HIGHER JET EVALUATION TRANSVERSALITY OF J-HOLOMORPHIC CURVES

Title & Authors
HIGHER JET EVALUATION TRANSVERSALITY OF J-HOLOMORPHIC CURVES
Oh, Yong-Geun;

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, $\small{\omega}$)). Using this transversality result, we prove that there exists a subset $\small{\cal{J}^{ram}_{\omega}\;{\subset}\;\cal{J}_{\omega}}$ of second category such that for every $\small{J\;{\in}\;\cal{J}^{ram}_{\omega}}$, 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 $\small{J\;{\in}\;\cal{J}^{ram}_{\omega}}$ there are only a finite number of ramication profiles of J-holomorphic curves in a given homology class $\small{\beta\;{\in}\;H_2}$(M; $\small{\mathbb{Z}}$) and provide an explicit upper bound on the number of ramication proles in terms of $\small{c_1(\beta)}$ and the genus g of the domain surface.
Keywords
higher jet evaluation transversality;holomorphic jets;ramication profiles;distributions with points support;
Language
English
Cited by
References
1.
J.-F. Barrard, Courbes pseudo-holomorphes equisingulieres en dimension 4, Bull. Soc. Math. France 128 (2000), no. 2, 179-206.

2.
A. Floer, The unregularized gradient ow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775-813.

3.
M. Hirsch, Differential Topology, Springer-Verlag, 1976.

4.
L. Hormander, The Analysis of Linear Partial Differential Operators. II, Springer- Verlag, Berlin, 1983.

5.
I. M. Gelfand and G. E. Shilov, Generalized Functions, vol 2, Academic Press, New York and London, 1968.

6.
D. McDuff, Examples of symplectic structures, Invent. Math. 89 (1987), no. 1, 13-36.

7.
D. McDuff, Singularities of J-holomorphic curves in almost complex 4-manifolds, J. Geom. Anal. 2 (1992), no. 3, 249-266.

8.
Y.-G. Oh, Seidel's long exact sequence on Calabi-Yau manifolds, submitted, arXiv: 1002.1648.

9.
Y.-G. Oh and K. Zhu, Embedding property of J-holomorphic curves in Calabi-Yau man- ifolds for generic J, Asian J. Math. 13 (2009), no. 3, 323-340.

10.
J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic Curves in Symplectic Geometry, 165-189, Progr. Math., 117, Birkhauser, Basel, 1994.