Bergman kernels and subadjunction

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative…

Variation of Bergman kernels of adjoint line bundles

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.

Canonical measures and the dynamical systems of Bergman kernels

In this article, we construct the canonical semipositive current or the canonical measure ($=$ the potential of the canonical semipositive current) on a smooth projective variety of nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a Kähler-Einstein metric and coincides the one constructed independently by J. Song and G. Tian. The major difference between their work and the present article is that they found the canonical measure in terms of Käher-Ricci flows, while I found the canonical measure in terms of the…

Curvature of vector bundles and subharmonicity of Bergman kernels

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…

极化阿贝尔簇的Bergman核

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。

退化情形下的 Bergman 核

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。

An explicit estimate of the Bergman kernel for positive line bundles

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.

Boundary asymptotics of the relative Bergman kernel metric for curves

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…

Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $ϕ$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $ϕ^t$ be the restriction of $ϕ$ to $D_t$ and denote by $K_t(z,ζ)$ the Bergman kernel of $D_t$ with the weight function $ϕ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $ϕ=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setti…

Bergman kernels and local holomorphic Morse inequalities

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.

Toeplitz算子、次乘法滤过与加权Bergman核

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。

关于Bergman核的局部化及其在Toeplitz理论中的应用

On the Localization of the Bergman Kernel and applications to Toeplitz theory

研究Bergman核在大线丛和Radon测度下的局部化现象，确认Zelditch猜想，并展示Toeplitz算子在组合下构成代数，其谱在Bernstein-Markov测度下均匀分布。