EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

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doi:10.1017/S1474748014000115 c Cambridge University Press 2014

EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

DAMIEN GAYET^1,2 AND JEAN-YVES WELSCHINGER³

1Univ. Grenoble Alpes, IF, F-38000 Grenoble, France ([email protected])

2CNRS, IF, F-38000 Grenoble, France

3Universit´e de Lyon, CNRS UMR 5208, Universit´e Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France

([email protected])

(Received 19July2013; revised 9April2014; accepted9 April2014; first published online 12 May 2014)

Abstract LetX be a smooth complex projective manifold of dimensionnequipped with an ample line bundle Land a rankk holomorphic vector bundleE. We assume that16k6n, that X,E and Lare defined over the reals and denote byRX the real locus ofX. Then, we estimate from above and below the expected Betti numbers of the vanishing loci inRXof holomorphic real sections ofE⊗L^d, whered is a large enough integer. Moreover, given any closed connected codimensionksubmanifoldΣofRⁿwith trivial normal bundle, we prove that a real section ofE⊗L^d has a positive probability, independent of d, of containing around√

dⁿconnected components diffeomorphic toΣin its vanishing locus.

Keywords: real projective manifold; ample line bundle; random polynomial; Betti numbers 2010Mathematics subject classification: 14P25; 32Q15; 60D05

1. Introduction

Let X be a smooth complex projective manifold of positive dimension n equipped with an ample line bundle L and let E be a holomorphic vector bundle of rank k over X. From the vanishing theorem of Kodaira and Serre, we know that the dimension Nd

of the complex vector space H⁰(X,E⊗L^d) of global holomorphic sections of E⊗L^d grows as a polynomial of degree n in d. We will assume throughout this paper that 16k6n and that X, E and L are defined over the reals. We denote by RX the real locus of X and byRH⁰(X,E⊗L^d)the real vector space of real holomorphic sections of E⊗L^d; see (5). Its dimension equalsN_d. The discriminant locusR1_d ⊂RH⁰(X,E⊗L^d) of sections which do not vanish transversally is a codimension 1 submanifold ford large enough, and for everyσ in its complement, the real vanishing locusRCσ ofσ is a smooth codimensionksubmanifold ofRX. The topology ofRCσ drastically depends on the choice ofσ ∈RH⁰(X,E⊗L^d)\R1_d. Whenn=k=1, X =CP¹, L =O_C_P1(1)and E=O_CP1

for example,σ is a real polynomial of degree d in one variable and RCσ the set of its real roots.

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The spaceRH⁰(X,E⊗L^d)inherits classical probability measures. Indeed, leth_E be a Hermitian metric onE andhL be a Hermitian metric of positive curvature onL, bothhE

andh_L being real, that is invariant under theZ/2Z-Galois action ofEand L. We denote byh_E,d=h_E⊗h^d_Lthe induced metric onE⊗L^d. Then, the vector spaceRH⁰(X,E⊗L^d) becomes Euclidean, with theL²-scalar product defined by

∀σ, τ ∈RH⁰(X,E⊗L^d), hσ, τi = Z

XhE,d(σ, τ )dx,

wheredxdenotes any chosen volume form onX (our results being asymptotic ind, they turn out not to depend on the choice of dx). It thus inherits a Gaussian probability measureµ_R whose density atσ ∈RH⁰(X,E⊗L^d)with respect to the Lebesgue measure is(1/√

π^N^d)e^−k^σ^k².

What is the typical topology of RCσ for σ ∈RH⁰(X,E⊗L^d)chosen at random for dµ_R? We do not know, but can estimate its average Betti numbers. To formulate our results, let us denote, for every i ∈ {0, . . . ,n−k}, by bi(RCσ,R)=dimHi(RCσ,R)the ith Betti number ofRCσ and by

E(b_i)= Z

RH⁰(X,E⊗L^d)\R1_d

b_i(RCσ,R)dµ_R(σ )

its expected value.

1.1. Upper estimates

As in [14], for every i ∈ {0, . . . ,n−k}, we denote by Sym_R(i,n−k−i) the open cone of real symmetric matrices of sizen−k and signature(i,n−k−i), byµ_R the classical Gaussian measure on the space of real symmetric matrices and by e_R(i,n−k−i) the numbers

e_R(i,n−k−i)= Z

Sym_R(i,n−k−i)|detA|dµ_R(A); (1) see §3.1. We then denote by Vol_h_L(RX)the volume of RX for the Riemannian metric induced by the K¨ahler metric g_h_L defined by the curvature form ofh_L; see (3) and (4).

Theorem 1.1.1. Let X be a smooth real projective manifold of dimension n, (L,h_L) be a real holomorphic Hermitian line bundle of positive curvature over X and (E,h_E)be a rankkreal holomorphic Hermitian vector bundle, with16k6n,k6=n. Then, for every 06i 6n−k,

lim sup

d→∞

√1

dⁿE(bi)6 n−1

k−1

e_R(i,n−k−i) Vol_h_L(RX) Vol_{F S}(RP^k). Moreover, whenk=n,(1/√

dⁿ)E(b₀)converges toVol_h_L(RX)/Vol_{F S}(RPⁿ)asd grows to infinity.

In fact, the right hand side of the inequality given by Theorem 1.1.1 also involves the determinant of random matrices of size k−1 and the volume of the Grassmann manifold of (k−1) linear subspaces of Rⁿ⁻¹ (see Theorem 3.1.2), but these can be

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computed explicitly. Note that when E is the trivial line bundle, Theorem1.1.1reduces to Theorem 1.1 of [14].

Theorem 1.1.1 relies on Theorem 3.1.3, which establishes the asymptotic equidistribution of clouds of critical points; see§3.1. We obtain a similar result in a complex projective setting, for critical points of Lefschetz pencils; see Theorem3.5.1.

