Higher jet evaluation transversality of $J$-holomorphic curves

TL;DR

This paper proves higher jet evaluation transversality for $J$-holomorphic curves in symplectic manifolds, leading to finiteness results on ramification profiles and dimension reductions in moduli spaces for generic almost complex structures.

Contribution

It establishes a general stratawise higher jet evaluation transversality result and derives finiteness and dimension reduction properties for $J$-holomorphic curves with ramification.

Findings

Existence of a generic set of almost complex structures with transversality properties.

Dimension reduction of moduli spaces depending on ramification profiles.

Finiteness of ramification profiles in a given homology class.

Abstract

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 $(M,\omega)$. Using this transversality result, we prove that there exists a subset $\CJ_\omega^{ram} \subset \CJ_\omega$ of second category such that for every $J \in \CJ_\omega^{ram}$, 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 $J \in \CJ_\omega^{ram}$ there are only a finite number of ramification profiles of $J$-holomorphic curves in a given homology class $\beta \in H_2(M;\Z)$ and provide an explicit upper bound on the number of ramification profiles in terms of $c_1(\beta)$ and the genus $g$ of the domain surface.