Hodge polynomials of moduli spaces of stable pairs on K3 surfaces

Virtual curve counts have been defined for threefolds by integration against virtual classes on moduli spaces of stable maps (Gromov-Witten theory), ideal sheaves (Donaldson-Thomas theory), and stable pairs (Pandharipande-Thomas theory). The first two theories are proven to be equivalent for toric threefolds, and all three are conjecturally equivalent for arbitrary threefolds. One may ask whether there is such a correspondence for surfaces. In particular, the Gromov-Witten theory of K3 surfaces has recently been computed by Maulik, Pandharipande, and Thomas it is governed by quasimodular forms and is closely related to invariants obtained from the moduli spaces of rank r = 0 stable pairs with n = 1 sections. We compute the Hodge polynomials of the moduli spaces of stable pairs for higher rank r &ge 0 and level n &ge 1, and explore the modularity properties and relationship to Gromov-Witten theory.