On the first order asymptotics of partial Bergman kernels

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

DANCOMAN AND GEORGEMARINESCU

On the first order asymptotics of partial Bergman kernels

Tome XXVI, n^o5 (2017), p. 1193-1210.

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On the first order asymptotics of partial Bergman kernels

Dan Coman⁽¹⁾ and George Marinescu⁽²⁾

ABSTRACT. — We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metric along the hypersurface. Finally, we study the asymptotics of the partial Bergman kernel function on a given compact set and near the vanishing locus.

RÉSUMÉ. — Nous montrons, sous des hypothèses très générales, que le noyau de Bergman partiel des sections s’annulant sur une hypersurfaces analytique décroît exponentiellement dans un voisinage du lieu d’annula- tion. Pour un fibré ample, nous montrons une estimée uniforme du noyau de Bergman associé à une métrique singulière le long d’une hypersurface.

Finalement nous étudions les asymptotiques du noyau de Bergman sur un compact près du lieu d’annulation.

1. Introduction

Partial Bergman kernels were recently studied in different contexts, es- pecially Kähler geometry [11, 12, 13] or random polynomials [2, 15].

(*)Reçu le 3 janvier 2016, accepté le 18 août 2016.

Keywords:Bergman kernel function, singular Hermitian metric.

2010Mathematics Subject Classification:32L10, 32A60, 32C20, 32U40, 81Q50.

(1) Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA — dcoman@syr.edu

D. Coman is partially supported by the NSF Grant DMS-1300157.

(2) Universität zu Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln, Deutschland & Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania — gmarines@math.uni-koeln.de

G. Marinescu acknowledges the support of Syracuse University, where part of this paper was written.

Funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.

Article proposé par Vincent Guedj.

Let us consider the following general setting.

(A) (X, ω) is a compact Hermitian manifold of dimension n, Σ is a smooth analytic hypersurface ofX, andt >0 is a fixed real number.

(B) (L, h) is a singular Hermitian holomorphic line bundle on X with singular metrichwhich has locally bounded weights.

We define the space

H₀⁰(X, L^p) :=H⁰ X, L^p⊗ O − btpcΣ

(1.1) of holomorphic sections of thep-th tensor power L^p vanishing to order at leastbtpcalong Σ, where bxcdenotes the integral part of x∈ R. Set dp = dimH⁰(X, L^p) andd0,p= dimH₀⁰(X, L^p). We introduce on H⁰(X, L^p) the L² inner product (·,·)p induced by the metric hp =h^⊗p and the volume form ωⁿ/n! , see (2.1). This inner product is inherited byH₀⁰(X, L^p). The (full) Bergman kernel function is defined by taking an orthonormal basis {S_j^p: 16j6dp}of (H⁰(X, L^p),(·,·)p) and setting

Pp(x) =

d_p

X

j=1

|S^p_j(x)|²_hp, |S_j^p(x)|²_hp:=hS_j^p(x), S_j^p(x)ih_p, x∈X. (1.2) By considering an orthonormal basis{S_j^p: 16j6d0,p}of (H₀⁰(X, L^p),(·,·)p), we define thepartial Bergman kernel functionP_0,p by

P0,p(x) =

d0,p

X

j=1

|S_j^p(x)|²_hp, x∈X. (1.3) Note that this definition is independent of the choice of basis, cf. (2.2).

The asymptotics of the Bergman kernel function for a positive line bundle (L, h) [4, 16], see also [10] for a comprehensive study, is very important in understanding the Yau–Tian–Donaldson conjecture. On the other hand, partial Bergman kernels are useful in connection to the slope semi-stability with respect to a submanifold [14]. On a toric variety X (and for a toric Σ) this study was carried out in [11]. In this context it is shown that the partial Bergman kernel has an asymptotic expansion, having rapid decay of order p^−∞ (see §2.1 for this notation) in a neighborhood U(Σ) of Σ, and giving the full Bergman kernel function to orderp^−∞ outside the closure of U(Σ). Moreover [11] gives a complete distributional asymptotic expansion on X, whose leading term has an additional Dirac delta measure plus a dipole measure over∂U(Σ). These results were generalized in [13] and [17] to the case when the data in question are invariant under anS¹-action.

In general, if no symmetry is assumed, it was shown in [2, Theorem 4.3]

that if the bundleL⊗O(−Σ) is ample, there exists a neighborhoodU(Σ) of Σ,

such thatP_0,p(x) has exponential decay onU(Σ) andp⁻ⁿP_0,p(x) converges toc₁(L, h)ⁿ/ωⁿ inL¹ outside the closure ofU(Σ).

