923 research outputs found
Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
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 adde
Non-connected toric Hilbert schemes
We construct small (50 and 26 points, respectively) point sets in dimension 5
whose graphs of triangulations are not connected. These examples improve our
construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size,
but also in that their toric Hilbert schemes are not connected either, a
question left open in that article. Additionally, the point sets can easily be
put into convex position, providing examples of 5-dimensional polytopes with
non-connected graph of triangulations.Comment: 18 pages, 2 figures. Except for Remark 2.6 (see below) changes w.r.t.
version 2 are mostly minor editings suggested by an anonimous referee of
"Mathematische Annalen". The paper has been accepted in that journal. Most of
the contents of Remark 2.6 have been deleted, since there was a flaw in the
argumen
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