_ℝ H(π^-1_ℝ X(x)) is  formed by  pairs (α,β)∈ℝ H^0(X;ℒ^d)^2  such that x∈ℝcrit(u_αβ). In the next section we will identify this space with an intersection of some quadrics in the vector space ℝ H^0(X;ℒ^d)^2.

In the complex case, the same argument gives us, for any continuous function φ on X 
  
   
    𝔼[ν_αβ](φ)=∫_Xφ(x)∫_π_H(π^-1_X(x))1/|(π_H^-1)^*Jac_N(π_X)|dμ_|π_H(π^-1_X(x))_h.


  
 

 §.§ Computation of the normal jacobian

  In this section we compute the normal jacobian that appears in (<ref>) and (<ref>). We follow the notations of Sections <ref>, <ref> and <ref>.  The main result of this section is the following proposition:
   Following the notation of Sections <ref> and <ref>, under the hypothesis of Theorem <ref>, we have:
 
    𝔼[ℝν_αβ](φ)=
    ∫_x∈ℝ Xφ(x)R_d(x)_h,
 
where 
    R_d(x)=√(π d)^n(∫_Q|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))dμ_Q +O(1/√(d)))

and Q⊂ℝ^2(n+1)+n(n+1) is the product of the intersection of quadrics

    Q̃={(a_0,b_0,…,a_n,…,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0  ∀ i=1,…,n}

with the vector space ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n and
 
    dμ_Q=
    e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q

where _Q is the Riemannian volume form of  Q.


The remaining part of this section is devoted to the proof of Proposition <ref>. Our main tool will be the peak sections defined in Section  <ref>.
 
 We fix a point x∈  X (resp. x∈ℝ X) and we want to compute the integral 
    ∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))

that appears in (<ref>). 
We recall that the tangent space of ℐ (resp. ℝℐ) at (α,β,x) is 

    {(α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX|(α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}



    (resp.  {(α̇,β̇,ẋ)∈ℝ H^0(X;ℒ^d)^2× T_xℝ X|
    (α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)=0}.)

We remark that dπ_H|(α,β,x) is (almost everywhere) an isometry, because on ℐ we put the (singular) metric π_H^*⟨· ,·⟩_L^2. For any x∈ X (resp. x∈ℝ X) we will compute  the normal Jacobian (π_H^-1)^*Jac_N(π_X) (resp. (π_ℝ H ^-1)^*Jac_N(π_ℝ X)) at a point (α,β)∈π_H(π_X^-1(x)) by using the following two linear maps:
  
  
    A_αβ:H^0(X;ℒ^d)× H^0(X;ℒ^d)→ T_x^*X⊗ℒ_x^2d

  
  
    (resp.   A_αβ:ℝ H^0(X;ℒ^d)×ℝ H^0(X;ℒ^d)→ℝ(T^*X⊗ℒ^2d)_x)
  
and 
 
    B_αβ:T_x X→ T_x^*X⊗ℒ_x^2d

  
    (resp.   B_αβ:T_xℝ X→ℝ(T^*X⊗ℒ^2d)_x)

 defined by 
    A_αβ(α̇,β̇)=
       (α̇∇β-β∇α̇
    +α∇β̇-β̇∇α)(x)
 and 
    B_αβ(ẋ)=
    (α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)(x)

On T_x^*X⊗ℒ_x^2d (resp. ℝ(T^*X⊗ℒ^2d)_x) we have the Hermitian (resp. scalar) product induced by h.



   Following the notation of Sections <ref> and <ref>,  for any x∈ X (resp. x∈ℝ X) and any (α,β)∈π_ H(π^-1_X(x)) (resp. π_ℝ H(π^-1_ℝ X(x))), we have 
    ((π_H^-1)^*Jac_Nπ_X)(α,β)=Jac_N(A_αβ)/Jac(B_αβ),
 where A_αβ and B_αβ are the linear maps defined in (<ref>) and (<ref>).
  
  
 Recall that a vector (α̇,β̇,ẋ)∈ H^0(X;ℒ^d)^2× T_xX is in the tangent space of ℐ at (α,β,x) if and only if 

    (α̇∇β-β∇α̇
    +α∇β̇-β̇∇α
    +α∇^2_(ẋ,.)β-β∇^2_(ẋ,.)α)
    (x)=0.

