arXiv research

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.

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

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

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/040…

Master thesis proves Bergman kernel asymptotics for positive line bundles.

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 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…

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…

We study the asymptotic of the Bergman kernel of the spin$^c$ Dirac operator on high tensor powers of a line bundle.

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…

Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.

Study shows quantum behavior near infinity in metric asymptotics.

The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.

Analytic Kähler potentials yield analytic Bergman kernels.

The study computes Bergman kernels and point process asymptotics on Kähler manifolds.

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

The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.

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

The paper discusses convergence of Bergman kernels on complex manifolds.

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

We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem.

Study Bergman kernels on Kähler orbifolds with specific properties.

New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.

Study Bergman and spectral kernels for non-compact complex manifolds.

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated w…

Estimates Bergman kernels on Kähler manifolds with Ricci bounds.

We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formul…

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 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…

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

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…

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…

The paper improves $L^2$-estimates for Dirac-Dolbeault operators on complex manifolds.

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

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 abstract discusses embedding theorems for pseudo-Kähler manifolds.

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…

Formula for Bergman kernel of complex hyperbolic manifolds proved.

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.

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 …

Study continuity of Bergman kernels on degenerating varieties.

We establish the cancellation of the first $2j$ terms in the diagonal asymptotic expansion of the restriction to the $(0,2j)$-forms of the Bergman kernel associated to the spin${}^c$ Dirac operator on high tensor powers of a positive line bundle twisted by a (non necessarily holomorphic) complex vector bundle, over a c…

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

Explicit formula for Bergman kernel of abelian varieties proved.

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two o…

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

We establish the cancellation of the first |2j-q| terms in the diagonal asymptotic expansion of the restriction to the (0, 2j)-forms of the Bergman kernel associated to the modified spin^c Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a (non necessarily holomorphic) complex vecto…