1.2. Lower estimates and topology

Let Σ be a closed submanifold of codimension k of Rⁿ, 16k6n, which we do not assume to be connected. For everyσ ∈RH⁰(X,E⊗L^d)\R1_d, we denote byNΣ(σ )the maximal number of disjoint open subsets ofRX having the property that each such open subsetU⁰contains a codimensionk submanifoldΣ⁰ such thatΣ⁰⊂RCσ and(U⁰, Σ⁰)is diffeomorphic to(Rⁿ, Σ ). We then set

E(NΣ)=Z

NΣ(σ )dµ_R(σ ) (2) and we associate with Σ, in fact with its isotopy class in Rⁿ, a constant cΣ which is positive if and only ifΣ has trivial normal bundle inRⁿ; see (14) for its definition and Lemma 2.2.3. The latter measures`a la Donaldson the amount of transversality that a polynomial mapRⁿ→R^k vanishing along a submanifold isotopic toΣ may have.

Theorem 1.2.1. Let X be a smooth real projective manifold of dimension n, (L,h_L) be a real holomorphic Hermitian line bundle of positive curvature over X and (E,hE) be a rank k real holomorphic Hermitian vector bundle, with 16k6n. Let Σ be a closed submanifold of codimension k of Rⁿ with trivial normal bundle, which does not need to be connected. Then,

lim inf

d→∞

√1

dⁿE(NΣ)>cΣVolhL(RX).

In particular, when Σ is connected, Theorem 1.2.1 bounds from below the expected number of connected components diffeomorphic to Σ in the real vanishing locus of a random section σ ∈RH⁰(X,E⊗L^d). The constant cΣ does not depend on the choice of the triple (X, (L,h_L), (E,h_E)); it only depends on Σ. When k=1 and E =O_X, Theorem1.2.1coincides with Theorem 1.2 of [16]. ComputingcΣ for explicit submanifolds Σ yields the following lower bounds for the Betti numbers.

Corollary 1.2.2. Under the hypotheses of Theorem 1.2.1, for everyi ∈ {0, . . . ,n−k}, lim inf

d→∞

√1

dⁿE(b_i)>exp(−e⁸⁴⁺⁶ⁿ)Vol_h_L(RX).

1.3. Some related results

The case where X =CP¹, E=O_CP1 and L =O_CP1(1)was first considered by M. Kac in [18] for a different measure. In this case and with our measure, Kostlan [19] and Shub and Smale [34] gave an exact formula for the mean number of real roots of a polynomial, as well as the mean number of intersection points of n hypersurfaces in RPⁿ. Still in

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RPⁿ, Podkorytov [27] computed the mean Euler characteristics of random algebraic hypersufaces, and B¨urgisser [4] extended this result to complete intersections. In [13], we proved the exponential rarefaction of real curves with a maximal number of components in real algebraic surfaces. In [14, 15], we bounded from above the mean Betti numbers of random real hypersurfaces in real projective manifolds and in [16], we gave a lower bound for them.

A similar probabilistic study of complex projective manifolds has been performed by Shiffman and Zelditch (see [2, 30, 33] for example, and also [3, 36]). In particular, the asymptotic equidistribution of critical points of random sections over a fixed projective manifold has been studied in [8,9,22], and also [1,5,11], while we studied critical points of the restriction of a fixed Morse function on random real hypersurfaces; see [14,15].

A similar question concerns the mean number of components of the vanishing locus of eigenfunctions of the Laplacian. It has been studied on the round sphere by Nazarov and Sodin [25] (see also [35]), Lerario and Lundberg [20] and Sarnak and Wigman [28]. For a general Riemannian setting, Zelditch proved in [38] the equidistribution of the vanishing locus, whereas critical points of random eigenfunctions of the Laplacian were addressed by Nicolaescu in [26].

Section 2is devoted to lower estimates and the proof of Theorem 1.2.1. In this proof, the L²-estimates of H¨ormander play a crucial rˆole (see §2.3), and we follow the same approach as in [16] (see also [12] for a similar construction). Section 3 is devoted to upper estimates and the proof of Theorem1.1.1.

2. Lower estimates for the expected Betti numbers 2.1. Statement of the results

2.1.1. Framework. Let us first recall our framework. We denote by X a smooth complex projective manifold of dimension n defined over the reals, by cX :X → X the induced Galois antiholomorphic involution and byRX =Fix(c_X)the real locus ofXwhich we implicitly assume to be non-empty. We then consider an ample line bundleL overX, also defined over the reals. It comes thus equipped with an antiholomorphic involution cL :L →L which turns the bundle projection map π: L→ X into a Z/2Z-equivariant one, so that c_X◦π=π◦c_L. We equip L in addition with a real Hermitian metric h_L, being thus invariant undercL, which has a positive curvature formωlocally defined by

ω= 1

2iπ∂∂¯loghL(e,e) (3)

for any non-vanishing local holomorphic section e of L. This metric induces a K¨ahler metric

ghL =ω(. ,i. ) (4)

onX, which reduces to a Riemannian metricghL onRX. Let finallyE be a holomorphic vector bundle of rank k, 16k6n, defined over the reals and equipped with an antiholomorphic involution c_E and a real Hermitian metric h_E. For every d>0, we denote by

RH⁰(X,E⊗L^d)= {σ ∈ H⁰(X,E⊗L^d)|(c_E⊗c_Ld)◦σ =σ◦c_X} (5)

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the space of global real holomorphic sections ofE⊗L^d. It is equipped with theL²-scalar product defined by the formula

∀(σ, τ )∈RH⁰(X,E⊗L^d), hσ, τi = Z

XhE,d(σ, τ )(x)dx, (6) where h_E,d =h_E⊗h^d_L. Here, dx denotes any volume form of X. For instance, dx can be chosen to be the normalized volume form dVhL =ωⁿ/R

Xωⁿ.This L²-scalar product finally induces a Gaussian probability measure µ_R on RH⁰(X,E⊗L^d) whose density with respect to the Lebesgue one atσ ∈RH⁰(X,E⊗L^d)is written as(1/√

π^N^d)e^−k^σ^k², where Nd =dimH⁰(X,E⊗L^d). It is with respect to this probability measure that we consider random real codimensionk submanifolds (as in the works [19] and [14–16,34]).

