RESEARCH ARTICLE

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

  • Xin ZHANG
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  • Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received date: 28 May 2018

Accepted date: 28 Aug 2018

Published date: 29 Oct 2018

Copyright

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

Abstract

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Cite this article

Xin ZHANG . Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds[J]. Frontiers of Mathematics in China, 2018 , 13(5) : 1189 -1214 . DOI: 10.1007/s11464-018-0727-7

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