On the vanishing of characteristic numbers (Q2263815)

Let \(X\) be a closed connected oriented Riemannian manifold of dimension \(4n\), let \(A\) be a Killing vector field on \(X\) (that is, the flow generated by \(A\) acts by isometries). Let \(\text{zero}(A)\) denote the zero-set of \(A\), and let \(\pm \sqrt{-1}\lambda_ j\) be the eigenvalues of the skew-adjoint transformation induced by \(A\) in the normal bundle of \(\text{zero}(A)\). These \(\lambda_ j\) are constant on each connected component and are called the normal eigenvalues of \(\text{zero}(A)\). In this situation, for a symmetric polynomial \(f\) homogeneous of degree at most \(n\), one has the Atiyah-Bott-Singer localization formula (Theorem 8.11) in [\textit{M. F. Atiyah} and \textit{I. M. Singer}, Ann. Math. (2) 87, 546--604 (1968; Zbl 0164.24301)]. In particular, this formula expresses the Pontrjagin numbers of the manifold \(X\) in terms of the Pontrjagin numbers and normal eigenvalues of \(\text{zero}(A)\). The author's main aim in the paper under review is to prove that if \(X\) has a Killing vector field \(A\) of \`\` pure type\'\'\, (meaning that the normal eigenvalues \(\lambda_ j\) of \(\text{zero}(A)\) are all equal) such that the maximal real dimension of the connected components in \(\text{zero}(A)\) is smaller than \(n\), then the vanishing statements, for symmetric polynomials \(f\) of degree smaller than \(n\), given by the Atiyah-Bott-Singer localization formula imply (Proposition 3.5) that the Pontrjagin numbers of the whole manifold \(X\) vanish. Analogous results for the Chern numbers of compact almost-Hermitian manifolds are also presented.