Xu Wang
We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tian's partial $C^0$-estimate.
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On the Bergman Kernel of complex hyperbolic manifolds
本文推导了极化复双曲流形的Bergman核公式，将其表示为流形中的测地环路求和，并研究了Bergman核函数的最大最小值及离对角线Bergman核的估计。
On the Bergman kernel of polarized abelian varieties
本文研究极化阿贝尔簇的Bergman核，推导出显式公式，并探讨了正线 bundle 的同构性与Bergman核函数的极值点性质。
Bergman kernels on degenerations
本文研究极化簇平坦退化中的 Bergman 核，定义逐纤维 Bergman 核并证明其连续性及 Fubini–Study 流的一致收敛性。
We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical…
Generalizing the recent result of Berndtsson, we prove the Nakano semipositivity of the direct image of relative pluricanonical systems and the direct image of relative adjoint (singular) hermitian line bundle with semipositive curvature. This gives a new proof of Kawamata's theorem on the semipositivity of the direct image of relative pluricanonical systems.
Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian
本文研究极化Kähler流形上两类Bergman核，建立无对数奇点的充要条件，回答Lu-Tian问题，并证明紧致局部齐性情形下圆丛的Bergman对数平坦性。
In a previous paper, \cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of $L^2$-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, $V$, over a complex manifold is ample in the sense of Hartshorne, then $V\gr…
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
Semipositive line bundles on punctured Riemann surfaces: Bergman kernels and random zeros
本文研究了穿孔黎曼曲面上半正线丛的高张量幂的Bergman核展开及随机零点，证明了半经典极限下高斯全纯截面零点的等分布、大偏差、中心极限定理和数方差等结果。
Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak inequalities hold on any hermitian manifold X, regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in the standard complex space C^n.
Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct image sheaf. In general, following Schumacher's idea, we use the Lie derivative to compute the variation. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
On slice regular Bergman spaces and fiber bundles
本文将纤维丛理论应用于切片正则函数研究，证明了四元数切片正则Bergman空间是坐标球丛的底空间，并从纤维丛视角探讨了其相关性质。
We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle
In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.
On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence.
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds
In this note, we give a positive answer to a question raised by Jean-Pierre Demailly in 2013, and show the additivity of the approximation functional of closed positive $(1,1)$-currents induced by Bergman kernels.
We prove the asymptotic of the logarithmic Bergman kernel. And as an application, we calculate the conditional expectation of density of zeros of Gaussian random sections of powers of a positive line bundle that vanish along a fixed smooth subvariety.
In this note, we explain that Ross-Thomas' result on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result. This result plays an important role in the companion paper to prove an orbifold version of Donaldson Theorem.
We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings on hyperbolic Riemann surfaces and on complete Kaehler manifolds satisfying certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces when the top manifold is simply-connected.
Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ of dimension $n,$ and let $e^{-ϕ}$ be a continuous metric on $L.$ Fixing a measure $dμ$ on $X$ gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of $L.$ We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair $(K,ϕ),$ where $K$ is the support of $dμ,$ as long as $dμ$ is stably Bernstein-Markov with respect to $(K,ϕ).$ Here the Bergman measure denotes $dμ$ times the restriction to the diagonal of…
Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface
本研究推广了Montgomery关于线丛高张量幂上Bochner拉普拉斯算子的结果，特化到黎曼曲面时得到半正线丛的Bergman核展开与几何量子化结果，证明借助了Bochner拉普拉斯算子与次黎曼拉普拉斯算子的关联。
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.
We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
In this paper, a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech meaure on the Teichmuller space of genus $g\ge 2$ is obtained.
We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.
Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels
本文证明极化复射影流形上次乘法滤过的权重算子是Toeplitz算子，并细化加权Bergman核的渐近分析，改进跳跃测度收敛结果。
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/0404494) of Dai-Liu-Ma. We compute the second coefficient b_1 in the asymptotic expansion using the method of our paper "Generalized Bergman kernels on symplectic manifolds" (math.DG/0411559).
We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.
Bergman Kernels of the Cheng-Yau metrics on quasi-projective manifolds
该研究探讨拟射影流形上Cheng-Yau度量的Bergman核在无穷远光滑除子附近的渐近性，发现近除子点存在量子现象，丰富了复几何与度量几何的相关理论。
Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is infinitely differentiable but not necessarily real analytic. The bounds obtained are the best possible for this class of metrics.
Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i ω$, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive definite inner products $\langle \, , \rangle$ on $\mathcal{O}(E)$ functions FS$(\langle \, ,\rangle) \in \mathcal{H}_ω=\{u \in C^{\infty}(X):ω+i\partial\overline{\partial} u >0\}$. We prove that FS is an injective immersion, but its image in general is not closed in $\mathcal{H}_ω$. To obtain a closed range, FS has to be extended to certain degenerate inner p…
In this paper, we investigate a restricted version of Bergman kernels for high powers of a big line bundle over a smooth projective variety. The geometric meaning of the leading term is specified. As a byproduct, we derive some integral representations for the restricted volume.
