Hodge polynomials of the moduli space of SL (2,C) - character varieties
Seminar Room 1, Newton Institute
Let X be a compact Riemann surface of genus g. SL(2,C)-character varieties of X are rich objects which lie in the intersection of algebraic geometry, complex geometry, and differential geometry. While they are diffeomorphic to moduli spaces of Higgs bundles, as algebraic varieties they are very different. Character varieties are affine, while Higgs moduli spaces are foliated by the fibers of the Hitchin map which are compact algebraic subvarieties. There has been much work investigating the mixed Hodge structures on the cohomology groups of these character varieties, and their Hodge polynomials have been computed using number theoretical techniques.
Our goal is to compute the Hodge polynomial of SL(2,C)-character varieties, by stratifying these spaces in such a way that Hodge structure theory gives simpler formulas for the strata, so we are allowed to compute the whole polynomials in terms of the Hodge polynomials of the strata. This is work in progress with V. Muñoz and P. Newstead.