Our first result is that under the very general hypotheses (A) and (B) above (in particular, without any positivity condition), the partial Bergman kernel function decays exponentially in a neighborhood of the divisor Σ.

Theorem 1.1. — Assume that conditions (A)–(B) are fulfilled. Then there exist a neighborhoodUtofΣand a constanta∈(0,1)such thatP0,p6 a^p onUt forp >2t⁻¹. In particular P0,p=O(p^−∞)asp→ ∞on Ut.

For more precise statements see Theorem 3.1 and Corollary 3.3. Theo- rem 1.1 can be formulated for non-compact manifoldsX (see Theorem 3.4), in which case the exponential decay of the partial Bergman kernel holds in a neighborhood of the intersection of Σ with any given compact subset of X. This includes for instance the case of classical Bergman spaces of L²- holomorphic functions on domains inCⁿ.

An object which is closely related to the partial Bergman kernel is the Bergman kernel for a singular metric. The full asymptotic expansion on compact subsets of the regular part of the metric was established in [8, Theorem 1.8]. We are here concerned with asymptotics at arbitrary points with dependence on the distance to the singular set. More precisely, we will consider the following situation.

Let S_Σ ∈ H⁰(X,O(Σ)) be a canonical holomorphic section of the line bundle O(Σ), vanishing to first order on Σ. We fix a smooth Hermitian metrichΣonO(Σ) such that

%:= log S_Σ

_h

Σ <0 onX. (1.4)

We consider a function ξ : X → R∪ {−∞}, smooth on X \Σ, such that ξ=tρin a neighborhoodU of Σ. Let dist(·,·) be the distance onXinduced byω. Our main result is the following:

Theorem 1.2. — Let (X, ω),(L, h),Σbe as in(A)–(B), and assume ω is Kähler andhis smooth. Consider the singular Hermitian metriceh=he^−2ξ onL and assume thatc1(L,eh)>εω for some constant ε >0. Let Pep be the Bergman kernel function of H₍₂₎⁰ (X, L^p,eh_p, ωⁿ/n!), where eh_p :=eh^⊗p. Then there exists a constantC >1such that for everyx∈X\Σand everyp∈N withpdist(x,Σ)^8/3> C we have

Pep(x) pⁿ

ω_xⁿ c1(L,eh)ⁿ_x −1

6Cp^−1/8. (1.5)

Theorem 1.2 can be interpreted in two ways. First, ifxruns in a compact setK ⊂X\Σ, we have a concrete boundp0=Cdist(K,Σ)^−8/3 such that

forp > p₀ the estimate (1.5) holds. By [8, Theorem 1.8] we have Pe_p(x) = P∞

r=0b_r(x)p^n−r+O(p^−∞) as p→ ∞ locally uniformly on X \Σ. Hence, there existsp₀(K)∈NandC_K such that forp > p₀(K) we have

Pep(x) pⁿ

ωⁿ_x c1(L,eh)ⁿ_x −1

6CKp⁻¹ onK.

However,p0(K) is not easy to determine.

We can also recast Theorem 1.2 as a uniform estimate inpfor the singular Bergman kernel on compact sets ofX\Σ whose distance to Σ decreases as p^−3/8. Indeed, set Kp={x∈X : dist(x,Σ)>(C/p)^3/8}. Then (1.5) holds onKp for everyp.

We consider now the global behavior of the partial Bergman kernel. Given a compact setK⊂X\Σ we set

t0(K) := supn

t >0 :∃ η∈C^∞(X,[0,1]), suppη⊂X\K, η= 1 near Σ, andc1(L, h) +t dd^c(η%) is a Kähler current onXo

. (1.6) A consequence of Theorems 1.1 and 1.2 is the following result about the asymptotics of the partial Bergman kernel:

Theorem 1.3. — Let (X, ω),(L, h),Σ be as in (A)–(B), and assume ω is Kähler, h is smooth, and c₁(L, h) >εω for some constant ε > 0. Let K⊂X\Σbe a compact set and lett∈(0, t₀(K)). Then there exist constants C >1,M >1 and a neighborhoodU_tofΣ, all depending ont, such that for x∈U_twe have

M e^t%(x)<1 and P_0,p(x)6(M e^t%(x))^p forp >2/t, (1.7) P0,p(x)> pⁿ

C exp(2tp%(x)) forpdist(x,Σ)^8/3> C, (1.8) where the function%is defined in (1.4). Moreover, we have uniformly on K, P0,p(x) =Pp(x) +O(p^−∞), p→ ∞, (1.9) and in particular,

P_0,p(x) =b0(x)pⁿ+b1(x)pⁿ⁻¹+O(pⁿ⁻²), p→ ∞, (1.10) where

b0= c₁(L, h)ⁿ

ωⁿ , b1= b₀

8π(r^X−2∆ logb0), (1.11) andr^X,∆, are the scalar curvature, respectively the Laplacian, of the Rie- mannian metric associated toc1(L, h).

