arXiv research

Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.

Infinite volumes of Bergman spaces on product manifolds.

Study shows quantum behavior near infinity in metric asymptotics.

Study shows convergence of Bergman kernels on covering spaces of Kähler manifolds.

Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.

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

Study on symmetric domains with Bergman metric properties.

We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix m…

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…

New metric defined for bounded symmetric domains.

Develops a universal Hermitian projective calculus for complex hyperbolic two-space

The paper studies the curvature behavior near the boundary of certain domains.

Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.

Proves cohomology theorems for tropical varieties.

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spa…

We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac operator assoc…

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plan…

Let X be a projective manifold. We prove that the Mabuchi Energy of X is bounded below on all degenerations in B (the space of Bergman metrics) if and only if it is bounded below uniformly on B.

In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the va…

We shall give a definition of the curvature operator for a family of weighted Bergman spaces $\{\mathcal H_t\}$ associated to a smooth family of smoothly bounded strongly pseudoconvex domains $\{D_t\}$. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature fo…

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h 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 corr…

In this paper we study Kaehler manifolds that are strongly not relative to any projective Kaehler manifold, i.e. those Kaehler manifolds that do not share a Kaehler submanifold with any projective Kaehler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results whic…

We compute the Hessian of quantized Ding functionals and give an elementary proof for the convexity of quantized Ding functionals along Bergman geodesics from the view point of projective geometry. We study also the asymptotic behavior of the Hessian using the Berezin-Toeplitz quantization.

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSC…

We prove the existence of commutative $C^*$-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space $\mathbb{P}^n(\mathbb{C})$. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an assoc…

Quantizes semipositive line bundles on complex manifolds.

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.

We compute the leading and sub-leading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.

Explicit formula for Bergman kernel of abelian varieties proved.

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

Study resolvents of Bochner Laplacians on compact manifolds.

Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric $ω$ with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of …

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…

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.