The stringy E-function of the moduli space of rank 2 bundles over a Riemann surface of genus 3

Kiem, Young-Hoon

doi:10.1090/S0002-9947-02-03125-2

The stringy E-function of the moduli space of rank 2 bundles over a Riemann surface of genus 3
HTML articles powered by AMS MathViewer

by Young-Hoon Kiem PDF

Trans. Amer. Math. Soc. 355 (2003), 1843-1856 Request permission

Abstract:

We compute the stringy E-function (or the motivic integral) of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E-functions of the GIT quotients of linear representations.

References

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
Victor V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 1–32. MR 1672108
A. Craw. An introduction to motivic integration. Preprint, math.AG/9911179.
Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI 10.1007/s002220050284
Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144
J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94 (French). MR 999313, DOI 10.1007/BF01850655
Richard Earl and Frances Kirwan, The Hodge numbers of the moduli spaces of vector bundles over a Riemann surface, Q. J. Math. 51 (2000), no. 4, 465–483. MR 1806453, DOI 10.1093/qjmath/51.4.465
Johannes Huebschmann, Poisson geometry of flat connections for $\textrm {SU}(2)$-bundles on surfaces, Math. Z. 221 (1996), no. 2, 243–259. MR 1376296, DOI 10.1007/PL00004249
Frances Kirwan, On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface, Proc. London Math. Soc. (3) 53 (1986), no. 2, 237–266. MR 850220, DOI 10.1112/plms/s3-53.2.237
Frances Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. 24 (1986), no. 2, 221–275. MR 884188, DOI 10.1007/BF02384399
Frances Clare Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), no. 1, 41–85. MR 799252, DOI 10.2307/1971369
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
E. Looijenga. Motivic measures. Sém. Bourbaki 1999/2000, Astérisque, No. 276 (2002), 267-297.
P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546290
Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque, vol. 96, Société Mathématique de France, Paris, 1982 (French). Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. MR 699278
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158