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

Yong Wang, Aihui Sun

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.

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

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.

Paper Structure

This paper contains 4 sections, 435 equations.

Table of Contents

  1. Introduction
  2. Rescaling $(D^B_p)^2$ and Taylor expansion
  3. The first coefficient in the asymptotic expansion
  4. The second coefficient in the asymptotic expansion