In this master thesis, we give a new proof on the pointwise asymptotic expansion for Bergman kernel of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfy local spectral gap condition. The main point is to introduce a suitable semi-classical symbol space and related symbolic calculus inspired from recent work of Hsiao and Savale. Particularly, we establish the existence of pointwise asymptotic expansion on the positive part for certain semi-positive line bundles.
This paper is a sequel to \cite{Xu}. In this paper, an estimation of the Bergman Kernel of Kähler hyperbolic manifold is given by the $L^2$ estimate and the Bochner formula. As an application, an effective criterion of the very ampleness of the canonical line bundle of Kähler hyperbolic manifold is given, which is a generalization of Yeung's result.
The main purpose of the following article is to give a proof of Y. Kawamata's celebrated subadjunction theorem in the spirit of our previous work on Bergman kernels. We will use two main ingredients : an $\displaystyle L^{2\over m}$--extension theorem of Ohsawa-Takegoshi type (which is also a new result) and a more complete version of our former results.
On Bergman kernel functions and weak Morse inequalities
该研究利用渐近Bergman核函数与Bochner-Kodaira-Nakano公式，给出了多种流形及拟凸域上弱全纯Morse不等式的简单统一证明。
We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard L${}^2$ Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a l…
Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.
In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkhäuser, 2007).
Semi-Classical Asymptotics of Bergman and Spectral Kernels for (0,q)-forms
本文提出一种新的缩放方法，在局部谱间隙条件下研究复流形上线丛张量幂的谱核与Bergman核，建立了点态渐近的简单证明，并获得了函数情形下指数衰减的Bergman核首项，且该渐近在曲率退化时仍有效。
Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.
The main result of the present article is a (practically optimal) criterium for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg and F. Campana. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon--J. McKernan, S. Takayama and H. Tsuji concerning the extension of twisted pluricanonical forms. More applications will be offered in the sequel of this a…
We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.
Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$ on a Kähler manifold $X$, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of $L_p$, under a natural convergence assumption on the sequence of curvatures $c_1(L_p,h_p)$. We then apply this to study the asymptotic distribution of common zeros of random sequences of $m$-tuples of sections of $L_p$ as $p\to\infty$.
Actually we will discuss some topics related to Bergman kernel on Cartan- Hartogs domain.
Fiberwise Bergman kernels, vector bundles, and log-subharmonicity
本文研究全纯向量丛上奇异Hermitian度量边界点处模相关的Bergman核，证明其对数次调和性，并应用于加权$L^2$积分下界估计及强开性有效性结果的重新证明。
Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels. The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric associated to the fixed metric, as well as in terms of the Monge-Ampere measure of the fixed metric itself on a certain support set. It is also shown that the …
Let $G \subset \mathbb{C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$. The inequality is optimal as equality holds for the egg domains $\{|z|^2+|w|^{2s}<1\},$ $s \in \mathbb{Z}_+$, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain $G \subset \mathbb{C}^2$ cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to…
We prove the convergence of the Bergman kernels and the $L^2$-Hodge numbers on a tower of Galois coverings $\{X_j\}$ of a compact Kähler manifold $X$ converging to an infinite Galois (not necessarily universal) covering $\widetilde{X}$. We also show that, as an application, sections of canonical line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into some projective space, if so do sections of $K_{\widetilde{X}}$.
We investigate Lp regularity of weighted Bergman projections and zeros of weighted Bergman kernels for the weights that are radially symmetric and comparable to 1 on the unit disc.
Bergman Kernel Asymptotics for Semipositive Line Bundles near Curvature-Degenerate Points
本文研究半正线丛Bergman核在曲率退化点附近的渐近展开，通过谱假设和拟齐次结构获得完整展开与快速衰减，并应用于分支覆盖的拉回。
We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal-Bargmann spaces of the complex plane are briefly discussed.
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincar{é} 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{é} model near the singularity tends to one up to arbitrary negative powers of the tensor power.
Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics.
On the $p$-Bergman kernel with respect to a functional $ξ$
本文将p-Bergman核与ξ-Bergman核推广为关于泛函ξ的p-Bergman核，建立其性质，研究高阶Bergman核的L^p版本与ξ-Bergman核的关系，并推广了Blocki和Zwonek的先前结果。
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $kφ$ in a $\frac{1}{\sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
Bergman kernels and Poincaré series
该研究通过证明Bergman核与群作用的关系，扩展了相对Poincaré级数不消失的结论，适用于一般有限体积的局部对称空间。
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