In particular, the vector ẋ is uniquely determined by 
    ẋ=-(α∇^2β-β∇^2α)(x)^-1∘(α̇∇β-β∇α̇
    +α∇β̇-β̇∇α)(x).

The map ((π_H^-1)^*dπ_X)(α,β) sends (α̇,β̇) to ẋ and then, by (<ref>) and by the definition of A_αβ and B_αβ, we have  
    ((π_H^-1)^*dπ_X)(α,β)=-B_αβ^-1∘ A_αβ.
 
Passing to the normal Jacobian and using that π_H is a local isometry, we get ((π_H^-1)^*Jac_Nπ_X)(α,β)=Jac(B_αβ)^-1Jac_N(A_αβ).

Fix  real holomorphic coordinates (x_1,…,x_n) in a neighborhood of a point x∈ℝ X such that (∂/∂ x_1,…,∂/∂ x_n) is an orthonormal basis of T_xX (resp. T_xℝ X). We want to compute the integral 
    ∫_π_ℝ H(π^-1_ℝ X(x))1/|(π_ℝ H^-1)^*Jac_N(π_ℝ X)|dμ_|π_ℝ H(π^-1_ℝ X(x))

that appears in (<ref>). 

For any  (α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) we have

    α=∑_i=0^n a_iσ_i+∑_1⩽ k⩽ l⩽ na_klσ_kl+τ


    β=∑_i=0^n b_iσ_i+∑_1⩽ k⩽ l⩽ nb_klσ_kl+τ'

where τ, τ'∈ J^2_x and σ_i,σ_kl are the peak section of Lemma <ref>.

We remark that (α,β)∈π_H(π^-1_X(x)) if and 
only if a_0b_i-a_ib_0=0 ∀ i=1,…,n, and also that the definition of Jac_N(π_X) involves only the 2-jets of sections. With this remark in mind we define the following spaces:


  * K_2≑( J^2_x× J^2_x)⊂ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2);

  * H_2≑ Vect{(σ_i,0),(σ_kl,0),(0,σ_i),(0,σ_kl)}⊂ H^0(X:ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) for i=0,…,n and 1≤ l≤ k≤ n;

  * Q=H_2∩π_H(π^-1_X(x)).

We see Q as the product of the intersection of quadrics:

    Q̃={(a_0,b_0,…,a_n,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0  ∀ i=1,…,n}

with the vector space ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n.

Let π_2:K_2^⊥→ H_2 be the orthogonal projection. A consequence of Proposition <ref> is that, for large d, the map π_2 is invertible. 



 Following the notation of Section <ref> and <ref>, let A_αβ and B_αβ be the linear applications defined in (<ref>) and (<ref>). Then, in the complex case, under the hypothesis of Theorem <ref>, 

    (π_2^-1)_*Jac_N(A_αβ)=(πδ^2_ℒd^2n+1((a_ia̅_j+b_ib̅_j)E_ij
    +(|a_0|^2+ |b_0|^2)Id+O(1/√(d)))_ij)


    (π_2^-1)_*Jac(B_αβ)=|(πδ_ℒ√(d)^2n+2((a_0b_ij-b_0a_ij)Ẽ_ij
    +O(1/√(d)))_ij)|^2

and, in the real case, under the hypothesis of the Theorem <ref>,


    (π_2^-1)_*Jac_N(A_αβ)=
    √((πδ_ℒ^2d^2n+1((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id+O(1/√(d)))_ij))


    (π_2^-1)_*Jac(B_αβ)=(πδ_ℒ√(d)^2n+2((a_0b_ij-b_0a_ij)Ẽ_ij
    +O(1/√(d)))_ij)


where Ẽ_ij for 1⩽ i⩽ j⩽ n and E_ij for i,j=1,…,n are the matrices defined in Definition <ref>. 