2.1.2. The lower estimates. The aim of §2 is to prove Theorem1.2.1. In addition to Theorem 1.2.1, we also get the following Theorem 2.1.1, which is a consequence of Proposition2.4.2below.

Theorem 2.1.1. Under the hypotheses of Theorem 1.2.1, for every06 <1, lim inf

d→∞ µ_R

σ ∈RH⁰(X,E⊗L^d)| NΣ(σ )>cΣVolh_L(RX)√

dⁿ >0.

In fact, the positive lower bound given by Theorem2.1.1can be made explicit; see (30).

Let us now denote, for every 16k6n, byH_n,k the set of diffeomorphism classes of smooth closed connected codimensionksubmanifolds of Rⁿ. For everyi ∈ {0, . . . ,n−k} and every [Σ] ∈H_n,k, we denote by bi(Σ )=dimHi(Σ;R) its ith Betti number with real coefficients and bymi(Σ )itsith Morse number. This is the infimum over all Morse functions f on Σ of the number of critical points of index i of f. Then, we setc_[Σ]= sup_Σ_∈[_Σ_]cΣ and

E(m_i)= Z

m_i(RCσ)dµ_R(σ ).

Corollary 2.1.2. Let X be a smooth real projective manifold of dimension n, (L,hL) be a real holomorphic Hermitian line bundle of positive curvature over X and (E,h_E) be a rank k real holomorphic Hermitian vector bundle, with 16k6n. Then, for every i∈ {0, . . . ,n−k},

lim inf

d→∞

√1

dⁿE(bi) >

X

[Σ]∈H_n,k

c_[Σ]bi(Σ )

VolhL(RX)and likewise (7)

lim inf

d→∞

√1

dⁿE(m_i) >

X

[Σ]∈H_n,k

c_[Σ]m_i(Σ )

Vol_h_L(RX). (8)

Note that in Corollary 2.1.2, we could have chosen one representative Σ in each diffeomorphism class[Σ] ∈H_n,kand obtained the lower estimates (7), (8) with constants cΣ instead ofc_[Σ]. But it turns out that in the proof of Corollary 2.1.2we are free to choose the representative that we wish in every diffeomorphism class and that the higher cΣ is, the better the estimates (7), (8) are. This is why we introduce the constantc_[Σ], which is positive if and only if[Σ]has a representativeΣ with trivial normal bundle in Rⁿ; see (14) and Lemma 2.2.3.

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2.2. Closed affine real algebraic submanifolds

We introduce here the notion of a regular pair (see Definition 2.2.1), and the constant cΣ associated with any isotopy class of the smooth closed codimensionk submanifoldΣ ofRⁿ; see (14).

Definition 2.2.1. LetUbe a bounded open subset ofRⁿandP ∈R[x1, . . .x_n]^k,16k6n. The pair(U,P)is said to be regular if and only if

1. zero is a regular value of the restriction of P to U, 2. the vanishing locus of P in U is compact.

Hence, for every regular pair (U,P), the vanishing locus of P does not intersect the boundary ofU and it meetsU in a smooth compact codimensionksubmanifold.

In the sequel, for every integerpand every vectorv∈R^p, we denote by|v|its Euclidean norm, and for every integer p and q, and every linear map F:R^p→R^q, we denote by F^∗ the adjoint of F, defined by the property

∀v∈R^p,∀w∈R^q, hF(v), wi = hv,F^∗(w)i, and denote bykFkits operator norm, that is

kFk = sup

v∈R^p\{0}|F(v)|/|v|. We will also use the norm

kFk2=√

Tr F F^∗.

These norms satisfykFk6kFk2.Finally, ifP=(P₁, . . . ,P_k)∈R[x1, . . . ,x_n]^k, we denote bykPk_L² its L²-norm defined by

kPk²_L2 =Z

Cⁿ

|P(z)|²e⁻^π^|z|²dz=

k

X

i=1

Z

Cⁿ

|Pi(z)|²e⁻^π^|z|²dz=

k

X

i=1

kPik²_L2. (9)

Definition 2.2.2. For every regular pair (U,P) given by Definition 2.2.1, we denote by T_(U,P) the set of(δ, )∈(R^∗₊)²such that

1. there exists a compact subset K ofU satisfyinginfx∈U\K|P(x)|> δ, 2. for every y∈U,|P(y)|< δ⇒ ∀w∈R^k, |(d_|yP)^∗(w)|>|w|.

Hence, for every regular pair(U,P)given by Definition2.2.1, (δ, )belongs toT_(U,P) provided that theδ-sublevel of P does not intersect the boundary ofU while inside this δ-sublevel, and P is in a sense -far from having a critical point. This quantifies how much transversally P vanishes in a way similar to the one used by Donaldson in [7].

Then, for every regular pair (U,P), we set R(U,P)=max(1,sup_y∈U|y|), so U is contained in the ball centered at the origin and of radius R(U,P). Finally, we set

τ_(U,P)=24kρ_R_(U,P)kPk²_L2 inf

(δ,)∈T_(U,P)

1 δ²+πn

²

∈R^∗₊, (10)

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where, for every R>0,

ρ_R=inf

R⁺

g_R, (11)

g_R:s∈R^∗₊7→ (R+s)²ⁿ

s²ⁿ e^π(R⁺^s)², (12) and so

e^π^R² 6ρ_R 64ⁿe^4π^R². (13) This constant τ_(U,P) is the main ingredient in the definition of cΣ; see (14). The lower τ(U,P) is, the largercΣ is and the better the estimates given by Theorem1.2.1are. Note thatτ_(U,P) remains small wheneverδ, are not too small, that is when P vanishes quite transversally inU.

