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

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Given a symplectic three-fold (M,ω) we show that for a generic almost complex structure J which is compatible with ω, there are finitely many J-holomorphic curves in M of any genus g≥ 0 representing a homology class β in ₂(M,) with c₁(M).β=0, provided that the divisibility of β is at most 4 (i.e. if β=nα with α∈ H₂(M,) and n∈ then n≤ 4). Moreover, each such curve is embedded and 4-rigid.