Intersection cohomology of rank two character varieties of surface groups

Abstract

For G = GL2, SL2, PGL2 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.

Lagrangian fibrations

“…However, this may fail if N is singular, e.g. if f is a normalization of a nodal cubic, even if N has Q-factorial log terminal singularities, see for instance [Mau21,Thm. 5.11].…”

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

“…The P = W conjecture by de Cataldo-Hausel-Migliorini [4] predicts that the perverse filtration associated with the Hitchin fibration h n,d : M n,d → A n,d matches the double indexed weight filtration associated with the mixed Hodge structure on M ′ n,d . Moreover, this striking phenomenon is expected to hold more generally without the coprime assumption of n, d if we work with intersection cohomology [5,11,31]:…”