Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in $C{P}^2$

Abstract

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $C{P}^2$ coincides with the whole $C{P}^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve are improved. We introduce a natural holomorphic structure on the pulled back tangent bundle, in which the differential of a ps.-hol. map in an analytic morphism and describe cusps of ps.-hol. curves using Bennequin index.