The gluing construction for normally generic J-holomorphic curves

Abstract

Under an assumption of normal genericity, we show that a stable J-holomorphic curve has, in the space of homologous curves of the same genus, a locally Euclidean neighbourhood of the expected dimension given by Riemann-Roch. In dimension 4, the normal genericity condition is satisfied in by every curve in CP2 (for an almost complex structure homotopic with the standard one) which has only nodes as singularities. This leads in particular to a solution of the symplectic isotopy problem for surfaces of degree 3.

24 pages. This paper is, apart for very minor corrections, unchanged from the ENS Lyon preprint of March 2000. Since its appearance, V. Shevchishin made much progress on the symplectic isotopy problem (cf. math.SG/0010262)

References in corpus (2)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Higher genus symplectic invariants and sigma model coupled with gravity

Cited by in corpus (9)

• Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
• Pseudoholomorphic curves and the symplectic isotopy problem
• Kodaira dimension and symplectic sums
• Symplectic tori in rational elliptic surfaces
• Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four
• Moduli space of $J$-holomorphic subvarieties
• The topology of the space of J-holomorphic maps to CP^2
• Symplectic log Calabi-Yau surface---deformation class
• A four-dimensional mapping class group relation