Now, letΣ be a closed submanifold of codimensionkofRⁿ, not necessarily connected.

We denote byI_Σ the set of regular pairs (U,P)given by Definition2.2.1, such that the vanishing locus of P inU contains a subset isotopic toΣ in Rⁿ.

Lemma 2.2.3. LetΣ be a closed submanifold of codimensionk>0 ofRⁿ, not necessarily connected. Then,I_Σ is non-empty if and only if the normal bundle ofΣ inRⁿ is trivial.

Proof. If (U,P)∈I_Σ, then P:Rⁿ →R^k contains in its vanishing locus a codimension k submanifold Σb which is isotopic toΣ in Rⁿ. The normal bundle of Σ in Rⁿ is thus trivial if and only if the normal bundle ofΣbinRⁿ is trivial. But the differential of P at every point ofΣbprovides an isomorphism between the normal bundle of Σbin Rⁿ and the productΣb×R^k.

Conversely, ifΣhas a trivial normal bundle inRⁿ, it has been proved by Seifert [29] (see also [24]) that there exist a polynomial map P :Rⁿ→R^k and a tubular neighborhood U ofΣ inRⁿ such that P⁻¹(0)∩U is isotopic toΣ inU. The strategy of the proof is to first find a smooth functionU →R^k in a neighborhood ofΣ which vanishes transversally along Σ and then to suitably approximate the coordinates of this function by some polynomial; see [24,29]. The pair(U,P)then belongs toI_Σ by Definition2.2.1.

We then setcΣ =0ifΣ does not have a trivial normal bundle inRⁿ and cΣ = sup

(U,P)∈IΣ

mτ_(U,P)

2ⁿVol(B(R_(U,P)))

otherwise, (14)

where Vol(B(R_(U,P))) denotes the volume of the Euclidean ball of radius R_(U,P) in Rⁿ, and where, for everyτ >0,

mτ = sup

[√τ ,+∞[

fτ, (15)

with fτ :a∈ [√

τ ,+∞[ 7→1/√π (1−(τ/a²))R_+∞

a e⁻^t²dt.For large values ofmτ, such as the ones which appear in§2.6, the estimate

cΣ >e^−2τ^(U,P) (16)

holds; compare (2.8) of [16].

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2.3. H¨ormander sections

Our key tool for proving Theorems1.1.1and1.2.1has been developed by L. H¨ormander.

We introduce in this part,§2.3, the material that we need. For every positived and every σ ∈RH⁰(X,E⊗L^d), we set

kσk²_L2(hL)= Z

Xkσk²_h_E,ddV_h_L, wheredV_h_L =ωⁿ/R

Xωⁿ; compare (6). Let us choose a field ofh_L-trivializations of L on RX given by Definition 3.2 of [16]. It provides in particular, for every x∈RX, a local holomorphic chartψ_x :(W_x,x)⊂X →(V_x,0)⊂Cⁿisometric at x, and a non-vanishing holomorphic section e of L defined over W_x such that φ= −logh_L(e,e)vanishes at x and is positive elsewhere. Moreover, there exists a positive constantα₁such that

∀y∈V_x,|φ◦ψ_x⁻¹(y)−π|y|²|6α₁|y|³. (17) Restricting Wx if necessary, we choose a holomophic trivialization (e1, . . . ,ek) of E_|Wx

which is orthonormal atx. This provides a trivialization(e₁⊗e^d, . . . ,e_k⊗e^d)ofE⊗L^d_|W_x. In this trivialization, the restriction ofσ to W_x is written as

σ =

k

X

j=1

f_σ^je_j⊗e^d (18)

for some holomorphic functions fσ^j :Wx →C. We write fσ =(f_σ¹, . . . , f_σ^k)and we set

|σ| = |fσ|, (19)

so on W_x, kσk²_h_E,d =

P_k

j=1 f_σ^ke_j

2

hEe⁻^dφ and kσ (x)k²_h_E,d = |σ (x)|² since the frames (e₁, . . . ,e_k) and e are orthonormal at the point x, and so in particular φ (x)=0. For everyz∈Wx, we define

kd_|zσk2= kd_|y(fσ◦ψ_x⁻¹)k2, (20) kd|zσk = kd|y(fσ◦ψ_x⁻¹)k, (21) and

(d_|zσ )^∗=(d_|y(fσ◦ψ_x⁻¹))^∗, (22) where y=ψ_x(z). Finally, we denote, for every small enoughr >0, by B(x,r)⊂W_x the ball centered atx and of radiusr for the flat metric ofVx pulled back byψ_x, so

B(x,r)=ψ_x⁻¹(B(0,r)). (23)

Proposition 2.3.1. Let X be a smooth real projective manifold of dimension n, (L,hL) be a real holomorphic Hermitian line bundle of positive curvature over X and (E,h_E) be a rank k real holomorphic Hermitian vector bundle, with 16k6n. We choose a field ofhL-trivializations onRX. Then, for every regular pair(U,P), every large enough integerd, everyx inRX and every local trivialization of E orthonormal atx, there exist σ_(U,P)∈RH⁰(X,E⊗L^d)and an open subset U_d of B(x,R_(U,P)/√

d)∩RX such that

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1. kσ_(U,P)k_L²_(h_L) becomes equivalent to ((kPk_L²)/√

δ_L) as d grows to infinity, where kPk_L² is defined by (9) and δ_L =R

Xωⁿ,

2. (U_d, σ_(U,P)⁻¹ (0)∩U_d)is diffeomorphic to(U,P⁻¹(0)∩U)⊂Rⁿ,

3. for every (δ, )∈T_(U,P) given by Definition 2.2.2, there exists a compact subset K_d⊂U_d such that

Uinfd\Kd|σ_(U,P)|>δ 2

√dⁿ,

while for every yinUd,

|σ_(U,P)(y)|< δ 2

√dⁿ⇒ ∀w∈R^k, |(d_|yσ_(U,P))^∗(w)|>

2

√dⁿ⁺¹|w|. (24)

