arXiv research

The paper studies invariant weighted Bergman metrics on domains.

Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.

Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of…

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.

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

The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.

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

The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.

Infinite volumes of Bergman spaces on product manifolds.

Study Bergman kernels on Kähler orbifolds with specific properties.

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

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…

Sharp estimates for Bergman metrics derived from Kähler quantization.

Let $φ\in C^\infty(\Complex^n)$ be a given real valued function. We assume that $\pr\ddbarφ$ is non-degenerate of constant signature $(n_-,n_+)$ on $\Complex^n$. When $q=n_-$, it is well-known that the Bergman kernel for $(0,q)$ forms with respect to the $k$-th weight $e^{-2kφ}$, $k>0$, admits a full asymptotic expansi…

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

The Bergman kernel and period map for curves are studied.

We prove that the Bergman and the Teichmuller metrics are equivalent on Teichmuller spaces.

Let $D$ be a homogeneous bounded domain of $\mathbb{C}^n$ and $\mathcal{A}$ a set of (anti--Wick) symbols that defines a commutative algebra of Toeplitz operators on every weighted Bergman space of $D$. We prove that if $\mathcal{A}$ is rich enough, then it has an underlying geometric structure given by a Lagrangian fo…

Study on symmetric domains with Bergman metric properties.

The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.

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…

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

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping t…

The Bergman space and conformally flat 2-disk operads are linked to vertex operator algebras.

We compute the first four coefficients of the asymptotic off-diagonal expansion of the Bergman kernel for the N-th power of a positive line bundle on a compact Kaehler manifold, and we show that the coefficient b_1 of the N^{-1/2} term vanishes when we use a K-frame. We also show that all the coefficients of the expans…

The paper proves new inequalities on the unit ball in higher dimensions.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be thos…

Master thesis proves Bergman kernel asymptotics for positive line bundles.

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existen…

Study curvature in holomorphic fibration fields.

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a Kähler metric $αg(μ;ν)$ $(α>0)$ on $D_{n,m}(μ)$, …

Formula for Bergman kernel of complex hyperbolic manifolds proved.

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. W…

Study continuity of Bergman kernels on degenerating varieties.

New metric defined for bounded symmetric domains.

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

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…

The paper connects Bergman geometry with information geometry.

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

The paper discusses convergence of Bergman kernels on complex manifolds.

Explicit formula for Bergman kernel of abelian varieties proved.

We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges.…

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

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…

New Schwarz Lemma for Bergman metrics in bounded domains.

Proper holomorphic isometries between Bergman domains are biholomorphisms.