Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
Open Access
- 18 April 2006
- journal article
- Published by Proceedings of the National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 103 (16) , 6120-6124
- https://doi.org/10.1073/pnas.0601337103
Abstract
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.Comment: 8 pages, references and an announcement of a proof of a conjecture of Kac are addeKeywords
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