On character varieties of singular manifolds

Abstract

In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of G-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group G, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case G = SL2(k), the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios.

Virtual Classes of Character Stacks

“…Hence, this map is actually the multiplication map by a polynomial of Z[q] and this polynomial turns out to be precisely the E-polynomial of the representation variety e(R G (W )). This work has been substantially extended in [15] and [16], where the TQFT was adapted to work also in the parabolic setting, even for non-generic parabolic structures, in [17] to surfaces with conic singularities, in [46] for non-orientable surfaces and in [20] for G the group of upper triangular matrices of rank ≤ 4.…”