The first terms in the expansion of the Bergman kernel in higher degrees: mixed curvature case

TL;DR

This paper derives the initial terms in the asymptotic expansion of the Bergman kernel for (0, 2j)-forms on symplectic manifolds with mixed curvature, extending previous results to a broader setting.

Contribution

It establishes the cancellation of certain terms in the expansion and provides explicit formulas for the leading coefficients in a generalized mixed curvature context.

Findings

Cancellation of the first |2j-q| terms in the expansion

Explicit formulas for the first and second leading coefficients

Generalization of previous results to mixed curvature cases

Abstract

We establish the cancellation of the first |2j-q| terms in the diagonal asymptotic expansion of the restriction to the (0, 2j)-forms of the Bergman kernel associated to the modified spin^c Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a (non necessarily holomorphic) complex vector bundle, over a compact symplectic manifold. Moreover, we give a local formula for the first and the second (non-zero) leading coefficients which generalizes the Puchol-Zhu's results.