arXiv research

Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.

We provide a simple proof of a result of Rouby-Sjöstrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the Kähler potential is real analytic then the Bergman kernel is an \textit{analytic kernel} meaning that its amplitude is an \textit{analytic symbol} and its phase is given by the polarization…

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 $k\varphi$ …

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the Kähler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevr…

We study the asymptotic properties of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is in Gevrey class $G^a$ for some $a>1$, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of…

We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^…

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 …

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

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 Kä…

It is shown that geodesics in the space of Kähler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for Kähler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to Kähler man…

We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated…

Formula for Bergman kernel of complex hyperbolic manifolds proved.

Study continuity of Bergman kernels on degenerating varieties.

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

Study Bergman metric on Cartan-Hartogs domains and their duals.

Explicit formula for Bergman kernel of abelian varieties proved.

Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is def…

The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.

The paper studies invariant weighted Bergman metrics on domains.

New Schwarz Lemma for Bergman metrics in bounded domains.

Proper holomorphic isometries between Bergman domains are biholomorphisms.

The Bergman metric on symmetrized bidisc has negative curvature properties.

Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.

The Bergman measure converges to the Zhang measure on a hybrid space.

Study on convergence rate of Bergman metrics on Kähler manifolds.

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtai…

Infinite volumes of Bergman spaces on product manifolds.

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.

Study shows Bergman metric is non-Einstein for certain domains.

Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.

New method for spectral and Bergman kernels under local spectral gap condition.

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

Estimates Bergman kernel for positive line bundles.

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

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 …

Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.

Study shows Bergman kernel quotient approaches one for punctured surfaces.

Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.

Quantifies nearly spherical subsets in complex ball geometry.

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 metr…

Study on when Bergman metrics of domains are induced by balls.

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.

Rigidity theorem for Bergman metric on Hartogs domains over bounded homogeneous domains.

Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.

Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.

Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.

We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.

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 b…