The Hodge theory of character varieties

This is an expanded version of notes from my talk at the conference “Classical Algebraic Geometry Today”, at MSRI in Berkeley, January 25-29, 2009. The talk reported on joint work with T. Hausel at Oxford and L. Migliorini at Bologna, written up in [de Cataldo et al. 2011]. Following the recommendation of the editors, this article is designed to be accessible to nonspecialists and to give a small glimpse into an active area of research. The reader is referred to the introduction of the paper just cited for more details on what follows. Let C be a nonsingular complex projective curve. We consider the following two moduli spaces associated with the moduli space of stable holomorphic rank two Higgs bundles on C of degree one (see Section 3) and the character variety, the moduli space of irreducible complex dimension two representations of subject to the condition that a loop around the chosen point is sent to —Id. There is an analogous picture associated with any complex reductive Lie group G and the above corresponds to the case. In [de Cataldo et al. 2011] only the cases are dealt with. Both M and M' are quasiprojective irreducible and nonsingular of some even dimension 2d. While M depends on the complex structure of C, M! does not. There is a proper flat and surjective map, the Hitchin map, with general fibers abelian varieties of dimension d; in particular, M is not affine: it contains complete subvarieties of positive dimension. On the other hand, M' is easily seen to be affine (it is a GIT quotient of an affine variety).