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

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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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