Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform.

Journal article

A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This technique in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence, simple unified proofs are obtained for formulas of Poincaré polynomials of toric hyperkähler varieties (recovering results of Bielawski-Dancer and Hausel-Sturmfels), Poincaré polynomials of Hilbert schemes of points and twisted Atiyah-Drinfeld-Hitchin-Manin (ADHM) spaces of instantons on C2 (recovering results of Nakajima-Yoshioka), and Poincaré 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.

Authors: Hausel, T

• Publication status: Published
• Journal: Proceedings of the National Academy of Sciences of the United States of America
• Volume: 103
• Issue: 16
• Pages: 6120-6124
• Publication date: 2006-04-01
• DOI: 10.1073/pnas.0601337103
• EISSN: 1091-6490
• ISSN: 0027-8424
• Language: English
• Keywords: math.SG; math.CO; math.AG; math.RT