Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Abstract

Given a smooth positive measure µ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on (0, 1)-forms. Thanks to an appropriate version of the Bochner-Kodaira-Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold.Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam.We also show in an appendix that the "twisted basic identities" of, e.g., [MV15] are standard basic identities with respect to conformally Kähler metrics.

The ∂-complex on weighted Bergman spaces on Hermitian manifolds

“…and therefore, (3.4) follows immediately from the well-known identity for the∂-complex (see, e.g., [2] or [6]).…”

“…For the case of general compactly supported v, we can use the partition of unity to reduce to the case above; we omit the details. Thus, if additionally (M, h) is a complete manifold (so that the Andreotti-Vesentini density lemma applies) and u ∈ dom(D * ), then we have a local expression for D * u as follows (see, e.g., [6, (6.20)] or [2]):…”