Refined curve counting on complex surfaces

Abstract

We define refined invariants which “count” nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of $K3$ and abelian surfaces. We also give a refinement of the Caporaso–Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at $y=-1$ to the Itenberg–Kharlamov–Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.