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).