Proof. We proceed as in the proof of Proposition 3.4 of [16]. Let (U,P) be a regular pair, x∈RX and d large enough. We setUd =ψ_x⁻¹((1/√

d)U)⊂B(x,R(U,P)/√ d) and Kd =ψ_x⁻¹((1/√

d)K).Letχ :Cⁿ→ [0,1]be a smooth function with compact support in B(0,R_(U,P)), which equals 1 in a neighborhood of the origin. Then, letσ be the global smooth section ofE⊗L^d defined by σ_|_X_\_W_x =0and

σ_|W_x =(χ◦ψ_x) k

X

j=1

Pj(√

dψ_x)ej⊗e^d

,

whereP =(P₁, . . . ,P_k)is now considered as a functionCⁿ→C^k. From theL²-estimates of H¨ormander (see [17] or [21]), there exists a global sectionτ ofE⊗L^dsuch that∂τ¯ = ¯∂σ andkτk_L²_(h_E,d)6k¯∂σk_L²_(h_E,d)ford large enough. This sectionτ can be chosen orthogonal to holomorphic sections and is then unique, and in particular real. Moreover, there exist positive constantsc₁ andc₂, which do not depend on x, such thatkτkL²(hE,d)6c₁e^−c²^d andsup_U_d(|τ| + kτk2)6c₂e^−c²^d; see Lemma 3.5 of [16]. We then setσ_(U,P)=√

dⁿ(σ−τ ).

It has the desired properties, as can be checked along the same lines as in the proof of Proposition 3.4 of [16] and thanks to Lemma 2.3.2.

Lemma 2.3.2. Let U be an open subset of Rⁿ, 16k6n, f :U →R^k be a function of classC¹ and(δ, )∈(R^∗₊)² be such that

1. there exists a compact subset K of U such thatinfU\K|f|> δ, 2. for every y inU,|f(y)|< δ⇒ ∀w∈R^k, |(d_|yf)^∗(w)|>|w|.

Then, for every functiong:U →R^k of classC¹such thatsup_U|g|< δ andsup_Ukdgk<

, zero is a regular value of f +g and (f+g)⁻¹(0) is compact and isotopic to f⁻¹(0) inU.

Proof. The proof is analogous to that of Lemma 3.6 of [16], sincek(dg)^∗k = kdgk. The following Lemma2.3.3 establishes the existence of peak sections for higher rank vector bundles.

Lemma 2.3.3(Compare Lemma 1.2 of [37]). Let X be a smooth real projective manifold of dimension n, (L,h_L) be a real holomorphic Hermitian line bundle of positive curvature

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over X and (E,h_E) be a rank k real holomorphic Hermitian vector bundle, with 16 k6n. Let x∈RX,(p1, . . . ,pn)∈Nⁿ,i ∈ {1, . . . ,k}and p⁰>p1+ · · · +pn. There exists d₀∈Nindependent ofxsuch that for everyd >d₀, there existsσ ∈RH⁰(X,E⊗L^d)with the property thatkσkL²(hL)=1 and if(y₁, . . . ,y_n)are local real holomorphic coordinates in the neighborhood of x and (e1, . . .ek) is a local real holomorphic trivialization of E orthonormal at x, we can assume that in a neighborhood ofx,

σ (y1, . . . ,yn)=λy₁^p¹· · ·yn^pⁿei⊗e^d(1+O(d^−2p⁰))+O(λ|y|^2p⁰), (25) where λ⁻²=R

B(x,(logd/√

d))|y₁^p¹· · ·yn^pⁿ|²ke^dk²_hd

LdVhL, with dVhL =ωⁿ/R

Xωⁿ and where e is a local trivialization of L whose potential −logh_L(e,e) reaches a local minimum atx with Hessianπ ω(.,i.).

Proof. The proof goes along the same lines as that of Lemma 1.2 of [37]. Letη be a cutoff function onRwithη=1in a neighborhood of0, and

ψ=(n+2p⁰)η dkzk²

log²d

log dkzk²

log²d

in the coordinatesz onX. Then,i∂∂ψ¯ is bounded from below by−Cω, whereC is some uniform constant independent of d and x. Let s∈C^∞(X,E⊗L^d) be the real section defined by

s=η dkzk²

log²d

y₁^p¹· · ·yn^pⁿei⊗e^d.

Then, from Theorem 5.1 of [6], for d large enough and not depending on x, there exists a real sectionu∈C^∞(X,E⊗L^d)such that∂u¯ = ¯∂s, and satisfying the H¨ormander L²-estimates

Z

Xkuk²_h_E_,de⁻^ψdVhL 6 Z

Xk¯∂sk²_h_E_,de⁻^ψdVhL.

The presence of the singular weight e⁻^ψ forces the jets of u to vanish up to order 2p⁰ at x. As in Lemma 1.2 of [37], we conclude that the real holomorphic section σ =(s−u)/ks−uk_L²_(h_E,d) satisfies the required properties.

In this first section we will only need peak sections given by Lemma 2.3.3 with P_n

i=1p_i =0, whereas in the second one we will need those given withP_n

i=1p_i 62.

Definition 2.3.4. Fori∈ {1, . . . ,k},letσ₀ⁱ be the section given by Lemma2.3.3withp⁰=3 andp₁= · · · =p_n=0. Likewise, for every j ∈ {1, . . . ,n}, letσⁱ_j be a section given by (25) with p⁰=3, pj =1 and pl =0 forl ∈ {1, . . . ,n} \ {j}. Finally, for every 16l 6m6n, letσ_lmⁱ be a section given by (25) with p⁰=3, p_j =0for every j∈ {1, . . . ,n} \ {l,m}and p_l =p_m=1 ifl6=m, while p_l =2 otherwise.

The asymptotic values of the constants λin (25) are given by Lemma2.3.5(compare Lemma 2.1 of [37]).

