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Background cohomology of a non-compact Kähler $ G$-manifold


Author: Maxim Braverman
Journal: Trans. Amer. Math. Soc. 367 (2015), 2235-2262
MSC (2010): Primary 32L10
DOI: https://doi.org/10.1090/S0002-9947-2014-06314-9
Published electronically: July 18, 2014
MathSciNet review: 3286513
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Abstract: For a compact Lie group $ G$ we define a regularized version of the
Dolbeault cohomology of a $ G$-equivariant holomorphic vector bundle over non-compact Kähler manifolds. The new cohomology is infinite dimensional, but as a representation of $ G$ it decomposes into a sum of irreducible components, each of which appears in it with finite multiplicity. Thus equivariant Betti numbers are well defined. We study various properties of the new cohomology and prove that it satisfies a Kodaira-type vanishing theorem.


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Additional Information

Maxim Braverman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

DOI: https://doi.org/10.1090/S0002-9947-2014-06314-9
Received by editor(s): April 8, 2012
Received by editor(s) in revised form: February 21, 2013, and October 10, 2013
Published electronically: July 18, 2014
Additional Notes: This research was supported in part by the NSF grant DMS-1005888.
Article copyright: © Copyright 2014 American Mathematical Society