Generalized Bergman kernels on symplectic manifolds

Abstract

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 of them. As consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut-Lebeau.

48 pages. Add two references on the Hermitian scalar curvature

References in corpus (3)

• On the asymptotic expansion of Bergman kernel
• Embeddability of some strongly pseudoconvex CR manifolds
• Number variance of random zeros on complex manifolds

Cited by in corpus (56)

• Bergman Kernel from Path Integral
• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Toeplitz operators on symplectic manifolds
• Berezin-Toeplitz quantization on Kaehler manifolds
• Exponential Estimate for the asymptotics of Bergman kernels
• A closed formula for the asymptotic expansion of the Bergman kernel
• On canonical metrics on Cartan-Hartogs domains
• Equidistribution for sequences of line bundles on normal Kaehler spaces
• Balanced metrics on some Hartogs type domains over bounded symmetric domains
• Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface
• Equidistribution of zeros of holomorphic sections in the non compact setting
• On the full asymptotic of analytic torsion
• Convergence of Fubini-Study currents for orbifold line bundles
• Remarks on the naturality of quantization
• Asymptotics of Bergman kernels
• Generalized Bergman kernels on symplectic manifolds of bounded geometry
• Hölder singular metrics on big line bundles and equidistribution
• Donaldson's $Q$-operators for symplectic manifolds
• Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells
• The log-term of the disc bundle over a homogeneous Hodge manifold
• Rawnsley's $\epsilon$-function on some Hartogs type domains over bounded symmetric domains and its applications
• Local trace formulae and scaling asymptotics in Toeplitz quantization, II
• Scaling asymptotics for quantized Hamiltonian flows
• On the spectrum of non degenerate magnetic Laplacian
• Equidistribution results for singular metrics on line bundles
• On a property of Bergman kernels when the Kähler potential is analytic
• Off-diagonal asymptotic properties of Bergman kernels associated to analytic Kähler potentials
• Number variance of random zeros on complex manifolds
• Quantitative upper bounds for Bergman kernels associated to smooth Kähler potentials
• Abstract Bergman kernel expansion and its applications
• Szegő kernel asymptotics on some non-compact complete CR manifolds
• $G$-invariant Szegö kernel asymptotics and CR reduction
• The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds
• Semiclassical eigenvalue asymptotics for the Bochner Laplacian of a positive line bundle on a symplectic manifold
• On the approximation of positive closed currents on compact Kaehler manifolds
• Local scaling asymptotics for the Gutzwiller trace formula in Berezin-Töplitz quantization
• The first terms in the expansion of the Bergman kernel in higher degrees
• The second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator
• Geometric quantization of symplectic maps and Witten's asymptotic conjecture
• Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula
• Quantization, dequantization, and distinguished states
• Geometric quantization results for semi-positive line bundles on a Riemann surface
• On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization
• Berezin-Toeplitz quantization for eigenstates of the Bochner-Laplacian on symplectic manifolds
• Asymptotic properties of Bergman kernels for potentials with Gevrey regularity
• Fubini-Study forms on punctured Riemann surfaces
• Remarks on the canonical metrics on the Cartan-Hartogs domains
• Asymptotics of G-equivariant Szegő kernels
• Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds
• Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces
• On the coefficients of the equivariant Szegő kernel asymptotic expansions
• The regular quantizations of certain holomorphic bundles
• The first terms in the expansion of the Bergman kernel in higher degrees: mixed curvature case
• The spectral density function of the renormalized Bochner Laplacian on a symplectic manifold
• Balanced metrics on the Fock-Bargmann-Hartogs domains
• Convergence speed of weighted Bergman kernels towards extremal functions