Intersection Cohomology of Rank 2 Character Varieties of Surface Groups

Abstract

For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

A blowing-up formula for the intersection cohomology of the moduli of rank 2 Higgs bundles over a curve with trivial determinant

“…Recently, IP t (M) was obtained in [Ma21] by using other way than ours. The author of [Ma21] first calculated E-polynomial of the compactly supported intersection cohomology of M and then proved the purity of IH * (M) from the observation of semiprojectivity of M. He used the purity of IH * (M) and the Poincaré duality to calculate IP t (M).…”

P=W conjectures for character varieties with symplectic resolution

“…-There is no degeneration from K(A, v) with Mukai vector v = (0, nX, n(g − 1)) with g > 2 to M Dol (X, SL n ) for dimensional reason. However, K(A, v) and M Dol (X, SL n ) have the same type of singularities: they are stably isosingular in the sense of [59,Def. 2.6 & Th.…”

Intersection cohomology of character varieties for punctured Riemann surfaces

“…Felisetti [Fel21] computed the intersection cohomology in rank n = 2 and genus g = 2. Mauri [Mau21a] generalized the computation to rank n = 2 and arbitrary genus. Felisetti-Mauri [FM22] proved the P = W conjecture for intersection cohomology in genus g = 1 and arbitrary rank n, and in genus g = 2 and rank n = 2.…”