Let e be a local trivialization of ℒ at x as in Section <ref> and let  (σ_i)_i=0,…,n, 
  (σ_kl)_1⩽ k⩽ l⩽ n be as in Lemma <ref> (resp. Lemma <ref>).
For any (α,β) ∈ H^0(X;ℒ^d)^2 (resp. ℝ H^0(X;ℒ^d)^2) we have 

    α=∑_i=0^n a_iσ_i+∑_1⩽ k⩽ l⩽ na_klσ_kl+τ


    β=∑_i=0^n b_iσ_i+∑_1⩽ k⩽ l⩽ nb_klσ_kl+τ'

where τ, τ'∈ J^2_x. In particular, we have

    α(x)=a_0σ_0(x),    β(x)=b_0σ_0(x),
 
    ∇α(x)=∑_i=0^na_i∇σ_i(x),    ∇β(x)=∑_i=0^nb_i∇σ_i(x),
 
    ∇^2α(x)=∑_i=0^na_i∇^2σ_i(x)+∑_k,l
    a_kl∇^2σ_kl(x),  ∇^2β(x)=∑_i=0^nb_i∇^2σ_i(x)+∑_k,l
    b_kl∇^2σ_kl(x).

As basis for T_xX and T^*_xX⊗ℒ^2d_x (resp.  T_xℝ X and ℝ(T^*_xX⊗ℒ^2d_x)) we choose (∂/∂ x_1,…,∂/∂ x_n) and (dx_1⊗ e^2d,…,dx_n⊗ e^2d) respectively.
We choose (σ_i,0) and  (0,σ_i), i=0,…,n, as
a basis of a complement of J_x^1× J^1_x. 
Thanks to Lemma <ref>, this basis is asymptotically orthonormal for the L^2-Hermitian product ⟨·,·⟩_L^2. By definition it is an orthonormal basis for the scalar product (π_2^-1)_*⟨·,·⟩_L^2 restricted to H_2 . 
Then we obtain, using Lemma <ref>,

    ⟨ A_αβ(σ_0,0),dx_j⊗ e^2d⟩
    =b_j√(π)δ_ℒ√(d)^2n+1+O(√(d)^2n);


    ⟨ A_αβ(σ_i,0),dx_j⊗ e^2d⟩=
    -b_0√(π)δ_ℒ√(d)^2n+1δ_ij+O(√(d)^2n)   for   i=1,…,n;


    ⟨ A_αβ(0,σ_0),dx_j⊗ e^2d⟩=
    -a_j√(π)δ_ℒ√(d)^2n+1+O(√(d)^2n);


    ⟨ A_αβ(0,σ_i),dx_j⊗ e^2d⟩=
    a_0√(π)δ_ℒ√(d)^2n+1δ_ij+O(√(d)^2n)
      for   i=1,…,n;


    ⟨ B_αβ(∂/∂ x_i),dx_j⊗ e^2d⟩
    =(a_0b_ij-b_0a_ij)πδ_ℒ√(d)^2n+2+O(√(d)^2n+1)   for   i≠ j;
 

    ⟨ B_αβ(∂/∂ x_k),dx_k⊗ e^2d⟩
    =√(2)(a_0b_kk-b_0a_kk)πδ_ℒ√(d)^2n+2+O(√(d)^2n+1).

where the Hermitian (resp. scalar) product ⟨·,·⟩  on T^*_xX⊗ℒ^2d_x  (resp. ℝ(T^*_xX⊗ℒ^2d_x)) is  induced by the Hermitian metric h on ℒ.
 
What we have just computed are the coefficients of the matrices of A_αβ and B_αβ with respect to our choice of basis and with respect to the scalar product (π_2^-1)_*⟨·,·⟩_L^2. We recall that B_αβ is a square matrix and that Jac_N(A_αβ)=√(Jac(A_αβA_αβ^*)).
 More precisely, as d→∞ , A_αβ is equivalent to the following matrix: 


    √(π)δ_ℒ√(d)^2n+1[  b_1 -b_0    0    …    0 -a_1  a_0    0    …    0;  b_2    0 -b_0    …    0 -a_2    0  a_0    …    0;    …    …    …    …    …    …    …    …    …    …;  b_n    0    0    … -b_0 -a_n    0    0    …  a_0 ]

 and B_αβ to the following one: 

