Generating functions for the number of curves on Abelian surfaces

Abstract

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we also count the number of curves in the fixed linear system |C| passing through g-2 generic points. These two numbers, (defined appropriately) only depend on n and g where 2n=C^2+2-2g and not on the particular X or C (n is the number of nodes when a curve is nodal and reduced). Gottsche conjectured that certain quasi-modular forms are the generating functions for the number of curves in a fixed linear system. Our theorem proves his formulas and shows that (a different) modular form also arises in the problem of counting curves without fixing a linear system. We use techniques that were developed in our earlier paper for similar questions on K3 surfaces. The techniques include Gromov-Witten invariants for families and a degeneration to an elliptic fibration. One new feature of the Abelian surface case is the presence of non-trivial Pic^0(X). We show that for any surface S the cycle in the moduli space of stable maps defined by requiring that the image of the map lies in a fixed linear system is homologous to the cycle defined by requiring the image of the map meets b_1 generic loops in S representing the generators of the first integral homology group (mod torsion).