Refined curve counting on algebraic surfaces
Seminar Room 1, Newton Institute
Let $L$ be ample line bundle on an an algebraic surface $X$. If $L$ is sufficiently ample wrt $d$, the number $t_d(L)$ 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 $t_d(L)$ 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 $\chi_y$-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.