Family Gromov-Witten Invariants for Kahler Surfaces

The usual Gromov-Witten invariants are zero for K"{a}hler surfaces with $p_g\ge 1$. In this paper we use analytic methods to define Family Gromov-Witten Invariants for K"{a}hler surfaces. We prove that these are well-defined invariants of the deformation class of the K"{a}hler structure.