Cohomology of the Mumford Quotient

Details Cohomology of the Mumford Quotient Cohomology of the Mumford Quotient

1998-09-24

arxiv.org/abs/math/9809146v3

Mathematics

Symplectic Geometry

A mistake in the proof of Theorem 3.1.b is corrected. The definition of the integration map is slightly changed. To appear in

Scientific paper

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.

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