Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$

Pages 1329-1407 from Volume 175 (2012), Issue 3 by Mark Andrea A. de Cataldo, Tamás Hausel, Luca Migliorini

Abstract

For $\mathrm{G}=\mathrm{GL}_2,\mathrm{PGL}_2, \mathrm{SL}_2$ we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted $\mathrm{G}$-Higgs bundles on a compact Riemann surface $C$ agrees with the weight filtration on the rational cohomology of the twisted $\mathrm{G}$ character variety of $C$ when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

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Keywords
, , , ,
DOI
https://doi.org/10.4007/annals.2012.175.3.7
MR
2912707
zbMATH
06051272
Milestones
Received: 13 May 2010
Revised: 29 March 2011
Accepted: 3 June 2011
Published online: 1 May 2012

Authors

Mark Andrea A. de Cataldo

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651

Tamás Hausel

Mathematical Institute, University of Oxford, 24-20 St. Giles', Oxford OX1 3LB, United Kingdom

Luca Migliorini

Dipartimento de Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy