Off-diagonal decay of Bergman kernels: On a conjecture of Zelditch

TL;DR

This paper investigates Zelditch's conjecture that exponential off-diagonal decay of Bergman kernels implies the metric's real analyticity, proving it within a symmetric framework across arbitrary dimensions.

Contribution

The paper proves Zelditch's conjecture for a broad class of metrics with symmetry, extending previous work to higher dimensions and providing evidence for the conjecture.

Findings

Proved the conjecture within a symmetric framework

Extended analysis to manifolds of arbitrary dimensions

Provided examples where smooth metrics lack exponential decay

Abstract

Consider a complex line bundle over a compact complex manifold equipped with an infinitely differentiable metric with strictly positive curvature form. Assign to positive tensor powers of this bundle the associated product metrics and Bergman projection operators. Zelditch has conjectured that if the Bergman kernels, away from the diagonal, decay exponentially fast to zero as the power tends to infinity, then the metric must be real analytic. Moreover, this is conjectured even if exponential decay is assumed to hold merely for some arbitrarily sparse subsequence of powers tending to infinity. Previously the author has constructed examples in which the metric is infinitely differentiable, but exponential decay does not hold. These examples are within a framework in which a certain degree of symmetry is present. In the present paper we prove the conjecture for all structures within this framework, thus providing evidence in favor of the conjecture. The analysis is carried out for base manifolds of arbitrary dimensions, rather than only for dimension one as in the author's previous work.