Quotient of Bergman kernels on punctured Riemann surfaces

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincar{\'e} metric near the punctures, and a holomorphic line bundle that polarizes the metric. We show that the quotient of the Bergman kernel of high tensor powers of the line bundle and of the Bergman kernel of the Poincar{\'e} model near the singularity tends to one up to arbitrary negative powers of the tensor power.


Introduction
In this paper we study the asymptotics of Bergman kernels of high tensor powers of a singular Hermitian line bundle over a Riemann surface under the assumption that the curvature has singularities of Poincaré type at a finite set.We show namely that the quotient of these Bergman kernels and of the Bergman kernel of the Poincaré model near the singularity tends to one up to arbitrary negative powers of the tensor power.In our previous paper [5] (see also [4]) we obtained a weighted estimate in the C m -norm near the punctures for the difference of the global Bergman kernel and of the Bergman kernel of the First author is partially supported by ANR contract ANR-14-CE25-0010.Second author is partially supported by NNSFC No. 11829102 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.Third author is partially supported by CRC TRR 191.
Poincaré model near the singularity, uniformly in the tensor powers of the given bundle.Our method is inspired by the analytic localization technique of Bismut-Lebeau [7].
There exists a well-known expansion of the Bergman kernel on general compact manifolds [8,11,14,20,22,23,27,28] with important applications to the existence and uniqueness of constant scalar curvature Kähler metrics [17,27] as part of the Tian-Yau-Donaldson's program.Coming to our context, a central problem is the relation between the existence of special complete/singular metrics and the stability of the pair (X, D) where D is a smooth divisor of a compact Kähler manifold X; see e.g. the suggestions of [26, §3.1.2]for the case of "asymptotically hyperbolic Kähler metrics", which naturally generalize to higher dimensions the complete metrics ω Σ studied here.Moreover, the technique developed here can be extended to the higher dimensional situation in the case of Poincaré type Kähler metrics with reasonably fine asymptotics on complement of divisors, see the construction of [2, §1.1] and [3,Theorem 4].
The Bergman kernel function of a singular polarization is of particular interest in arithmetic situations [9,10,6].In [5] we applied the precise asymptotics of the Bergman kernel near the punctures in order to obtain optimal uniform estimates for the supremum of the Bergman kernel, relevant in arithmetic geometry [1,21,18].There are also applications to "partial Bergman kernels", see [13].
We place ourselves in the setting of [5] which we describe now.Let Σ be a compact Riemann surface and let D = {a 1 , . . ., a N } ⊂ Σ be a finite set.We consider the punctured Riemann surface Σ = Σ D and a Hermitian form ω Σ on Σ.Let L be a holomorphic line bundle on Σ, and let h be a singular Hermitian metric on L such that: (α) h is smooth over Σ, and for all j = 1, . . ., N , there is a trivialization of L in the complex neighborhood V j of a j in Σ, with associated coordinate z j such that |1| 2 h (z j ) = log(|z j | 2 ) .(β) There exists ε > 0 such that the (smooth) curvature R L of h satisfies iR L ≥ εω Σ over Σ and moreover, iR L = ω Σ on V j := V j {a j }; in particular, ω Σ = ω D * in the local coordinate z j on V j and (Σ, ω Σ ) is complete.
Here ω D * denotes the Poincaré metric on the punctured unit disc D * , normalized as follows: (1.1) ω D * := idz ∧ dz |z| 2 log 2 (|z| 2 ) • For p ≥ 1, let h p := h ⊗p be the metric induced by h on L p | Σ , where L p := L ⊗p .We denote by H 0 (2) (Σ, L p ) the space of L 2 -holomorphic sections of L p relative to the metrics h p and ω Σ , endowed with the obvious inner product.The sections from H 0 (2) (Σ, L p ) extend to holomorphic sections of L p over Σ, i. e., (see [22, (6.2.17)]) In particular, the dimension d p of H 0 (2) (Σ, L p ) is finite.