    πδ_ℒ√(d)^2n+2[ √(2)(a_0b_11-b_0a_11)       a_0b_12-b_0a_12                     …       a_0b_1n-b_0a_1n;       a_0b_21-b_0a_21 √(2)(a_0b_22-b_0a_22)                     …       a_0b_2n-b_0a_2n;                     …                     …                     …                     …;       a_0b_n1-b_0a_n1       a_0b_n2-b_0a_n2                     … √(2)(a_0b_nn-b_0a_nn) ]


A direct computation shows us that A_αβA_αβ^* is the matrix 
    (πδ^2_ℒd^2n+1)
    ((a_ia̅_j+b_ib̅_j)E_ij
    +(|a_0|^2+|b_0|^2)Id+O(1/d)).
 The results follows.
 


By Proposition <ref> we have


    (π_2^-1)_*1/Jac_N(π_X)=
     (π d)^n
    (
    Jac_ℝ((a_0b_ij-b_0a_ij)Ẽ_ij)/((a_ia̅_j+b_ib̅_j)E_ij+
    (|a_0|^2+|b_0|^2)Id)
    +O(1/√(d)))




    (π_2^-1)_*1/Jac_N(π_ℝ X)=
     √(π d)^n
     (
     ((a_0b_ij-b_0a_ij)Ẽ_ij)/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))
    +O(1/√(d))).

We want to integrate this quantity over π_ℝ H(π^-1_ℝ X(x)).
We recall that the measure dμ_|π_ℝ H(π^-1_ℝ X(x)) is the following one: first we restrict the scalar product ⟨·,·⟩_L^2 on ℝ H^0(X;ℒ^d)^2 to π_ℝ H(π^-1_ℝ X(x)), that is a codimension n submanifold, then  we  consider the Riemannian measure associated with this metric, and finally we multiply it by the  factor 1/π^N_de^-α_L^2^2-β_L^2^2, where N_d= H^0(X;ℒ^dℒ^d).
Then  (<ref>) is equal to


    ∫_π_ℝ H(π^-1_ℝ X(x))|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_|π_ℝ H(π^-1_ℝ X(x))
    =∫_K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))⊕ K_2|Jac(B_αβ)|/| Jac_N(A_αβ)|dμ_|π_H(π^-1_ℝ X(x))=
    
    =∫_K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))
    =∫_Q(π_2^-1)_*|Jac(B_αβ)|/|Jac_N(A_αβ)|(π_2*dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))).

By Proposition <ref>, the pushforward measure (π_2)_*(μ_| K_2^⊥) on H_2 coincides with the Gaussian measure associated with the orthonormal basis {(σ_i,0),(σ_kl,0),(0,σ_i),(0,σ_kl)}_1≤ i≤ n 
 1≤ k≤ l≤ n up to a O(1/√(d)) term. As a consequence we have that (π_2*dμ_| K_2^⊥∩π_ℝ H(π^-1_ℝ X(x))) is equal to 
    dμ_Q=e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q
 up to a O(1/√(d)) term, where _Q is the Riemannian volume form of  Q.
We then have that (<ref>) is equal to 


    ∫_Q(π_2^-1)_*|Jac(B_αβ)|/|Jac_N(A_αβ)|dμ_Q+O(1/√(d))=
    
    =∫_[ a_i,b_i,a_ij,b_ij;   a_0b_i-b_0a_i=0 ]√(π d)^n
     (|((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))dμ_Q +O(1/√(d))).

Putting (<ref>) in (<ref>) and using Proposition <ref>, we obtain Proposition <ref>.

 §.§ Computation of the universal constant

The purpose of this section is the explicit computation of the function 
R_d(x) that appears in Proposition <ref>. We use the notation of Section <ref>.

To understand R_d(x), we have to compute 

    √(π d)^n∫_Q| ((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q.

