The Index Theorem for Quasi-Tori

TL;DR

This paper extends the Index theorem for holomorphic line bundles from complex tori to quasi-tori, using $L^2$-methods and advanced isomorphisms to handle non-compact cases.

Contribution

It generalizes the Index theorem to quasi-tori, including non-linearizable line bundles, employing $L^2$-techniques and the Kazama-Dolbeault isomorphism.

Findings

Index theorem holds for holomorphic line bundles on quasi-tori.

Vanishing of certain cohomology groups depends on hermitian form signature.

The approach applies to both linearizable and non-linearizable bundles.

Abstract

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert-Morimoto decomposition of quasi-tori as well as the K\"unneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using $L^2$-methods coupled with the Kazama-Dolbeault isomorphism and Bochner-Kodaira formulas.