  On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form $\a$, there is
an almost complex structure associated with $\a$. We show how this implies
vanishing theorems for the Gromov-Witten invariants of $X$. This extends the
approach, used in \cite{lp} for K\"{a}hler surfaces, to higher dimensions.