The main result of this section is the following computation:

 Let Q be as in Proposition <ref>. Then 
    √(π d)^n∫_Q| ((a_0b_ij-b_0a_ij)Ẽ_ij)|/√(det((a_ia_j+b_ib_j)E_ij+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q

 is equal to 
    {[ n!!/(n-1)!!e_ℝ(n)π/2√(d)^n                if n is odd;    n!!/(n-1)!!e_ℝ(n)√(d)^n              if n is even. ].

where Ẽ_ij and E_ij are the matrices of Definition <ref> and e_ℝ(n) is the expected value of the determinant of (the absolute value of) the real symmetric matrices.

We recall that Q⊂ℝ^2(n+1)+n(n+1) is the product of the intersection of quadrics:

    Q̃={(a_0,b_0,…,a_n,…,b_n)∈ℝ^2(n+1)| a_0b_i-a_ib_0=0  ∀ i=1,…,n}

with 
 ℝ^n(n+1) of coordinates a_ij and b_ij for 1≤ i≤ j≤ n.
We consider the parametrization 
ψ:ℝ^(n+2)→Q̃ defined by 
    ψ(a,b,t_1,…,t_n)=(a,b,at_1,bt_1,…,at_n,bt_n).

 We have Jac (ψ)=√(1+∑_it_i^2)√((a^2+b^2))^n.


A computation gives us

    JacψJacψ^t=[ 1+∑_i=1^nt_i^2              0           t_1a           t_2a              …              …           t_na;              0 1+∑_i=1^nt_i^2           t_1b           t_2b              …              …           t_nb;           t_1a           t_1b        a^2+b^2              0              …              …              0;           t_2a           t_2b              0        a^2+b^2              …              …              0;              …              …              …              …              …              …              …;              …              …              …              …              …              …              …;           t_na           t_nb              0              0              …              …        a^2+b^2 ]

We develop the last line and we obtain

    (a^2+b^2)[ 1+∑_i=1^nt_i^2              0           t_1a           t_2a              …              …         t_n-1a;              0 1+∑_i=1^nt_i^2           t_1b           t_2b              …              …         t_n-1b;           t_1a           t_1b        a^2+b^2              0              …              …              0;           t_2a           t_2b              0        a^2+b^2              …              …              0;              …              …              …              …              …              …              …;              …              …              …              …              …              …              …;         t_n-1a         t_n-1b              0              0              …              …        a^2+b^2 ]


    +(-1)^nt_nb·[ 1+∑_i=1^nt_i^2           t_1a           t_2a              …              …           t_na;              0           t_1b           t_2b              …              …           t_nb;           t_1a        a^2+b^2              0              …              …              0;           t_2a              0        a^2+b^2              …              …              0;              …              …              …              …              …              …;              …              …              …              …              …              0;         t_n-1a              0              0              …        a^2+b^2              0 ]


    +(-1)^n+1t_na·[              0           t_1a           t_2a              …              …           t_na; 1+∑_i=1^nt_i^2           t_1b           t_2b              …              …           t_nb;           t_1b        a^2+b^2              0              …              …              0;           t_2b              0        a^2+b^2              …              …              0;              …              …              …              …              …              …;              …              …              …              …              …              0;         t_n-1b              0              0              …        a^2+b^2              0 ]

 For the second matrix we have:


    [ 1+∑_i=1^nt_i^2           t_1a           t_2a              …              …           t_na;              0           t_1b           t_2b              …              …           t_nb;           t_1a        a^2+b^2              0              …              …              0;           t_2a              0        a^2+b^2              …              …              0;              …              …              …              …              …              …;              …              …              …              …              …              0;         t_n-1a              0              0              …        a^2+b^2              0 ]



    =(1+∑_i=1^nt_i^2)[    t_1b    t_2b       …       …       …    t_nb; a^2+b^2       0       …       …       0       0;       0 a^2+b^2       …       …       0       0;       …       …       …       …       …       …;       …       …       … a^2+b^2       0       0;       0       …       …       0 a^2+b^2       0 ]


    =(-1)^n+1(1+∑_i=1^nt_i^2)t_nb(a^2+b^2)^n-1.

where the first equality is obtained by developping the first column and remarking that, in the development, each time we clear the i-th line, the (i-1)-th column and the last column are linearly equivalent.
Similarly, 

    [              0           t_1a           t_2a              …              …           t_na; 1+∑_i=1^nt_i^2           t_1b           t_2b              …              …           t_nb;           t_1b        a^2+b^2              0              …              …              0;           t_2b              0        a^2+b^2              …              …              0;              …              …              …              …              …              …;              …              …              …              …              …              0;         t_n-1b              0              0              …        a^2+b^2              0 ]


    =(-1)^n(1+∑_i=1^nt_i^2)t_na(a^2+b^2)^n-1.

