Chern numbers and the indices of some elliptic differential operators

Details Chern numbers and the indices of some elliptic differential operators Chern numbers and the indices of some elliptic differential operators

2010-04-13

arxiv.org/abs/1004.2142v2

Pacific J. Math. 251 (2011), 173-182

Mathematics

Differential Geometry

9 pages, to appear in Pacific Journal of Mathematics

Scientific paper

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.

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