Abstract
Consider a holomorphic $#x2102;×-action on a possibly noncompact Kähler manifold. We show that the highercohomology groups appearing in the geometric quantization of thesymplectic quotient are isomorphic to the invariant parts of thecorresponding cohomology groups of the original manifold. Fornon-Abelian group actions on compact Kähler manifolds, this resultwas proved recently by Teleman. Our approach is applying the holomorphicinstanton complex to the prequantum line bundles over the symplecticcuts. We also settle a conjecture of Zhang and the present author on theexact sequence of higher cohomology groups in the context of symplecticcutting.
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Wu, S. A Note on Higher Cohomology Groups of Kähler Quotients. Annals of Global Analysis and Geometry 18, 569–576 (2000). https://doi.org/10.1023/A:1006712005064
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DOI: https://doi.org/10.1023/A:1006712005064