Then we have

    JacψJacψ^t=(a^2+b^2)[ 1+∑_i=1^nt_i^2              0           t_1a           t_2a              …              …         t_n-1a;              0 1+∑_i=1^nt_i^2           t_1b           t_2b              …              …         t_n-1b;           t_1a           t_1b        a^2+b^2              0              …              …              0;           t_2a           t_2b              0        a^2+b^2              …              …              0;              …              …              …              …              …              …              …;              …              …              …              …              …              …              …;         t_n-1a         t_n-1b              0              0              …              …        a^2+b^2 ]


    -(1+∑_i=1^nt_i^2)t_n^2(a^2+b^2)^n.

Continuing to develop in the same way, we obtain by induction

    JacψJacψ^t=(1+∑_i=1^nt_i^2)^2(a^2+b^2)^n-(1+∑_i=1^nt_i^2)
    ∑_i=1^nt_i^2(a^2+b^2)^n=(1+∑_i=1^nt_i^2)(a^2+b^2)^n.

Passing to the square root we obtain the result.

 In the following we will not write the symbols Ẽ_ij and E_ij defined in Definition <ref> in order to simplify the notation. 

After this change of variables, we have:


    √(π d)^n∫_Q|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q=


    =√(π d)^n∫_a,b,a_ij,b_ij,t_i∈ℝ|(ab_ij-ba_ij)|/√(((a^2+b^2)((t_it_j)+Id)))×


    ×e^-(1+∑_it_i^2)(a^2+b^2)-∑_i,j(a_ij^2+ b_ij^2)/π^n+1+n(n+1)/2√((1+∑_it_i^2))√((a^2+b^2))^ndadbda_ijdb_ijdt_i

Now ((a^2+b^2)((t_it_j)_ij+Id))=(1+∑_it_i^2)(a^2+b^2)^n so we obtain



    √(π d)^n∫_a,b,a_ij,b_ij,t_i∈ℝ|((ab_ij-ba_ij)_ij)|
    e^-(1+∑_it_i^2)(a^2+b^2)-∑_i,j(a_ij^2+ b_ij^2)/π^n+1+n(n+1)/2dadbda_ijdb_ijdt_i.

It is more practical to see (a,b) as a complex number c∈ℂ and
also (a_ij,b_ij) as e_ij∈ℂ. With a slight abuse of notation, we denote dc and de_ij instead of  -1/2idcdc̅ and -1/2ide_ijde̅_ij. Then we have



    √(π d)^n∫_c∈ℂ,e_ij∈ℂ,t_i∈ℝ|((c̅e_ij))|
    e^-(1+∑_it_i^2)| c |^2-∑_i,j|e_ij|^2/π^n+1+n(n+1)/2dcde_ijdt_i.

Now, set c̃=(√(1+∑_it_i^2))c and then c̃=re^iϑ. We obtain


    √(π d)^n∫_c̃∈ℂ,e_ij∈ℂ|((c̅̃̅e_ij))|e^-|c̃|^2-∑_i,j|e_ij|^2/π^1+n(n+1)/2dc̃de_ij·∫_t_i∈ℝ1/π^n√(1+∑_it_i^2)^n+1dt_i=


    √(π d)^n∫_ϑ∈ (0,2π],e_ij∈ℂ|((e^-iϑe_ij))|e^-∑_i,j| e_ij|^2/π^n(n+1)/2de_ijdϑ·∫_r=0^+∞r^n+1e^-r^2/πdr 
    ·∫_t_i∈ℝ1/π^n√(1+∑_it_i^2)^n+1dt_i.

Then, we  have to compute the three integrals appearing in the last equation.


