Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
- Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom; and Department of Mathematics, University of Texas, Austin, TX 78712
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Communicated by Robion C. Kirby, University of California, Berkeley, and approved February 24, 2006 (received for review September 27, 2005)
Abstract
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.
Footnotes
- †E-mail: hausel{at}maths.ox.ac.uk
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Author contributions: T.H. performed research and wrote the paper.
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↵ ‖ Incidentally, the reliability polynomial measures the probability of the graph remaining connected when each edge has the same probability of failure, a concept heavily used in the study of reliability of computer networks (13).
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Conflict of interest statement: No conflicts declared.
- Abbreviations:
- ADHM,
- Atiyah–Drinfeld–Hitchin–Manin.
Abbreviation
- © 2006 by The National Academy of Sciences of the USA
