Sergey Mozgovoy

Motivic Donaldson–Thomas invariants and the Kac conjecture

We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.

Abelian Quiver Invariants and Marginal Wall-Crossing

We prove the equivalence of (a slightly modified version of) the wall-crossing formula of Manschot, Pioline and Sen and the wall-crossing formula of Kontsevich and Soibelman. The former involves abelian analogues of the motivic Donaldson–Thomas type invariants of quivers with stability introduced by Kontsevich and Soibelman, for which we derive positivity and geometricity properties.

Arithmetic of character varieties of free groups

Via counting over finite fields, we derive explicit formulas for the E-polynomials and Euler characteristics of GL(d)- and PGL(d)-character varieties of free groups. We prove a positivity property for these polynomials and relate them to the number of subgroups of finite index.

Intersection cohomology of moduli spaces of sheaves on surfaces

We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincaré polynomials of the moduli spaces to Donaldson–Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincaré polynomials for sheaves with rank two and three on ruled surfaces.