Hence, (1.7) and (1.8) show that onU_t the exponential decay estimate for the partial Bergman kernel function is sharp. Moreover, onKthe partial Bergman kernel function has the same asymptotics as the full Bergman ker- nel function up to orderO(p^−∞). This was established in [13, Theorem 1.1]

under the additional assumption that there is anS¹-action in a neighborhood of Σ. Our method is to estimate the partial Bergman kernel P0,p by above and below with the full Bergman kernelPpand the singular Bergman kernel Pep. On the set where the singular metricehequals h, the kernels Pep andPp

differ byO(p^−∞). This is shown in Theorem 5.1, which gives a general local- ization result for singular Bergman kernels. Theorem 5.1 is a straightforward consequence of [8].

However, in Theorem 1.3 we do not necessarily obtain apartition of the manifoldX in two sets, one with exponential decay (1.7) and one with “full asymptotics” (1.9), since in generalUt∪K6=X. In [2, 13, 11, 17] a partition with two different regimes was exhibited under further hypotheses. The ap- proach introduced by Berman [2, Section 4.1] was to consider the equilibrium metric (or extremal envelope)htofhwith poles along Σ (see also [13]). The metrichtexists fortsufficiently small thanks to the positivity of (L, h). The local plurisubharmonic (psh) potentials ofhthave Lelong numbertalong Σ.

It is shown in [13, Proposition 2.13] that the partial Bergman kernel func- tionP0,phas exponential decay in the forbidden region {ht> h} which is a neighborhood of Σ. Moreover, under additional symmetry assumptions, it is shown in [13, Theorem 1.1] thatP0,pis essentially equal to the full Bergman kernel outside the forbidden region.

In Theorem 1.1 we show that exponential decay holds near Σ with no assumption on the positivity of (L, h). Without positivity assumptions the equilibrium metric might not exist, but we give here a proof in the general case based only on the sub-average inequality for holomorphic functions.

Our approach in Theorem 1.3 differs from the envelope approach above in that we first fix a compact K disjoint from Σ and then construct an interval of smallt >0 for which the corresponding partial Bergman kernel P0,pdecays exponentially near Σ and is essentially equal to the full Bergman kernel onK.

2. Preliminaries

2.1. Bergman kernel function

Let (L, h) be a singular Hermitian holomorphic line bundle over a com- pact Hermitian manifold (X, ω). We denote byH⁰(X, L^p) the space of holo- morphic sections ofL^p:=L^⊗p.

LetH₍₂₎⁰ (X, L^p) = H₍₂₎⁰ (X, L^p, hp, ωⁿ/n!) be the Bergman space of L²- holomorphic sections of L^p relative to the metric hp := h^⊗p induced by h and the volume formωⁿ/n! onX, endowed with the inner product

(S, S⁰)p:=

Z

X

hS, S⁰ih_p

ωⁿ

n! , S, S⁰∈H₍₂₎⁰ (X, L^p). (2.1) SetkSk²_p= (S, S)_p, d_p = dimH₍₂₎⁰ (X, L^p). Ifhhas locally bounded weights (e. g.his smooth) we have of courseH₍₂₎⁰ (X, L^p) =H⁰(X, L^p). We have the following variational characterization of the partial Bergman kernel

P_0,p(x) = maxn

|S(x)|²_hp: S∈H₀⁰(X, L^p), kSkp= 1o

, (2.2) and similar characterizations hold for the full and singular Bergman kernel functionsP_p andPe_p.

Throughout the paper we also use the following terminology. For a se- quence of continuous functionsfp on a manifoldM we writefp =O(p^−∞) if for every compact subset K ⊂ M and any ` ∈N there exists CK,` >0 such that for allp∈Nwe havekfpkK6CK,`p<sup>−`</sup>.

2.2. Geometric set-up

We prepare here the geometric set-up needed for the proofs of our results, by constructing a special neighborhoodW of Σ.

Let (X, ω) be a compact Hermitian manifold of dimensionn. Let (U, z), z= (z1, . . . , zn), be local coordinates centered at a point x∈X. For r >0 andy∈U we denote by

∆ⁿ(y, r) ={z∈U :|z_j−y_j|6r, j= 1, . . . , n}

the (closed) polydisk of polyradius (r, . . . , r) centered at y. If ω is a Kähler form, the coordinates (U, z) are called Kähler aty∈U if

ωz= i 2

n

X

j=1

dzj∧dzj+O(|z−y|²) onU .

