Let L be ample line bundle on an an algebraic surface X. If L is sufficiently ample wrt d, the number td(L) of d-nodal curves in a general d-dimensional sub linear system of |L| will be finite. Kool-Shende-Thomas use the generating function of the Euler numbers of the relative Hilbert schemes of points of the universal curve over |L| to define the numbers td(L) as BPS invariants and prove a conjecture of mine about their generating function (proved by Tzeng using different methods).
We use the generating function of the χy-genera of these relative Hilbert schemes to define and study refined curve counting invariants, which instead of numbers are now polynomials in y, specializing to the numbers of curves for y=1. If X is a K3 surface we relate these invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas.

In the case of toric surfaces we find that the refined invariants interpolate between the Gromow-Witten invariants (at y=1) and the Welschinger invariants at y=−1. We also find that refined invariants of toric surfaces can be defined and computed by a Caporaso-Harris type recursion, which specializes (at y=1,−1) to the corresponding recursion for complex curves and the Welschinger invariants. This is in part joint work with Vivek Shende.