We denote by B p ( • , • ) and by B p ( • ) the (Schwartz-)Bergman kernel and the Bergman kernel function of the orthogonal projection B p from the space of L 2 -sections of L p over Σ onto H 0 (2) (Σ, L p ).They are defined as follows: if {S p ℓ } dp ℓ=1 is an orthonormal basis of H 0 (2) (Σ, L p ), then Note that these are independent of the choice of basis (see [22, (6.1.10)]or [12,Lemma 3.1]).Similarly, let B D * p (x, y) and B D * p (x) be the Bergman kernel and Bergman kernel function of D * , ω D * , C, log(|z| 2 ) p h 0 ) with h 0 the flat Hermitian metric on the trivial line bundle C.
Note that for k ∈ N, the where ∇ p,Σ is the connection on (T Σ) ⊗ℓ ⊗ L p induced by the Levi-Civita connection on (T Σ, ω Σ ) and the Chern connection on (L p , h p ), and the pointwise norm | • | h p ,ω Σ is induced by ω Σ and h p .In the same way we define the We fix a point a ∈ D and work in coordinates centered at a. Let e L be the holomorphic frame of L near a corresponding to the trivialization in the condition (α).By assumptions (α) and (β) we have the following identification of the geometric data in the coordinate z on the punctured disc D * 4r of radius 4r centered at a, via the trivialization e L of L, In [5,Theorem 1.2] we proved the following weighted diagonal expansion of the Bergman kernel: Theorem 1.1.Assume that (Σ, ω Σ , L, h) fulfill conditions (α) and (β).Then the following estimate holds: for any ℓ, k ∈ N, and every δ > 0, there exists C = C(ℓ, k, δ) > 0 such that for all p ∈ N * , and with norms computed with help of ω Σ and the associated Levi-Civita connection on D * 4r .Note that in [5,Theorem 1.1] we also established the off-diagonal expansion of the Bergman kernel B p ( • , • ).The main result of the present paper is the following estimate of the quotient of the Bergman kernels from (1.7): i.e., for any ℓ > 0 there exists C > 0 such that for any p ∈ N * we have Theorem 1.2 is related to estimates in exponentially small neighborhoods of the punctures obtained in [24,Theorem 1.6] and [25,Lemma 3.3].
For each p ≥ 2 fixed (|z| is smooth and strictly positive on D 4r , as follows from (2.7).By [5,Remark 3.2], any holomorphic L 2 -section of L p over Σ extends to a homomorphic section on Σ (see the inclusion (1.3)) vanishing at 0 in D 4r .Thus by the formula (1.4) for B p we see that the quotient Remark 1.4.Theorem 1.1 admits a generalization to orbifold Riemann surfaces.Indeed, assume that Σ is a compact orbifold Riemann surface such that the finite set D ⊂ Σ does not meet the (orbifold) singular set of Σ.Then by the same argument as in [5, Remark 1.3] (using [14,15]) we see that Theorems 1.2 and 1.3 still hold in this context.
Note that the C k -norm used in (1.7) is induced by ω D * , roughly the sup-norm with respect to the derivatives defined by the vector fields z log(|z| 2 ) ∂ ∂z and z log(|z| 2 ) ∂ ∂z , which vanish at z = 0. Hence the norm in (1.10) is stronger than the C k -norm used in (1.7), because the norm in (1.10) is defined by using derivatives along the vector fields ∂ ∂z and ∂ ∂z .
Let us mention at this stage that even if the results above follow from our work [5], relying more precisely on [5, Theorem 1.2], the proofs are by no means an obvious rewriting of [5, Theorem 1.2] (for instance), since B D * p ( • ) takes extremely small values arbitrarily near the origin.This can be seen in [5, §3.2] and it is specific to the non-compact framework.What estimate (1.8) says is that B p ( • ) follows such a behaviour very closely in the corresponding regions of Σ via the chosen coordinates.
