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

Martin Puchol and Jialin Zhu

Vol. 274 (2015), No. 2, 373–403

DOI: 10.2140/pjm.2015.274.373

Abstract

We establish the cancellation of the first $2j$ terms in the diagonal asymptotic expansion of the restriction to the $\left(0,2j\right)$-forms of the Bergman kernel associated to the spin${}^c$ Dirac operator on high tensor powers of a positive line bundle twisted by a (not necessarily holomorphic) complex vector bundle, over a compact Kähler manifold. Moreover, we give a local formula for the first and the second (nonzero) leading coefficients, as well as for the third assuming that the first two vanish.

Keywords

Bergman kernel, quantization, asymptotic expansion

Mathematical Subject Classification 2010

Primary: 32A25, 53D50

Milestones

Received: 10 January 2014

Revised: 22 July 2014

Accepted: 16 September 2014

Published: 1 April 2015