A Note on Higher Cohomology Groups of Kähler Quotients

Details A Note on Higher Cohomology Groups of Kähler Quotients A Note on Higher Cohomology Groups of Kähler Quotients

1998-09-30

arxiv.org/abs/math/9809192v1

Mathematics

Symplectic Geometry

plain LeTeX, 8 pages

Scientific paper

Consider a holomorphic torus action on a possibly non-compact K\"ahler manifold. We show that the higher cohomology groups appearing in the geometric quantization of the symplectic quotient are isomorphic to the invariant parts of the corresponding cohomology groups of the original manifold. For non-Abelian group actions on compact K\"ahler manifolds, this result was proved recently by Teleman and by Braverman. Our approach is applying the holomorphic instanton complex to the prequantum line bundles over the symplectic cuts. We also settle a conjecture of Zhang and the present author on the exact sequence of higher cohomology groups in the context of symplectic cutting.

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