Todd genera of complex torus manifolds

TL;DR

This paper proves that certain complex manifolds with specific symmetry properties have Todd genus one, and shows that quasitoric manifolds with invariant complex structures are equivalent to smooth toric varieties, answering a longstanding question.

Contribution

It establishes a new criterion for the Todd genus of complex manifolds and characterizes quasitoric manifolds with invariant complex structures as smooth toric varieties.

Findings

Todd genus equals one under given conditions

Quasitoric manifolds with invariant complex structures are smooth toric varieties

Negative answer to Buchstaber-Panov problem

Abstract

In this paper, we prove that the Todd genus of a compact complex manifold $X$ of complex dimension $n$ with vanishing odd degree cohomology is one if the automorphism group of $X$ contains a compact $n$-dimensional torus $\Tn$ as a subgroup. This implies that if a quasitoric manifold admits an invariant complex structure, then it is equivariantly homeomorphic to a compact smooth toric variety, which gives a negative answer to a problem posed by Buchstaber-Panov.