For the first term, we have

    ∫_ϑ∈ (0,2π],e_ij∈ℂ|((e^-iϑc_ij))|e^-∑_i,j |e_ij|^2/π^n(n+1)/2de_ijdϑ=


    =2π∫_e_ij∈ℂ|( e_ij)|e^-∑_i,j|e_ij|^2/π^n(n+1)/2de_ij=


    =2π∫_b_ij∈ℝ|(b_ij)|e^-∑_i,jb_ij^2/√(π)^n(n+1)/2db_ij=2π e_ℝ(n).

Here, e_ℝ(n)=∫_B∈(n,ℝ)| B| dμ_ℝ(B). For the explicit value of e_ℝ(n), see <cit.>.


For the second term, we consider the change of variable r^2=ρ and we obtain  
    ∫_r=0^+∞r^n+1e^-r^2/πdr=
    1/2∫_ρ=0^+∞ρ^n/2e^-ρ/πdρ=Γ(n/2+1)/2π

where Γ is the Gamma function.


For the third term we use spherical coordinates and we obtain

    ∫_t_i∈ℝ1/√(1+∑_it_i^2)^n+1dt_i=Vol(S^n-1)∫_t=0^+∞t^n-1/√((1+t^2))^n+1dt.

where 
    Vol(S^n-1)=2π^n/2/Γ(n/2)
 is the volume of the (n-1)-dimensional sphere.
For 

    ∫_t=0^+∞t^n-1/(√(1+t^2))^n+1dt=
    ∫_t=0^+∞√(t^2/(1+t^2))^n-11/1+t^2dt

we make the change t^2/1+t^2=1-u^2 and we obtain

    ∫_0^1 √(1-u^2)^n-2du.
 
Finally, set u=sin(θ) and we have


    ∫_0^π/2cos^n-1(θ) dθ.

The formula 
    ∫cos^n-1(θ)du=sin(θ)cos^n-2(θ)/n-1+n-2/n-1∫cos^n-3(θ)du

tells us that ∫_0^π/2cos^n-1(θ) dθ is equal to 

    (n-2)!!/(n-1)!!

if n is even and it is equal to 


    (n-2)!!/(n-1)!!π/2

if n is odd.


Putting together all the values of these three integrals and using that 
    Γ(n/2+1)/Γ(n/2)=n/2,
 we obtain Proposition <ref>.





 §.§ End of the proofs  of Theorems <ref>, <ref> and  <ref>

We use the notations of Sections <ref>, <ref> and <ref>.
By Proposition <ref>, we have 
 
    𝔼[ℝν_αβ](φ)=∫_ℝ Xφ(x)(∫_Q|Jac_N(A_αβ)|/| Jac(B_αβ)|dμ(a_i,b_i,a_kl,b_kl)+O(1/d))_h=
    
    =∫_ℝ Xφ(x)(∫_[ a_i,b_i,a_ij,b_ij;   a_0b_i-b_0a_i=0 ]√(π d)^n
     (
     (a_0b_ij-b_0a_ij)/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))
    +O(1/√(d)))dμ_Q )_h
 
where  dμ_Q=
e^-∑_ia_i^2-∑_i
b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q. By Proposition <ref>  we have that the inner term of the last equation

    √(π d)^n∫_[ a_i,b_i,a_ij,b_ij;   a_0b_i-b_0a_i=0 ]|(a_0b_ij-b_0a_ij)|/√(((a_ia_j+b_ib_j)+(a_0^2+b_0^2)Id))e^-∑_ia_i^2-∑_i
    b_i^2-∑_i,ja_ij^2-∑_i,jb_ij^2/π^n+1+n(n+1)/2_Q

 is equal to 
    {[ n!!/(n-1)!!e_ℝ(n)π/2√(d)^n                if n is odd;    n!!/(n-1)!!e_ℝ(n)√(d)^n              if n is even. ].

Theorem <ref> is then obtained by dividing Equation (<ref>) by √(d)^n and then by passing to the limit.  

Theorem <ref> is  Theorem <ref> for φ=1.

The proof of Theorem <ref> follows the lines the proof of Theorem <ref>. For the computation of the universal constant in this case, we  put φ=1 and we use Proposition <ref>.




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