Todd genera of complex torus manifolds

Ishida, Hiroaki; Masuda, Mikiya

doi:10.2140/agt.2012.12.1777

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Todd genera of complex torus manifolds

Hiroaki Ishida and Mikiya Masuda

Algebraic & Geometric Topology 12 (2012) 1777–1788

DOI: 10.2140/agt.2012.12.1777

Abstract

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 $T^{n}$ 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 and Panov.

Keywords

Todd genera, quasitoric manifolds, torus manifolds, complex manifold, toric manifold

Mathematical Subject Classification 2010

Primary: 57R91

Secondary: 32M05, 57S25

References

Publication

Received: 14 March 2012

Accepted: 21 June 2012

Published: 24 August 2012

Authors

Hiroaki Ishida
Osaka City University Advanced Mathematical Institute Osaka City University 3-3-138, Sugimoto, Sumiyoshi-ku Osaka-shi 558-8585 Japan

Mikiya Masuda
Department of Mathematics Osaka City University 3-3-138, Sugimoto, Sumiyoshi-ku Osaka-shi 558-8585 Japan

Volume 12, issue 3 (2012)