The first terms in the expansion of the Bergman kernel in higher degrees

TL;DR

This paper analyzes the asymptotic expansion of the Bergman kernel for spin${}^c$ Dirac operators on high tensor powers of line bundles over Kähler manifolds, revealing cancellation patterns and explicit formulas for leading coefficients.

Contribution

It establishes the cancellation of the first 2j terms in the expansion and provides explicit local formulas for the first two non-zero coefficients.

Findings

Cancellation of the first 2j terms in the expansion.

Explicit local formulas for the first and second leading coefficients.

Abstract

We establish the cancellation of the first $2j$ terms in the diagonal asymptotic expansion of the restriction to the $(0,2j)$-forms of the Bergman kernel associated to the spin${}^c$ Dirac operator on high tensor powers of a positive line bundle twisted by a (non necessarily holomorphic) complex vector bundle, over a compact K\"{a}hler manifold. Moreover, we give a local formula for the first and the second (non-zero) leading coefficients.