Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

On the vanishing of characteristic numbers

Pages: 185 – 204

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a10

Author

Ping Li (Department of Mathematics, Tongji University, Shanghai, China)

Abstract

In this article we introduce the notion of pure type for Killing vector fields on compact Riemannian and almost-Hermitian manifolds and present an application of the celebrated Atiyah-Bott-Singer localization formula for these Killing vector fields. Our central result is that if a $4n$-dimensional compact Riemannian manifold has a Killing vector field of pure type such that the dimension of its zero point set is less than $n$, then the vanishing statements for low-degree polynomials as given by the Atiyah-Bott-Singer localization formula imply the vanishing of Pontrjagin numbers of this manifold. An analogous result for the Chern numbers of compact almost-Hermitian manifolds is also established. The main strategy of our proof is to construct a family of lower-degree polynomials originating from the monomial symmetric polynomials.

Keywords

Atiyah-Bott-Singer localization formula, characteristic number, semi-free circle action, monomial symmetric polynomial

2010 Mathematics Subject Classification

57R20, 57R25, 58J20

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