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Lemma 2.3.5. For every i∈ {1, . . . ,k}, the sections given by Definition2.3.4 satisfy σ₀ⁱ/p

δ_Ldⁿ ∼

d→∞ e_i⊗e^d+O(kyk⁶), (26)

∀j∈ {1, . . . ,n}, σⁱ_j/p

π δ_Ldⁿ⁺¹ ∼

d→∞ y_je_i⊗e^d+O(kyk⁶), (27)

∀l,m∈ {1, . . . ,n},l6=m, σ_lmⁱ / πp

δ_Ldⁿ⁺²

d→∞∼ y_ly_me_i⊗e^d+O(kyk⁶), (28) and∀l∈ {1, . . . ,n}, σ_llⁱ/ πp

δ_Ldⁿ⁺²

d→∞∼

√1

2y_l²e_i⊗e^d+O(kyk⁶). (29) Moreover, these sections are asymptotically orthonormal as d grows to infinity, as follows from Lemma2.3.6.

Lemma 2.3.6(compare Lemma 3.1 of [37]). For everyx∈RX, the sections(σⁱ_j)_16i_6k

06j6n

and (σ_lmⁱ )_16i6k

16l6m6n

given by Definition2.3.4have L²-norm equal to 1 and their pairwise scalar products are dominated by an O(d⁻¹)term which does not depend on x. Likewise, their scalar product with every section of RH⁰(X,E⊗L^d)of L²-norm equal to 1 and whose 2-jet at x vanishes is dominated by an O(d^−3/2)term which does not depend onx. Proof. The proof goes along the same lines as that of Lemma 3.1 of [37].

Lemma 2.3.7. Denote by v the density of dV_h_L =ωⁿ/R

Xωⁿ with respect to the volume form dx chosen in (6), such that dVh_L =v(x)dx. Then the sections given by Definition2.3.4 times√

v(x)are still asymptotically orthonormal for (6).

Proof. This is a direct consequence of Lemmas 2.3.3 and 2.3.6 and the asymptotic concentration of the support of the peak sections nearx.

Remark 2.3.8. The complex analogues of Lemmas 2.3.3, 2.3.5 and 2.3.6 hold;

compare [37].

2.4. Proof of Theorem 1.2.1

We first compute the expected localC¹-norm of sections.

Proposition 2.4.1. Let X be a smooth real projective manifold of dimensionn,(L,hL)be a real holomorphic Hermitian line bundle of positive curvature over X and(E,hE)be a rank k real holomorphic Hermitian vector bundle, with 16k6n. We equip RX with a field ofhL-trivializations; see §2.3. Then, for every positive R,

lim sup

d→∞ sup

x∈RX

1 dⁿE

sup

B(x,√^R d)

|σ|² v(x)

66kδ_Lρ_R and

lim sup

d→∞ sup

x∈RX

1 dⁿ⁺¹E

sup

B(x,√^Rd)

kdσk²₂ v(x)

66πnkδ_Lρ_R,

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where v is given by Lemma 2.3.7 and ρ_R is given by (11); see (19) and (20) for the definitions of|σ| andkdσk2.

Note that a global estimate on the sup norm of L² random holomorphic sections is given by Theorem1.1 of [32].

Proof. The proof goes along the same lines as the proof of Proposition 3.7 of [16]. We first establish from the mean value inequality that for everyx∈RX, R>0ands>0,

E

sup

B(x,√^R d)

|σ|²

6 1

Vol

B √s

d

Z

B(x,^R+s√ d )

E(|σ|²)ψ_x^∗dy

ford large enough, not depending onx. Then, for everyz∈ B(x, (R+s)/√

d)∩RX, we writeσ =P_k

i=1a_iσ₀ⁱ+τ, whereτ ∈RH⁰(X,E⊗L^d)vanishes atzand(σ₀ⁱ)_i_=1,...k are the peak sections atzgiven by Definition2.3.4. In particular, by Lemma2.3.5, at the pointz, for everyi =1, . . . ,k, kσ₀ⁱkhE,d ∼

d→∞

√δ_Ldⁿ. Moreover, since(e1, . . . ,en)is orthonormal atx,

|σ₀ⁱ(z)|² = kσ₀ⁱ(z)k²_h_E_,d(1+O(|z−x|)e^{dφ (z)} 6 δ_Ldⁿe^π(^R+s⁾²(1+o(1))

from the inequalities (17), where theo(dⁿ)term can be chosen not to depend onx∈RX. Suppose that dy=dV_h_L. Then, by Lemma2.3.6, the peak sections are asymptotically orthogonal to each other for the scalar product defined by (6), and asymptotically orthogonal to the space of sectionsτ vanishing at x. We deduce that

E(|σ (z)|²)= E

k

X

i=1

aiσ₀ⁱ

2

(1+o(1))

= k

X

i=1

|σ₀ⁱ(z)|² 1

√π Z

R

a²e⁻^a²da(1+o(1))

6 1

2kδ_Ldⁿe^π(^R+s⁾²(1+o(1)).

Whenz∈/ B(x, (R+s/√

d))∩RX, the space of real sections vanishing atzbecomes of real codimension2k in RH⁰(X,E⊗L^d). Lethθ₁ⁱ, θ₂ⁱ, i ∈ {1, . . . ,k}ibe an orthonormal basis of its orthogonal complement. From Remark2.3.8, for everyi∈ {1, . . . ,k}, j ∈ {1,2},

lim sup

d→∞

1

dⁿ|θⁱ_j(z)|²62δLe^π(R+s)², an upper bound which does not depend onz. We deduce that

E(|σ (z)|²) = Z

R^2k

k

X

i=1

(a₀₁ⁱ θ₁ⁱ(z)+a₀₂ⁱ θ₂ⁱ(z))

2

e⁻^P^kⁱ⁼¹^(a⁰¹ⁱ ⁾²⁺^(a⁰²ⁱ ⁾² 1

π^k5^k_i₌₁daⁱ₀₁daⁱ₀₂

6 2δ_Ldⁿe^π(R⁺^s)²(1+o(1))

k

X

i=1

Z

R²

(a₀₁ⁱ )²+(a₀₂ⁱ )²+2|a₀₁ⁱ ||a₀₂ⁱ | . . .