Here is a general strategy of our approach for Theorems 1.2 and 1.3.We choose a special orthonormal basis {σ implies that the coefficients of the expansion p in two groups: 1 ≤ j, ℓ ≤ δ p ; max{j, ℓ} ≥ δ p + 1.The contribution corresponding to 1 ≤ j, ℓ ≤ δ p , will be controlled by using Lemma 2.1 (or 3.1).The contribution corresponding to max{j, ℓ} ≥ δ p + 1 will be handled by a direct application of Cauchy inequalities (2.23).It turns out that by suitably choosing c, A > 0 this contribution has uniformly the relative size 2 −αp compared to This paper is organized as follows.In Section 2, we establish Theorem 1.2 based on the off-diagonal expansion of Bergman kernel from [5, §6].In Section 3, we establish Theorem 1.3 by refining the argument from Section 2. In Section 4 we give some applications of the main results.
Notation: We denote ⌊x⌋ as the integer part of x ∈ R.
. By [5,Remark 3.2] the inclusion (1.3) identifies the space H 0 (2) (Σ, L p ) of L 2 -holomorphic sections of L p over Σ to the subspace of H 0 (Σ, L p ) consisting of sections vanishing at the punctures, so it induces an isomorphism of vector spaces where O Σ (−D) is the holomorphic line bundle on Σ defined by the divisor −D.By the Riemann-Roch theorem we have for all p with p deg(L) − N > 2g − 2, where deg(L) is the degree of L over Σ, and g is the genus of Σ.
The Bergman kernel function (1.4) satisfies the following variational characterization, see e.g.[12,Lemma 3.1], , for z ∈ Σ. (2.4)By the expansion of the Bergman kernel on a complete manifold [22, Theorem 6.1.1](cf.also [5, Theorem 2.1, Corollary 2.4]), there exist coefficients b i ∈ C ∞ (Σ), i ∈ N, such that for any k, m ∈ N, any compact set K ⊂ Σ, we have in the C m -topology on K, and hence (2.7) For any m ∈ N, 0 < b < 1 and 0 < γ < 1 2 there exists by [5,Proposition 3.3] Taking into account Theorem 1.1 and (2.8) we see that in order to prove Theorem 1.2 it suffices, after reducing to some V j and identifying the geometric data on D * 4r and Σ via (1.6), to show that for some (small) c > 0 and (large) A > 0, and for all l ≥ 0 there exists C = C(c, A, l) > 0 such that for all p ≥ 2, (2.9) sup We now start to establish (2.9).In the whole paper we use the following conventions.
4. Finally, we carefully compare B D * p with B p using the three steps of the above construction to get estimate (2.9); of particular importance are the following intermediate estimates which will be deduced from [5, §6]: Lemma 2.1.With the notations above, for all m ∈ N, there exists C = C(m) > 0 such that for all p ∈ N * , p ≥ 2, and all j, ℓ ∈ {1, . . ., δ p }, (2.15) The proof of Lemma 2.1 is postponed to Section 2.2.Notice that we take care of stating estimates uniform in j, ℓ ∈ {1, . . ., δ p }. Observe moreover that (2.14), (2.15) are integral estimates, whereas we want to establish pointwise estimates in the end, hence we need an extra effort to convert these (among others) into (2.9).
Let us see now how to build on (2.15) to get the desired (2.9).

C k -estimate of the quotient of Bergman kernels
The proof of Theorem 1.3 follows the same strategy as in Section 2 (use of the orthonormal basis (σ ), but with some play on the parameters (in particular, the truncation floor δ p of Step 1. in the outline of the proof of Theorem 1.2).Some precisions on this basis are also needed: we'll see more precisely that in some sense, and provided relevant choices along the construction, the head terms σ ℓ z ℓ of D * than sketched above.This section is organized as follows.In Section 3.1, we establish a refinement of the integral estimate Lemma 2.1 which is again deduced from [5].In Section 3.2, we establish Theorem 1.3 by using Lemma 3.1.

A refined integral estimate
To establish Theorem 1.3, we follow Steps 1. to 4. in the outline of the proof of Theorem 1.2 by modifying δ p , thus refining Lemma 2.1 to Lemma 3.1 below.
