Cohomology of the Mumford Quotient
Abstract
Let $X$ be a smooth projective variety acted on by a reductive group $G$. Let $L$ be a positive $G$-equivariant line bundle over $X$. We use the Witten deformation of the Dolbeault complex of $L$ to show, that the cohomology of the sheaf of holomorphic sections of the induced bundle on the Mumford quotient of $(X,L)$ is equal to the $G$-invariant part on the cohomology of the sheaf of holomorphic sections of $L$. This result, which was recently proven by C. Teleman by a completely different method, generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also shows, that the Morse-type inequalities of Tian and Zhang for symplectic reduction are, in fact, equalities.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- September 1998
- DOI:
- 10.48550/arXiv.math/9809146
- arXiv:
- arXiv:math/9809146
- Bibcode:
- 1998math......9146B
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry
- E-Print:
- A mistake in the proof of Theorem 3.1.b is corrected. The definition of the integration map is slightly changed. To appear in "Quantization of singular symplectic quotients"