Todd genus and $A_k$-genus of unitary $S^1$-manifolds

TL;DR

This paper proves that under certain topological and symmetry conditions, the Todd genus and $A_k$-genus of a compact unitary 2n-dimensional manifold with a circle action vanish, revealing new constraints on such manifolds.

Contribution

It establishes vanishing results for the Todd genus and $A_k$-genus of unitary manifolds with circle actions under specific Chern class and cohomology conditions.

Findings

Todd genus vanishes under the given conditions

$A_k$-genus also vanishes for the manifold

Results impose new topological constraints on unitary $S^1$-manifolds

Abstract

Assume that $M$ is a compact connected unitary 2n-dimensional manifold and admits a non-trivial circle action preserving the given complex structure. If the first Chern class of $M$ equals to $k_0x$ for a certain 2nd integral cohomology class $x$ with $|k_0|\geq n + 2$, and its first integral cohomology group is zero, this short paper shows that the Todd genus and $A_k$-genus of $M$ vanish.