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. . . 1

πe⁻^(a⁰¹ⁱ ⁾²⁻^(aⁱ⁰²⁾²da₀₁ⁱ da₀₂ⁱ 6 6δ_Ldⁿe^π(R+s)²(1+o(1)).

We deduce the first part of Proposition2.4.1by taking the supremum overRX, choosing swhich minimizesgR_(U,P) and taking thelim supasd grows to infinity.

In general, the Bergman section at x for the L²-product (6) associated with the volume form dx is equivalent to the Bergman section σ₀ at x for dV_h times √

v(x);

see Lemma2.3.7. The same holds true for theσ_j, and the result follows on replacingδ_L withv(x)δ_L.

The proof of the second assertion goes along the same lines; see the proof of Proposition 3.7 of [16] (and [31] for similar results).

As in [16], we then compute the probability of the presence of closed affine real algebraic submanifolds, inspired by an approach of Nazarov and Sodin [25]; see also [20]. Let (U,P)be a regular pair given by Definition2.2.1andΣ=P⁻¹(0)⊂U. Then, for every x∈RX, we set B_d =B(x,R_(U,P)/√

d)∩RX (see (23)), and denote by Prob_x,Σ(E⊗L^d) the probability thatσ ∈RH⁰(X,E⊗L^d)has the property that σ⁻¹(0)∩Bd contains a closed submanifoldΣ⁰ such that the pair(Bd, Σ⁰)is diffeomorphic to(Rⁿ, Σ ). That is, Probx,Σ(E⊗L^d)=µ_R{σ ∈RH⁰(X,E⊗L^d)|(σ⁻¹(0)∩Bd)⊃Σ⁰, (Bd, Σ⁰)∼(Rⁿ, Σ )}. We then setProbΣ(E⊗L^d)=inf_x∈RXProbx,Σ(E⊗L^d).

Proposition 2.4.2. Let X be a smooth real projective manifold of dimensionn,(L,h_L)be a real holomorphic Hermitian line bundle of positive curvature over X and(E,h_E)be a rankkreal holomorphic Hermitian vector bundle, with16k6n. Let(U,P)be a regular pair given by Definition2.2.1 andΣ=P⁻¹(0)⊂U.Then,

lim inf

d→∞ ProbΣ(E⊗L^d)>mτ_(U,P); see (15).

Proof. The proof is the same as that of Proposition 3.8 of [16] and is not reproduced here.

The proof of Theorem1.2.1(resp. Corollary2.1.2) then just goes along the same lines as that of Theorem 1.2 (resp. Corollary 1.3) of [16].

2.5. Proof of Theorem 2.1.1

Let (U,P) be a regular pair given by Definition 2.2.1. For every d >0, let 3_d be a maximal subset of RX with the property that two distinct points of 3_d are at distance greater than (2R_(U,P))/√

d. The balls centered at points of 3_d and of radius R_(U,P)/√

d are disjoint, whereas those of radius (2R_(U,P))/√

d cover RX. Note that if we use the local flat metric given by a trivial hL-trivialization, then the associated lattice has asymptotically the same number of balls as 3_d as d grows to infinity, so we can suppose from now on that the balls are defined for this local metric. For every

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σ ∈RH⁰(X,E⊗L^d), denote byNΣ(3_d, σ )the number ofx∈3_dsuch that the ballB_d = B(x, (R(U,P))/√

d)∩RX contains a codimensionksubmanifoldΣ⁰withΣ⁰⊂σ⁻¹(0)and (B_d, Σ⁰) diffeomorphic to (Rⁿ, Σ ). By definition of NΣ(σ ), NΣ(3_d, σ )6NΣ(σ ) (see

§1.2), while from Proposition2.4.2, for every0< <1,

|3_d|mτ_(U,P) 6 X

x∈3_d

Probx,Σ(E⊗L^d)

6

|3_d|

X

j=1

jµ_R{σ|NΣ(3_d, σ )= j}

6 mτ_(U,P)|3_d|µ_R

σ|NΣ(3_d, σ )6mτ_(U,P)|3_d| + |3_d|µ_R

σ|NΣ(3_d, σ )>mτ_(U,P)|3_d| . We deduce that

(1−)mτ_(U,P) 6µ_R

σ|NΣ(σ )>mτ_(U,P₎|3_d| (30) and the result follows on choosing a sequence(U_p,P_p)_p∈IΣ such that

p→∞lim mτ(Up,Pp)|3_d| =cΣVol_h_L(RX)√ dⁿ; see (14).

2.6. Proof of Corollary1.2.2

In this paragraph, for every positive integer p, S^p denotes the unit sphere in R^p+1. Corollary 1.2.2is a consequence of Theorem 1.2.1 and the following Propositions2.6.1 and2.6.3.

Proposition 2.6.1. For every 16k6n,c_Sn−k >exp(−e⁵⁴⁺⁵ⁿ).

Recall the following.

Lemma 2.6.2 (Lemma 2.2 of [16]). If P =P

(i1,...,in)∈Nⁿa_i₁,...,inzⁱ₁¹· · ·zⁱ_nⁿ ∈R[z1, . . . ,z_n], then

kPk²_L2 = Z

Cⁿ

|P(z)|²e⁻^π^|z|²dz= X

(i₁,...,in)∈Nⁿ

|a_i₁_,...,i_n|²i1! · · ·in! πⁱ¹^+···+iⁿ.

Proof of Proposition2.6.1. For every n>0, we set P_k(x₁, . . . ,x_n)=P_n

j=kx²_j−1. For every x∈Rⁿ andδ >0,

|Pk(x)|< δ⇔1−δ <

n

X

i=k

x_i²<1+δ⇒ kd_|xPkk²₂=4

n

X

i=k

x_i²>4(1−δ).

