Asymptotic Expansion of the Bergman Kernel via Perturbation of the Bargmann–Fock Model

Abstract

Abstract. We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the kth tensor powers of a positive line bundle L in a 1 √ kneighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential kϕ in a 1 √ k -neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

“…Near the diagonal, i.e. in a 1 √ k -neighborhood of the diagonal, one has a scaling asymptotic expansion for the Bergman kernel (see [ShZe02,MaMa07,MaMa13,LuSh15,HeKeSeXu16]). For d(x, y) ≫ log k k , where d is the Riemannian distance induced by ω, no useful asymptotics are known.…”

Scaling asymptotics for Szegő kernels on Grauert tubes

“…is called the Grauert tube of radius τ . The complexified exponential map can be used to identify (16) with the co-ball bundle of radius τ :…”

“…Related results in the line bundle setting. Scaling asymptotics in the line bundle setting have been proved in varying degrees of generality; see [2,29,24,25,23,16]. The simplest setup is a closed Kähler manifold M polarized by an ample line bundle L. Endow L with a Hermitian metric h so that the curvature two-form Ω is positive, and let ω = 1 2 Ω be the Kähler form on M .…”

Random Sections of Line Bundles Over Real Riemann Surfaces

“…The following theorem says that in the (scaled) normal coordinates U x around a point x ∈ X the geometry of L d |Ux → U x looks like the geometry of the Bargmann-Fock space (see Section 5.4.2), at least in a ball of size B(x; R log d √ d ) for large d and any fixed R > 0. The following theorem is the main theorem of [5] (see also [15,Theorem 4.18], [10,2]). We will use normal coordinates, see for example [15] or [14,Section 3.2].…”