Positivity of Symplectic Area for Perturbed J-holomorphic Curves

Abstract

In this paper we will prove that for a compact, symplectic manifold $(M, \omega)$ and for $\omega$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to redefine the Floer symplectic homology. In particular, in subsequent papers, we will prove the Arnold conjecture in both degenerate and non-degenerate cases with integer coefficients for general, symplectic manifolds.