Moreover from Lemma2.6.2,

kPkk²_L2 =1+2(n−k+1)

π² 6n−k+2.

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Now set P_S=(P₁, . . . ,P_k)with P_j(x)=x_j for16 j 6k−1, so kP_Sk²_L₂ 6(k−1)/π+(n−k+2)6n+162n.

Since for everyw=(w₁, . . . , w_k)∈R^k and everyx∈Rⁿ,

|d_|xP_S^∗(w)|²=

k−1

X

i=1

w_i²+w²_kkd_|xPkk²₂, we get thatkd_|xP_S^∗k²>min 1,4(1−δ)

if|P_k(x)|< δ. Choose US= {(x1, . . . ,xn)∈Rⁿ|

n

X

j=1

x²_j <4}.

Then if0< δ <1, Kδ=

x∈U_S |1−δ 6

n

X

i=k

x_i²61+δ and Xk−1

i=1

x²_k 61−1 2(1+δ)²

is compact in US and taking R_(U² _S_,P_S₎=4, we see that the pair (US,PS) is regular in the sense of Definition2.2.1. The submanifold P_S⁻¹(0)⊂US is isotopic inRⁿ to the unit sphereS^n−k. We deduce that(3/4,1)∈T_(U_S_,P_S₎. From (10) and (13) we deduce

τ_(U_S_,P_S₎624k4ⁿe^16π2n(2+πn)6e⁵³⁺⁵ⁿ. The estimatec_Sn−1 >exp(−e⁵⁴⁺⁵ⁿ)follows then from (16).

Proposition 2.6.3. For every 16k6n and every 06i 6n−k, c_Si×Sⁿ⁻ⁱ⁻^k >

exp(−e⁸²⁺⁶ⁿ).

Proof. For every16k6n and every06i 6n−k, we set Q_k((x₁, . . . ,x_i+1), (y₁, . . . ,y_n−i−1))=(|x|²−2)²+

n−k−i

X

j=1

y²_j−1.

For every(x,y)∈Rⁱ⁺¹×Rⁿ⁻ⁱ⁻¹and0< δ <1/2,

|Qk(x,y)|< δ ⇔1−δ < (|x|²−2)²+

n−k−i

X

j=1

y²_j <1+δ

⇒ kd_|_(x,y)Q_kk²₂=4^n−k−iX

j=1

y²_j+16|x|²(|x|²−2)²,

with|x|²>2−√

1+δ >1/2. Thuskd_|(x,y)Qkk²₂>4(1−δ); compare Lemma 2.6 of [16].

Moreover from Lemma 2.6.2, kQkk²_L2 613n²; compare §2.3.2 of [16]. Now set Q= (Q₁, . . . ,Q_k)with Q_j(x,y)=y_n−i−_j for16 j 6k−1, so

kQk²_L₂ 6(k−1)/π+13n²613(n+1)².

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For everyw=(w₁, . . . , w_k)∈R^k and every(x,y)∈Rⁱ⁺¹×Rⁿ⁻ⁱ⁻¹,

|d_|(x,y)Q^∗(w)|² =

k−1

X

i=1

w²_i +w_k²kd_|(x,y)Qkk²₂

> min(1,4(1−δ))|w|² if|Qk(x,y)|6δ <1/2.We choose

U = {(x,y)∈Rⁱ⁺¹×Rⁿ⁻ⁱ⁻¹| |x|²+ |y|²<6}, Kδ=n

(x,y)∈U|1−δ6(|x|²−2)²+Pn−k−i

j=1 y²_j 61+δ and Pk−1

j=1y_n−i−² _j 61−δo ,

andR²₍_U_,_Q₎=6. The pair(U,Q)is regular in the sense of Definition2.2.1andQ⁻¹(0)⊂U is isotopic in Rⁿ to the product Sⁱ×S^n−i−k of unit spheres in Rⁱ⁺¹ and Rⁿ⁻ⁱ^−k+1. We deduce that for every positive,(1/2−,1)∈T_(U,Q), and from (10) and (13), that

τ_(U,Q)624k4ⁿe^24π13(n+1)²(4+πn)6e⁸¹⁺⁶ⁿ. The estimatec_Si×Sⁿ⁻ⁱ^−k >exp(−e⁸²⁺⁶ⁿ)follows then from (16).

3. Upper estimates for the expected Betti numbers 3.1. Statement of the results

For every 16k6n, we denote by Gr(k−1,n−1)the Grassmann manifold of(k−1)- dimensional linear subspaces of Rⁿ⁻¹. The tangent space of Gr(k−1,n−1) at every H∈Gr(k−1,n−1) is canonically isomorphic to the space of linear maps L(H,H^⊥) from H to its orthogonal H^⊥ and we equip it with the norm

A∈L(H,H^⊥)7→ kAk2=p

T r(A^∗A)∈R⁺.

The total volume of Gr(k−1,n−1) for this Riemannian metric is denoted by Vol(Gr(k−1,n−1))and we set

V_k−1,n−1= 1

√π^{(k−1)(n−k)}Vol(Gr(k−1,n−1)) as its volume for the rescaled metric A∈L(H,H^⊥)7→(1/√

π )kAk2.Likewise, we equip M_k−1(R)with the Euclidean norm A∈ M_k−1(R)7→ kAk2=√

T r(A^∗A)and setdµ(A)= (1/π^k−1)e^−k^A^k²²d A as the associated Gaussian measure on Mk−1(R). Then, we set

Ek−1(|det|^n−k+2)=Z

M_k−1(_R)

|detA|^n−k+2dµ(A).

Remark 3.1.1.

1. The orthogonal group O_n−1(R) acts transitively on the Grassmannian Gr(k−1, n−1)with fixators isomorphic to O_k−1(R)×O_n−k(R).We deduce that