Refined tropical invariants and characteristic numbers

Abstract

We prove that the Göttsche-Schroeter and Schroeter-Shustin refined invariants specialize at $q=1$ to the enumeration of rational, resp. elliptic complex curves on arbitrary toric surfaces matching constraints that consist of points and of points with a contact element. Furthermore, we show that the refined invariant extends to the case of any genus $g\ge 2$ and either one contact constraint or points in Mikhalkin position, and it again specializes to the corresponding characteristic number at $q=1$. In the appendix we show the limitations of extending this count to a more general setting.

58 pages, 22 figures, one appendix. New references added, minor mistakes corrected, no change of the results