Moreover, (σ As will be seen, estimate (3.3) is directly related to the play on δ ′ p , whereas the echelon property as such is not, and (3.4), (3.5) are a direct consequence of (2.30) and (2.31).Moreover, no estimate is given on the σ (p) ℓ for ℓ ≥ δ ′ p + 1 in the above statement; as in the proof of Theorem 1.2, it turns out that we content ourselves with rather rough estimates on these tail sections.

Proof of Theorem 1.3
We show now how to establish Theorem 1.3 by using Lemma 3.1.It can be noticed here that while estimate (3.3) is essential in establishing Theorem 1.3, the echelon property is not, but helps nonetheless clarify some of the upcoming computations.
We first tackle the sum over q = 1, s ≥ 1 and m ≥ 2, focusing on the cases s, m ≤ δ ′ p ; then we deal with the sum over q ≥ 2, s ≥ 1 and m ≥ 1, focusing on q, s, m ≤ δ ′ p , before we also address the cases of "large indices" (max{q, s, m} ≥ δ ′ p + 1).
Higher k-order estimates are established along the same lines: (1) the sum over the set of indices in A j p where one of indices satisfies ≥ δ ′ p + 1, will be controlled by a polynomial in p times 2 −α ′ p β D * p (z) k ; (2) to handle the sum over the set of indices ≤ δ ′ p , we observe first that the contribution from the terms with sum of indices < 2k + 2 is zero, so we will increase κ to absorb the exponential factor in the estimates.Thus the analogue of (3.78) holds for k > 1.We exemplify this for the second derivative d 2 dz 2 to show how the above argument works.From (3.39), we get It is clear that the contribution of the indices with q + m + t < 5 is zero, so the trick (3.42) works even in the presence of a z −2 -term in (3.81).
Recall that by [5,Remark 3.2] the sections of H 0 (2) (Σ, L p ) extend to holomorphic sections of L p over Σ that vanish at the punctures and this gives an identification It is well known that  p is a holomorphic embedding for p large enough, namely for all p satisfying p deg(L) − N > 2g (see [19, p. 215]).Thus  p,(2) is also an embedding for p large enough, as the restriction of an embedding of Σ.
Proof.We have indeed by [22,Theorem 5.1.6We compare next the induced Fubini-Study metrics by  p,(2) on Σ and on D * , and show that they differ from each other (modulo the usual identification on D * 4r in (1.6) with the neighbourhood of a singularity of Σ) by a sequence of (1, 1)-forms which is O(p −∞ ) (at every order) with respect to any smooth reference metric on D r : the situation is just as good as in the smooth setting.
The infinite dimensional projective space CP ∞ is a Hilbert manifold modeled on the space ℓ 2 of square-summable sequences of complex numbers (a j ) j∈N endowed with the norm (a j ) = j≥0 |a j | 2 1/2 .Then CP ∞ = ℓ 2 \ {0}/C * and for a ∈ ℓ 2 we denote by [a] its class in CP ∞ .The affine charts are defined as usual by U j = {[a] : a j = 0}.The Fubini-Study metric ω FS,∞ is defined by ω FS,∞ = i 2π ∂∂ log a 2 to the effect that for a holomorphic map F : M → CP ∞ from a complex manifold M to CP ∞ we have F * ω FS,∞ = We finish with an application to random Kähler geometry, more precisely to the distribution of zeros of random holomorphic sections [12,16].
Let us endow the space H 0 (2) (Σ, L p ) with a Gaussian probability measure µ p induced by the unitary map H 0 (2) (Σ, L p ) ∼ = C dp given by the choice of an orthonormal basis (S p j ) dp j=1 .Given a section s ∈ H 0 (2) (Σ, L p ) ⊂ H 0 (Σ, L p ) we denote by [s = 0] the zero distribution on Σ defined by the zero divisor of s on Σ.If the zero divisor of s is given by m j P j , where m j ∈ N and P j ∈ Σ, then [s = 0] = m j δ P j , where δ P is the delta distribution at P ∈ Σ.We denote by • , • the duality between distributions and test functions.For a test function Φ ∈ C ∞ (Σ) and s as above we have [s = 0], Φ = m j Φ(P j ).