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, <TEX>$\omega$</TEX>)). Using this transversality result, we prove that there exists a subset <TEX>$\cal{J}^{ram}_{\omega}\;{\subset}\;\cal{J}_{\omega}$</TEX> of second category such that for every <TEX>$J\;{\in}\;\cal{J}^{ram}_{\omega}$</TEX>, the dimension of the moduli space of (somewhere injective) J-holomorphic curves with a given ramication prole goes down by 2n or 2(n - 1) depending on whether the ramication degree goes up by one or a new ramication point is created. We also derive that for each <TEX>$J\;{\in}\;\cal{J}^{ram}_{\omega}$</TEX> there are only a finite number of ramication profiles of J-holomorphic curves in a given homology class <TEX>$\beta\;{\in}\;H_2$</TEX>(M; <TEX>$\mathbb{Z}$</TEX>) and provide an explicit upper bound on the number of ramication proles in terms of <TEX>$c_1(\beta)$</TEX> and the genus g of the domain surface.