  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.
