Chern numbers and the indices of some elliptic differential operators

TL;DR

This paper demonstrates how various Chern numbers of certain complex manifolds can be derived from indices of twisted Dirac and signature operators, providing a new proof and divisibility results.

Contribution

It introduces a method to determine multiple Chern numbers from operator indices for compact, spin, almost-complex manifolds, extending previous results.

Findings

Chern numbers can be derived from indices of specific elliptic operators.

A direct proof of Libgober-Wood's result is provided.

Certain characteristic numbers satisfy divisibility properties.

Abstract

Libgober and Wood proved that the Chern number $c_{1}c_{n-1}$ of a $n$-dimensional compact complex manifold can be determined by its Hirzebruch $\chi_{y}$-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 Libgober-Wood's result, which was originally proved by induction.