Homology, Homotopy and Applications
Volume 16 (2014)
On the vanishing of characteristic numbers
Pages: 185 – 204
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.
Atiyah-Bott-Singer localization formula, characteristic number, semi-free circle action, monomial symmetric polynomial
2010 Mathematics Subject Classification
57R20, 57R25, 58J20