Higher jet evaluation transversality of $J$-holomorphic curves

In this paper, we establish general stratawise higher jet evaluation transversality of $J$-holomorphic curves for a generic choice of almost complex structures $J$ tame to a given symplectic manifold $(M,ω)$. Using this transversality result, we prove that there exists a subset $\CJ_ω^{ram} \subset \CJ_ω$ of second category such that for every $J \in \CJ_ω^{ram}$, the dimension of the moduli space of (somewhere injective) $J$-holomorphic curves with a given ramification profile goes down by $2n$ or $2(n-1)$ depending on whether the ramification degree goes up by one or a new ramification point is created. We also derive that for each $J \in \CJ_ω^{ram}$ there are only a finite number of ramification profiles of $J$-holomorphic curves in a given homology class $β\in H_2(M;\Z)$ and provide an explicit upper bound on the number of ramification profiles in terms of $c_1(β)$ and the genus $g$ of the domain surface.

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