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 $g\ge 0$ 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 $\alpha \in H_2(M,\mathbb{Z})$ and $n\in \mathbb{Z}$ then $n\le 4$). Moreover, each such curve is embedded and 4-rigid.

This is a revision of the original submission. The assumption on the homology class is imposed in order to fill the gap in the original version

References in corpus (4)

• Hodge integrals and degenerate contributions
• BPS states of curves in Calabi--Yau 3--folds
• Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$
• Super-rigidity and finiteness of embedded $J$-holomorphic curves on Calabi-Yau threefolds