On finiteness and rigidity of J-holomorphic curves in symplectic three-folds

Abstract

Given a symplectic three-fold $(M,\omega)$ we show that for a generic almost complex structure $J$ which is compatible with $\omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class $\beta$ in $\H_2(M,\Z)$ with $c_1(M).\beta=0$, provided that the divisibility of $\beta$ is at most 4 (i.e. if $\beta=n\alpha$ with $\alpha\in H_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$). Moreover, each such curve is embedded and 4-rigid.