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

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.