Chern numbers and the indices of some elliptic differential operators

Ping Li

Vol. 251 (2011), No. 1, 173–182

DOI: 10.2140/pjm.2011.251.173

Abstract

Libgober and Wood proved that the Chern number c₁c_n−1 of a compact complex manifold of dimension n can be determined by its Hirzebruch χ_y-genus. Inspired by the idea of their proof, we show that, for compact, spin, almost-complex manifolds, more Chern numbers can be determined by the indices of some twisted Dirac and signature operators. As a byproduct, we get a divisibility result of certain characteristic number for such manifolds. Using our method, we give a direct proof of the result of Libgober and Wood, which was originally proved by induction.

Keywords

Chern number, index, Dirac operator, signature operator

Mathematical Subject Classification 2000

Primary: 57R20

Secondary: 58J20

Milestones

Received: 11 April 2010

Revised: 4 November 2010

Accepted: 27 November 2010

Published: 22 April 2011

Proposed: Kefeng Liu