Hodge-Elliptic genera, K3 surfaces and Enumerative Geometry

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of K3 x E. In this note we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so called Hodge-Elliptic genus which is then related to the counting of refined BPS states of K3 x E. We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations.