Higher jet evaluation transversality of J-holomorphic curves

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 J(omega)(ram) subset of J(omega) of second category such that for every J is an element of J(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 is an element of J(omega)(ram) there are only a finite number of ramification profiles of J-holomorphic curves in a given homology class beta is an element of 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.