  We prove a structure theorem for the Gromov-Witten invariants of compact
Kahler surfaces with geometric genus $p_g>0$. Under the technical assumption
that there is a canonical divisor that is a disjoint union of smooth
components, the theorem shows that the GW invariants are universal functions
determined by the genus of this canonical divisor components and the
holomorphic Euler characteristic of the surface. We compute special cases of
these universal functions.