Since Σ is compact, we can find an open coverW ={Wj}16j6N of Σ, whereW_jare Stein contractible coordinate neighborhoods centered at points y_j∈Σ, such that

∆ⁿ(yj,2)⊂Wj, Σ⊂W :=

N

[

j=1

∆ⁿ(yj,1), Σ∩Wj=

z∈Wj:z1= 0 , forj = 1. . . , N ,

(2.3)

wherez= (z1, . . . , zn) are the coordinates onWj. Moreover, ifωis a Kähler form, we may also ensure that

∀x∈∆ⁿ(y_j,1), ∃ z=z(x) coordinates on ∆ⁿ(y_j,2)

centered atxand Kähler atx. (2.4) As in [5, §2.5, Lemma 2.7] one can easily prove the following:

Lemma 2.1. — Let(X, ω),(L, h),Σ,ehbe as in Theorem 1.2, and letW= {Wj}16j6N be an open cover of Σ verifying (2.3) and (2.4). There exist constants C1 > 1, C2 > 0 and r1 > 0 with the following property: if j ∈ {1, . . . , N}, x ∈ ∆ⁿ(yj,1) and z = z(x) are the coordinates on ∆ⁿ(yj,2) given by (2.4), then:

(1) ∆ⁿ_z(x, r1)b∆ⁿ(yj,2) and forr6r1 we have

n! dm6(1 +C1r²)ωⁿ, ωⁿ6(1 +C1r²)n! dm on ∆ⁿ_z(x, r), (2.5) wheredm= dm(z)is the Euclidean volume and∆ⁿ_z(x,·)is the open polydisk relative to the coordinatesz.

(2) (L,eh)has a weightϕx onWj with

ϕx=tlog|f|+ψx, ψx∈C^∞(Wj),

ψx(z) = ReFx(z) +ψ_x⁰(z) +ψex(z) on∆ⁿ(yj,2), (2.6) wheref is a defining function forΣ∩Wj,Fx(z)is a holomorphic polynomial of degree 62 in z, ψ⁰_x(z) = Pn

`=1λ`|z`|<sup>2</sup>, λ` = λ`(x), and

|ψex(z)|6C2|z|³, z∈∆ⁿ_z(x, r1). (2.7)

3. Exponential decay

We prove here Theorem 1.1. LetW ={W_j}_16j6N be the cover of Σ and W ⊃Σ be the neighborhood of Σ constructed in Section 2.2 (see (2.3)). For a functionϕ∈L^∞_loc(Wj) set

kϕk_∞,Wj = sup

|ϕ(w)|: w∈∆ⁿ(yj,2) .

Let (L, h) be a singular Hermitian holomorphic line bundle on X, where the metric h has locally bounded weights. Since L|Wj is trivial, we fix a holomorphic framee_j of L|_Wj, and denote byϕ_j the corresponding weight ofhonWj, i.e. |ej|h=e^−ϕj. Set

khk_∞=khk_∞,W := max

1,kϕjk_∞,Wj : 16j 6N , (3.1) and let%be the function defined in (1.4).

Theorem 3.1. — In the setting of Theorem 1.1, there exists a constant A>1depending only onρandWsuch that for anyS ∈H₀⁰(X, L^p),x∈W, andp>1, we have

|S(x)|²_hp6(Ae^ρ(x))^2btpce^4pkhk∞kSk²_p. Therefore, for everyx∈W andp>1,

P0,p(x)6(Ae^ρ(x))^2btpce^4pkhk∞. For the proof we need the following elementary lemma.

Lemma 3.2. — If k >0 andf ∈ O(∆(0,2)), where ∆(0,2)⊂C is the closed disk centered at0and of radius 2, then

Z

∆(0,2)

|f(ζ)|²dm(ζ)6 k+ 1 2^2k

Z

∆(0,2)

|ζ|^2k|f(ζ)|²dm(ζ). Proof. — Consider the power expansionf(ζ) =P∞

j=0a_jζ^joffin ∆(0,2).

