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

Given a symplectic three-fold (M,ω)(M,ω) we show that for a generic almost complex structure J compatible with ω there are finitely many J-holomorphic curves in M of genus g representing the homology class β   for every g≥0g≥0 and every β∈H2(M,Z)β∈H2(M,Z) such that c1(M)β=0c1(M)β=0 and the divisibility of β   is at most 4 (i.e. if β=nαβ=nα with α∈H2(M,Z)α∈H2(M,Z) and n∈Zn∈Z, then n≤4n≤4). Moreover, every such curve is embedded and 4-rigid.