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

Abstract

For G = GL2, PGL2, SL2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted 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.

Prym varieties of spectral covers

“…Proof If X a is integral, this is shown in [3, Proposition 3.6] using the irreducibility of the polynomial s a defined in (6). In the general case we first show the statement when X a is irreducible but non-reduced and secondly when X a is reducible.…”

E-Polynomials of the SL(2, ℂ)-Character Varieties of Surface Groups

“…A key aspect is that the Riemann-Hilbert map taking a flat connection to its monodromy is not algebraic, which is reflected in the fact that the mixed Hodge structure on the cohomology of the character variety is different from the Dolbeault and de Rham cases, where they are agree and are pure [11]. A explanation of this phenomenon in terms of the perverse filtration associated to the Hitchin map was conjectured and proved for n = 2 in [3]. Motivated by mirror symmetry considerations [12], Hausel and Rodriguez-Villegas [10] computed the E-polynomials of M B for G = GL(n, C), a specialization of the mixed Hodge polynomial that is built from the mixed Hodge numbers of the variety.…”

BPS States, Torus Links and Wild Character Varieties

“…The latter is constructed using relative Hodge theory [18] for the Hitchin map. This conjecture was proven in [17] for rank two Hitchin systems on curves without marked points. As in [16,14] this identification plays a central role in the string theoretic approach to the cohomology of wild character varieties.…”