Integrating in polar coordinates we obtain Z

∆(0,2)

|f(ζ)|²dm(ζ) = 2π

∞

X

j=0

|a_j|² Z 2

0

r^2j+1dr= 2π

∞

X

j=0

2^2j+2 2j+ 2|a_j|². On the other hand,ζ^kf(ζ) =P∞

j=ka_j−kζ^j, so Z

∆(0,2)

|ζ|^2k|f(ζ)|²dm(ζ)

= 2π

∞

X

j=k

2^2j+2

2j+ 2|a_j−k|²= 2π

∞

X

j=0

2^2j+2+2k 2j+ 2 + 2k|aj|²

> 2^2k k+ 12π

∞

X

j=0

2^2j+2

2j+ 2|a_j|²= 2^2k k+ 1

Z

∆(0,2)

|f(ζ)|²dm(ζ). Proof of Theorem 3.1. — Letx∈W. Fixj ∈ {1, . . . , N} such thatx∈

∆ⁿ(yj,1) and letej be the local frame ofL|W_j andϕjbe the corresponding weight of h as considered in (3.1). Let S ∈ H₀⁰(X, L^p). On Wj we write

S=se^⊗p_j , withs∈ O(Wj). Then we haves(z) =z^btpc₁ es(z), withes∈ O(Wj).

Using the sub-averaging inequality we get

|S(x)|²_hp=|x1|^2btpc|es(x)|²e^−2pϕj(x) 6|x₁|^2btpce^−2pϕj(x) 1

πⁿ Z

∆ⁿ(x,1)

|es(z)|²dm(z) 6|x1|^2btpce^−2pϕj(x)

Z

∆ⁿ(0,2)

|es(z)|²dm(z).

(3.2)

Applying Fubini’s theorem for the splittingz = (z1, z⁰) and Lemma 3.2 for the variablez1, we obtain

Z

∆ⁿ(0,2)

|es(z)|²dm(z) = Z

∆ⁿ⁻¹(0,2)

Z

∆(0,2)

|es(z1, z⁰)|²dm(z1) dm(z⁰) 6btpc+ 1

2^2btpc Z

∆ⁿ(0,2)

|z1|^2btpc|es(z)|²dm(z) 6C exp 2p sup

∆ⁿ(0,2)

ϕ_j

!Z

∆ⁿ(0,2)

|s(z)|²e^−2pϕj(z)ωⁿ n! , (3.3) whereC=C(W)>1 is chosen such that dm(z)6Cωⁿ/n! on each ∆ⁿ(yj,2) in the local coordinates ofWj, for j= 1, . . . , N. Combining (3.2) and (3.3) we get

|S(x)|²_h

p6C|x₁|^2btpcexp 2p sup

∆ⁿ(0,2)

ϕ_j−2pϕ_j(x)

!

kSk²_p. (3.4) Note that there exists a constantA⁰=A⁰(ρ, W)>1 such that

|x1|6A⁰e^ρ(x), x∈W . (3.5) SetA=A⁰C. The estimates (3.4) and (3.5) yield

|S(x)|²_hp6(C|x1|)^2btpce^4pkhk∞kSk²_p6(Ae^ρ(x))^2btpce^4pkhk∞kSk²_p. Taking into account (2.2) we immediately obtain the conclusion.

Corollary 3.3. — In the setting of Theorem 3.1 we let Ut:=n

x∈W : (Ae^ρ(x))^te^4khk∞<1o

. (3.6)

Then for anyx∈Ut andp >2t⁻¹ we have P0,p(x)6

(Ae^ρ(x))^te^4khk∞p

. (3.7)

In particularP0,p=O(p^−∞)asp→ ∞onUt.

Proof. — This follows immediately from Theorem 3.1, sinceAe^ρ(x) <1 forx∈U_t, and 2btpc>2tp−2> tpforp >2/t.

We conclude this section by giving a version of Theorem 3.1 in the case whenX is not compact. Let (X, ω) be a Hermitian manifold of dimension n, Σ be a smooth analytic hypersurface of X, t > 0 a fixed real number, and (L, h) a singular Hermitian holomorphic line bundle onX with singular metrichwhich has locally bounded weights.

As in the case of a compact manifold X, we introduce the Bergman space H₍₂₎⁰ (X, L^p) = H₍₂₎⁰ (X, L^p, hp, ωⁿ/n!) of L²-holomorphic sections of L^prelative to the metrichpinduced byhand the volume formωⁿ/n! onX, endowed with the inner product (2.1). Let H_(2),0⁰ (X, L^p)⊂H₍₂₎⁰ (X, L^p) be the Bergman space ofL²-holomorphic sections ofL^p vanishing to order at leastbtpcalong Σ,

H_(2),0⁰ (X, L^p) :=H₍₂₎⁰ X, L^p⊗ O − btpcΣ .

The spaces H₍₂₎⁰ (X, L^p) andH_(2),0⁰ (X, L^p) are not necessarily finite dimen- sional but their Bergman kernel functions P_p and P_0,p can be defined as in (1.2), (1.3), by means of at most countable orthonormal bases, see [6, Lemma 3.1].

We fix a compact setE⊂Xand consider an open coverW={Wj}16j6N

and a neighborhoodW of the compact set Σ∩E constructed as in (2.3).

