Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform

Details Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform

2005-11-07

arxiv.org/abs/math/0511163v2

Mathematics

Algebraic Geometry

8 pages, references and an announcement of a proof of a conjecture of Kac are added

Scientific paper

10.1073/pnas.0601337103

A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple unified proofs are obtained for formulas of Poincare polynomials of toric hyperkahler varieties, Poincare polynomials of Hilbert schemes of points and twisted ADHM spaces of instantons on C^2 and Poincare polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced.

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