Anton Mellit (IST Austria), Toric stratifications of character varieties

For a k-tuple of conjugacy classes in GL_n(C) we consider the corresponding genus 0 character variety, i.e. the moduli space of local systems on P^1 minus k punctures with prescribed local monodromies at the punctures. In the case when at least one of the conjugacy classes is regular semisimple and the eigenvalues are generic we construct a stratification of the character variety in such a way that each stratum is a product of symplectic torus of dimension 2(d-i) and affine space of dimension i, where 2d is the dimension of the variety. I will explain how this implies the so-called curious hard Lefschetz conjecture of Hausel, Letellier and Villegas, and how their conjecture on mixed Hodge polynomials of character varieties is related to the conjectures of Gorsky, Negut, Oblomkov, Rasmussen and Shende, which connect Khovanov-Rozansky homology of links with K-theory of the Hilbert scheme.