Finally definekhk∞ =khk∞,W as in (3.1), and the function %as in (1.4), such that % <0 in a neighborhood of E. The following theorem is proved exactly as Theorem 3.1:

Theorem 3.4. — Let(X, ω)be a Hermitian manifold of dimensionn,Σ be a smooth analytic hypersurface ofX,t >0a fixed real number, and(L, h) a singular Hermitian holomorphic line bundle onX with singular metrich which has locally bounded weights. Then for any compact set E ⊂X there exists a neighborhoodW of the compact setΣ∩E and a constantA>1such that for everyx∈W andp>1,

P_0,p(x)6(Ae^ρ(x))^2btpce^4pkhk∞. The constantAdepends only on ρandW above.

4. Singular Bergman kernel

In this section we prove Theorem 1.2 by using ideas of Berndtsson, who gave in [3, Section 2] a simple proof for the first order asymptotics of the

Bergman kernel function in the case of powers of an ample line bundle (see also [5, Theorem 1.3]).

We start by recalling the following version of Demailly’s estimates for the∂ operator [7, Théorème 5.1] (see also [5, Theorem 2.5]) which will be needed in our proofs.

Theorem 4.1. — Let (X, ω) be a compact Kähler manifold of dimen- sion n, and let B > 0 be a constant such that Ric_ω > −2πBω on X.

Let (L, h)be a singular Hermitian holomorphic line bundle on X such that c₁(L, h) > εω, and fix p₀ such that p₀ε > 2B. Then for all p > p₀ and all g ∈ L²_0,1(X, L^p, loc) with ∂g = 0 and R

X|g|²_h

pωⁿ < ∞ there exists u∈L²_0,0(X, L^p, loc)such that ∂u=g andR

X|u|²_h

pωⁿ 6pε²

R

X|g|²_h

pωⁿ. Proof of Theorem 1.2. — Let W ={Wj}16j6N be an open cover of Σ verifying (2.3) and (2.4). If j ∈ {1, . . . , N} andx∈∆ⁿ(y_j,1), letz=z(x) be the coordinates on ∆ⁿ(y_j,2) given by (2.4), and lete_j,xbe a holomorphic frame ofLonW_j such that|e_j,x|

e^h

=e^−ϕx, whereϕ_xis given by (2.6).

Assume now thatx∈∆ⁿ(yj,1)\Σ and define r_x:= supn

r∈(0, r₁] : ∆ⁿ_z(x, r)⊂∆ⁿ(y_j,2)\Σo . We have

ωx= i 2

n

X

`=1

dz`∧d¯z`,

c1(L,eh)x=dd^cϕx(0) =dd^cψx(0) =dd^cψ⁰_x(0) = i π

n

X

`=1

λ`dz`∧d¯z`. (4.1)

Sincec1(L,eh)x>εωx it follows thatλ` >ε, ` = 1, . . . , n. Moreover, there exists Hx ∈ O(∆ⁿ_z(x, rx)) such that ReHx = ReFx+tlog|f|. We define a new frame forLover ∆ⁿ_z(x, r_x) bye_x=e^Hxe_j,x. Hence

|e_x| e^h

= exp(ReH_x) exp(−ϕ_x) = exp(−ψ⁰_x−ψe_x).

We fix nowj ∈ {1, . . . , N} and x∈∆ⁿ(y_j,1)\Σ and we will estimate Pep(x). Letrp∈(0, rx/2) be an arbitrary number which will be specified later.

We start by estimating the norm of a sectionS ∈H₍₂₎⁰ (X, L^p,eh_p, ωⁿ/n!) atx.

WritingS=se^⊗p_x , where s∈ O(∆ⁿ_z(x, rx)), we obtain by the sub-averaging inequality for psh functions on ∆ⁿ_z(x, rp) = ∆ⁿ(0, rp),

|S(x)|² e^hp

=|s(0)|²6 R

∆ⁿ(0,rp)|s|²e^−2pψx0 dm R

∆ⁿ(0,r_p)e^−2pψx0 dm ·

We have further by (2.5), (2.7), Z

∆ⁿ(0,r_p)

|s|²e^−2pψ0xdm

6(1 +C₁r²_p) exp 2p sup

∆ⁿ(0,rp)

ψe_x Z

∆ⁿ(0,r_p)

|s|²e^{−2p(ψ0x+ψ}e^x)ωⁿ n!

6(1 +C1r²_p) exp 2C2p r_p³ kSk²_p. Set

E(r) :=

Z

|ξ|6r

e^−2|ξ|2dm(ξ) =π 2

1−e^−2r2 .

