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

Abstract

For G = GL 2 , PGL 2 , SL 2 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.

Hodge polynomials of SL $$(2,\mathbb{C })$$ -character varieties for curves of small genus

“…2 ) = (q − 1)e(X 2 )e(X 4,λ 0 ). (3) Z 3 2 , given by b = 0, c = 0. It gives a similar contribution e(Z 3 2 ) = e(Z 2 2 ).…”

“…Then e(M Id ) = q 6 + 17q 4 − 26q 3 + 67q 2 + 26q − 65 . e(M − Id ) = q 6 − 2q 4 − 30q 3 − 2q 2 + 1 , e(M J + ) = q 8 − 3q 6 − 4q 5 − 39q 4 − 4q 3 − 15q 2 , e(M J − ) = q 8 − 3q 6 + 15q 5 + 6q 4 + 45q 3 , e(M ξ λ ) = q 8 + q 7 − 2q 6 + 13q 5 − 26q 4 + 13q 3 − 2q 2 + q + 1 .…”

Prym varieties of spectral covers

“…, where p n is the smallest prime divisor of n. By studying the cohomology long exact sequence of the pair (M, M ne ) we see that the restriction map Further, we note that any generic (c n − 1)-dimensional subvariety Λ cn−1 of A 0 will be disjoint from ∪A 0 Γ thus a cohomology class η ∈ H * (M; Q) which is not invariant under Pic 0 (C)[n] (we call such classes variant) must satisfy η| h −1 (Λ cn−1 ) = 0 and so by [dCHM,Theorem 1.4.8] a variant class of degree i must have perversity at most i − c n . By [dCHM,Theorem 1.4.12] this already implies i ≥ 2c n (which as we noted above in (12) also follows from the cohomology long exact sequence of (M, M ne )). which contradicts [GHS,Theorem 1], which proves that there are no variant classes in the middle cohomology H dim(M) (M; Q).…”

Twisted spectral correspondence and torus knots

“…The twisted wild version of the non-Abelian Hodge correspondence with P = W conjecture [dCHM12,HMW16] identifies P n,ℓ,r (u, v) with the weighted Poincaré polynomial W P n,ℓ,r (u, v) of the corresponding variety of Stokes data S n,ℓ,r , which is defined by…”

“…As in [CDP14, CDDP15,DDP18], the relation between the topology of character varieties and BPS states in string theory is based on the P = W conjecture of de Cataldo, Hausel and Migliorini [dCHM12], and a spectral construction for Higgs bundles. In particular, the spectral correspondence for twisted irregular Higgs bundles used in the present context is one of the main novelties of this paper.…”