Geometry & Topology

Prym varieties of spectral covers

Tamás Hausel and Christian Pauly

Full-text: Open access

Abstract

Given a possibly reducible and non-reduced spectral cover π:XC over a smooth projective complex curve C we determine the group of connected components of the Prym variety Prym(XC). As an immediate application we show that the finite group of n–torsion points of the Jacobian of C acts trivially on the cohomology of the twisted SLn–Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder–Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted SLn stable bundle moduli space.

Article information

Source
Geom. Topol., Volume 16, Number 3 (2012), 1609-1638.

Dates
Received: 29 June 2011
Accepted: 8 June 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732444

Digital Object Identifier
doi:10.2140/gt.2012.16.1609

Mathematical Reviews number (MathSciNet)
MR2967059

Zentralblatt MATH identifier
1264.14061

Subjects
Primary: 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20]
Secondary: 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 14H40: Jacobians, Prym varieties [See also 32G20]

Keywords
Prym varieties Hitchin fibration Higgs bundles vector bundles on curves

Citation

Hausel, Tamás; Pauly, Christian. Prym varieties of spectral covers. Geom. Topol. 16 (2012), no. 3, 1609--1638. doi:10.2140/gt.2012.16.1609. https://projecteuclid.org/euclid.gt/1513732444


Export citation

References

  • M Atiyah, The geometry and physics of knots, Lezioni Lincee., Cambridge Univ. Press (1990)
  • M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523–615
  • A Beauville, M S Narasimhan, S Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989) 169–179
  • C Birkenhake, H Lange, Complex abelian varieties, second edition, Grundl. Math. Wissen. 302, Springer, Berlin (2004)
  • P-H Chaudouard, G Laumon, Le lemme fondamental pondéré II: Énoncés cohomologiques
  • M A de Cataldo, T Hausel, L Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$, Ann. of Math. 175 (2012) 1329–1407
  • J-M Drézet, Faisceaux cohérents sur les courbes multiples, Collect. Math. 57 (2006) 121–171
  • J-M Drézet, Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives, Math. Nachr. 282 (2009) 919–952
  • E Frenkel, E Witten, Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys. 2 (2008) 113–283
  • O García-Prada, J Heinloth, A Schmitt, On the motives of moduli of chains and Higgs bundles
  • A Grothendieck, Éléments de géométrie algébrique II: Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Etudes Sci. 8 (1961) 5–205
  • A Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, Publ. Math. Inst. Hautes Etudes Sci. (1967) 5–333
  • G Harder, M S Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75) 215–248
  • R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York (1977)
  • T Hausel, Global topology of the Hitchin system
  • T Hausel, Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998) 169–192
  • T Hausel, M Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003) 197–229
  • N Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91–114
  • B Iversen, Cohomology of sheaves, Universitext, Springer, Berlin (1986)
  • A Kapustin, E Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007) 1–236
  • F C Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31, Princeton Univ. Press (1984)
  • S L Kleiman, The Picard scheme, from: “Fundamental algebraic geometry”, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, RI (2005) 235–321
  • Q Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press (2002) Translated from the French by Reinie Erné, Oxford Science Publications
  • I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319–343
  • H Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series 18, American Mathematical Society (1999)
  • P E Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972) 337–345
  • B C Ngô, Fibration de Hitchin et endoscopie, Invent. Math. 164 (2006) 399–453
  • B C Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. (2010) 1–169
  • N Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991) 275–300
  • D Schaub, Courbes spectrales et compactifications de jacobiennes, Math. Z. 227 (1998) 295–312
  • C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math. (1994) 47–129
  • C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety II, Inst. Hautes Études Sci. Publ. Math. (1994) 5–79
  • C Simpson, The Hodge filtration on nonabelian cohomology, from: “Algebraic geometry–-Santa Cruz 1995”, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI (1997) 217–281