Sinceλ`>εwe obtain E(rp

√pε)ⁿ pⁿλ1. . . λn 6

Z

∆ⁿ(0,r_p)

e^−2pψx0 dm6 Z

Cⁿ

e^−2pψx0 dm= (π/2)ⁿ pⁿλ1. . . λn

·

Combining these estimates it follows that

|S(x)|² e^hp

6(1 +C₁r²_p) exp 2C₂p r³_p E(rp

√pε)ⁿ pⁿλ1. . . λnkSk²_p. (4.2) The singular Bergman kernel also satisfies a variational formula,

Pep(x) = maxn

|S(x)|² e^hp

: S ∈H₍₂₎⁰ (X, L^p,ehp, ωⁿ/n!), kSkp= 1o .

Hence (4.2) implies the following upper estimate for the singular Bergman kernel,

Pep(x)

pⁿλ₁. . . λ_n 6(1 +C1r_p²) exp 2C2p r³_p E(r_p√

pε)ⁿ , ∀r_p∈(0, r_x/2). (4.3) For the lower estimate ofPep, let 06χ61 be a smooth cut-off function onCⁿwith support in ∆ⁿ(0,2) such thatχ≡1 on ∆ⁿ(0,1), and setχp(z) = χ(z/rp). ThenF =χpe^⊗p_x is a section ofL^p and|F(x)|

e^hp

=|e^⊗p_x (x)|

e^hp

= 1.

We have

kFk²_p6 Z

∆ⁿ(0,2rp)

e^{−2p(ψ0x+ψ}e^x)ωⁿ n!

6(1 + 4C1r²_p) exp 16C2p r³_p Z

∆ⁿ(0,2rp)

e^−2pψ0xdm 6π

2

ⁿ (1 + 4C₁r_p²) exp 16C₂p r_p³ pⁿλ1. . . λn

·

(4.4)

Setα=∂F. Sincek∂χpk² =k∂χk²/r²_p, wherek∂χkdenotes the maximum of|∂χ|, we obtain as above

kαk²_p= Z

∆ⁿ(0,2rp)

|∂χ_p|²e^{−2p(ψ0x+ψ}e^x)ωⁿ n!

6 k∂χk² r²_p

π 2

n (1 + 4C₁r_p²) exp 16C₂p r_p³ pⁿλ₁. . . λ_n ·

There existsp₀ ∈ N such that for p > p₀ we can solve the ∂–equation by Theorem 4.1. We get a smooth sectionGofL^p with∂G=α=∂F and

kGk²_p6 2

pεkαk²_p6 2k∂χk² pεr_p²

π 2

ⁿ (1 + 4C1r_p²) exp 16C2p r_p³ pⁿλ1. . . λn

· (4.5) Note thatGis holomorphic on ∆ⁿ(0, rp) since∂G=∂F = 0 there. So the estimate (4.2) applies toGon ∆ⁿ(0, rp) and gives

|G(x)|² e^hp

6 (1 +C1r²_p) exp 2C2p r_p³ E(rp

√pε)ⁿ pⁿλ₁. . . λ_nkGk²_p

6 2k∂χk² pεr_p²E(r_p√

pε)ⁿ π

2 n

(1 + 4C₁r²_p)²exp 18C₂p r³_p .

LetS =F−G∈H₍₂₎⁰ (X, L^p,ehp, ωⁿ/n!). Then

|S(x)|² e^hp >

|F(x)|

e^hp

− |G(x)|

e^{hp 2}

=

1− |G(x)|

e^{hp 2}

>

"

1−π 2

^n/2

√2k∂χk(1 + 4C₁r²_p) r_p√

pε E(r_p√

pε)^n/2 exp 9C2p r³_p

#²

=:K1(rp). Moreover, by (4.4) and (4.5)

kSk²_p6(kFkp+kGkp)²6π 2

ⁿ K2(rp) pⁿλ1. . . λn

,

where

K2(rp) = (1 + 4C1r²_p) exp 16C2p r³_p

1 +

√2k∂χk rp

√pε ²

.

Therefore

Pep(x)>

|S(x)|² e^hp kSk²_p >

2 π

ⁿ

pⁿλ1. . . λn

K1(rp)

K2(rp)· (4.6)

Using now (4.1), (4.3) and (4.6) we deduce that for everyx∈

N

[

j=1

∆ⁿ(yj,1)\Σ, r_p< r_x/2 andp > p₀,

K₁(rp)

K₂(r_p) 6Pe_p(x) ωⁿ_x

pⁿc₁(L,eh)ⁿ_x 6K₃(r_p), (4.7) where

K₃(r_p) =

π/2 E(rp

√pε) n

(1 +C₁r²_p) exp 2C₂p r³_p .

