Refined curve counting on algebraic surfaces

Most of the results discussed in this talk are conjectural.
Let L be ample line bundle on an a projective algebraic surface S. Let g be
the genus of a smooth curve in the linear system |L|. If L is suffciently ample with
respect to d, the number of n_{L,d}of d-nodal curves in a general d-dimensional sublinear
system of |L| will be  finite. Kool-Shende-Thomas use relative Hilbert
schemes of points of the universal curve over |L| to define the numbers
n_{L,d} as BPS invariants and prove a conjecture of mine about their
generating function.

We use the generating function of the chi_y-genera of these relative
Hilbert schemes to define and study refined curve counting
invariants N_{L,g}(y), which are now polynomials in y, with N_{L,d}(1)=n_{L,d}. If S is a K3 surface
we relate these invariants to the Donaldson-Thomas invariants considered by
Maulik-Pandharipande-Thomas.

In the case of real toric surfaces we see that the refined
invariants interpolate between the Gromov-Witten invariants (at y = 1) and the Welschinger invariants
(which count real curves) at y = -1.