Hostname: page-component-76c49bb84f-xk4dl Total loading time: 0 Render date: 2025-07-08T08:34:36.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants

Part of: Families, fibrations Representation theory of rings and algebras

Published online by Cambridge University Press:  15 February 2011

Markus Reineke
Markus Reineke*
Affiliation:
Fachbereich C – Mathematik, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany (email: reineke@math.uni-wuppertal.de)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert-scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson–Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.

Keywords

quiver modulifunctional equationswall crossingDonaldson–Thomas invariants

MSC classification

Secondary: 16G20: Representations of quivers and partially ordered sets 14D20: Algebraic moduli problems, moduli of vector bundles

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 943 - 964
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Adriaenssens, J. and Le Bruyn, L., Local quivers and stable representations, Comm. Algebra 31 (2003), 1777–1797.CrossRefGoogle Scholar
[2]Behrend, K. and Fantechi, B., Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008), 313–345.CrossRefGoogle Scholar
[3]Bialynicki-Birula, A., Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.CrossRefGoogle Scholar
[4]Brun, V., Stubban, J. O., Fjeldstad, J. E., Tambs Lyche, R., Aubert, K. E., Ljunggren, W. and Jacobsthal, E., On the divisibility of the difference between two binomial coefficients, in Den 11te skandinaviske matematikerkongress, Trondheim, 1949 (Johan Grundt Tanums Forlag, Oslo, 1952), 42–54.Google Scholar
[5]Cheah, J., On the cohomology of Hilbert schemes of points, J. Algebraic Geom. 5 (1996), 479–511.Google Scholar
[6]Crawley-Boevey, W. and Van den Bergh, M., Absolutely indecomposable representations and Kac–Moody Lie algebras. With an appendix by Hiraku Nakajima, Invent. Math. 155 (2004), 537–559.Google Scholar
[7]Engel, J. and Reineke, M., Smooth models of quiver moduli, Math. Z. 262 (2009), 817–848, arXiv:0706.4306.CrossRefGoogle Scholar
[8]Gessel, I. M., Some congruences for generalized Euler numbers, Canad. J. Math. 35 (1983), 687–709.CrossRefGoogle Scholar
[9]Granville, A., Arithmetic properties of binomial coefficients I: binomial coefficients modulo prime powers, Canad. Math. Soc. Conf. Proc. 20 (1997), 253–275.Google Scholar
[10]Gross, M., Pandharipande, R. and Siebert, B., The tropical vertex, Duke Math. J. 153 (2010), 297–362.CrossRefGoogle Scholar
[11]King, A. D., Moduli of representations of finite dimensional algebras, Q. J. Math. Oxford 45 (1994), 515–530.CrossRefGoogle Scholar
[12]Kontsevich, M. and Rosenberg, A. L., Noncommutative smooth spaces, in The Gelfand mathematical seminars, 1996–1999, Gelfand Mathematical Seminars (Birkhäuser, Boston, 2000), 85–108.CrossRefGoogle Scholar
[13]Kontsevich, M. and Soibelman, Y., Stability structures, Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv:0811:2435.Google Scholar
[14]Kontsevich, M. and Soibelman, Y., Motivic Donaldson–Thomas invariants: summary of results, Preprint (2009), arXiv:0910.4315.Google Scholar
[15]Le Bruyn, L., Noncommutative compact manifolds constructed from quivers, AMA Algebra Montp. Announc. (1999), 5, Paper 1.Google Scholar
[16]Le Bruyn, L. and Procesi, C., Etale local structure of matrix invariants and concomitants, in Algebraic groups Utrecht 1986, Lecture Notes in Mathematics, vol. 1271 (Springer, Berlin, 1987), 143–175.CrossRefGoogle Scholar
[17]Levine, M. and Pandharipande, R., Algebraic cobordism revisited, Invent. Math. 276 (2009), 63–130.CrossRefGoogle Scholar
[18]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory, I, Compositio Math. 142 (2006), 1263–1285.CrossRefGoogle Scholar
[19]Reineke, M., Cohomology of non-commutative Hilbert schemes, Algebr. Represent. Theory 8 (2005), 541–561, math.AG/0306185.CrossRefGoogle Scholar
[20]Reineke, M., Moduli of representations of quivers, in Trends in representation theory of algebras, EMS Series of Congress Reports, ed. Skowronski, A. (EMS Publishing House, Zurich, 2008).Google Scholar
[21]Reineke, M., Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2010), 653–667.CrossRefGoogle Scholar
[22]Sloane, N. J. A., The on-line encyclopedia of integer sequences, http://oeis.org.Google Scholar
[23]Stanley, R. P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[24]Stienstra, J., Mahler measure variations, Eisenstein series and instanton expansions, in Mirror symmetry V, AMS/IP Studies in Advanced Mathematics Series, vol. 38 eds Yui, N., Yau, S.-T. and Lewis, J. D. (American Mathematical Society, Providence, RI, 2006), 139–150.Google Scholar
[25]Weist, T., Localization in quiver moduli spaces, Preprint (2009), arXiv:0903.5442.Google Scholar