On the vanishing of characteristic numbers

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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Published 30 November 2014