We take now r_p = p^−3/8, so p r³_p = p^−1/8 → 0 and p r²_p = p^1/4 → ∞ as p→ ∞. Note that there exists a constantC₃>0 such that

K1(p^−3/8)>1−C3p^−1/8,

K2(p^−3/8)61 +C3p^−1/8, K3(p^−3/8)61 +C3p^−1/8. It follows by (4.7) that there exists a constantC₄>0 such that

1−C4p^−1/86Pep(x) ω_xⁿ

pⁿc1(L,eh)ⁿ_x 61 +C4p^−1/8 (4.8) holds for every x ∈ SN

j=1∆ⁿ(yj,1)\Σ , p^−3/8 < rx/2 and p > p0. Now rx> cdist(x,Σ), for some constantc >0, so there exists a constantC5>0 such that (4.8) holds for p > C₅dist(x,Σ)^−8/3. This concludes the proof of (1.5) forx∈SN

j=1∆ⁿ(y_j,1)\Σ.

By [8, Theorem 1.8] there existC₆>0 andp⁰₀∈Nsuch that

Pep(x) ωⁿ_x

pⁿc1(L,eh)ⁿ_x −1

6C6

p ,

for x ∈ X \ SN

j=1∆ⁿ(y_j,1) and p > p⁰₀. The proof of Theorem 1.2 is

complete.

5. Estimates for the partial Bergman kernel

In this section we prove Theorem 1.3. Let t < t₀(K). By the defini- tion (1.6) of t₀(K), there exist η ∈ C^∞(X,[0,1]) and δ > 0 such that suppη⊂X\K, η= 1 near Σ, andc1(L, h) +tdd^c(η%)>δωin the sense of currents onX. Define

eht=hexp(−2tη%), eht,p=eh^⊗p_t .

Note that eh_t = h in a neighborhood of K and eh_t > h on X. Since Σ is smooth, it follows by (1.4) that H₀⁰(X, L^p) = H₍₂₎⁰ (X, L^p,eht,p, ωⁿ/n!). We denote the norm onH₍₂₎⁰ (X, L^p,eh_t,p, ωⁿ/n!) by

kSk²_t,p= Z

X

|S|² e^ht,p

ωⁿ n! =

Z

X

|S|²_hpexp(−2tpη%)ωⁿ n! ·

LetPet,p be the Bergman kernel function of H₍₂₎⁰ (X, L^p,eht,p, ωⁿ/n!). Recall thatkSkp is the norm given by the scalar product (2.1) onH₀⁰(X, L^p). Since

% <0 we have kSk²_t,p >kSk²_p for any S ∈ H₀⁰(X, L^p). LetS ∈H₀⁰(X, L^p) withkSk²_t,p61. ThenkSk²_p61, too, hence

|S|² e^ht,p

=|S|²_hpexp(−2tpη%)6P0,pexp(−2tpη%), and thus

Pet,p6P0,pexp(−2tpη%).

Denote now by Pp the Bergman kernel function of H⁰(X, L^p) endowed with the scalar product (2.1). Since H₀⁰(X, L^p) is isometrically embedded inH⁰(X, L^p) we haveP_0,p6P_p. Consequently we have shown:

Pe_t,pexp(2tpη%)6P_0,p6P_p onX, Pet,p 6P0,p6Pp near K.

(5.1) Let nowW be the neighborhood of Σ defined in (2.3) and letU_tbe defined as in (3.6), so that the exponential estimate (3.7) holds onU_tforp >2t⁻¹. By shrinkingUt we can assume thatη = 1 onUt. Setting M := e^4khk∞A^t we obtain (1.7). By Theorem 1.2 we have

Pe_t,p(x)>(1−Cp^−1/8)pⁿc₁(L,eh_t)ⁿ_x ω_xⁿ

for every p ∈ N with pdist(x,Σ)^8/3 > C. Note that c₁(L,eh_t) > δω in the sense of currents on X. Since c₁(L,eh_t) is smooth on X \Σ we have c₁(L,eh_t)ⁿ

ωⁿ >δⁿ onX\Σ. By increasingC if necessary, it follows that Pet,p(x)> pⁿ

C forp > Cdist(x,Σ)^−8/3. Hence

P_0,p(x)> pⁿ

C exp(2tp%(x)) forx∈U_tandp > Cdist(x,Σ)^−8/3. This proves (1.8). In order to prove (1.9) we need the following localization theorem for the Bergman kernel for singular Hermitian metrics. We refer to [1, Theorem 2.2] for a localization principle in the case of smooth metrics.