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 Morsetype inequalities of Tian and Zhang for symplectic reduction are, in fact, equalities.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 1998
 DOI:
 10.48550/arXiv.math/9809146
 arXiv:
 arXiv:math/9809146
 Bibcode:
 1998math......9146B
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry
 EPrint:
 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"