Pseudoholomorphic curves and the symplectic isotopy problem

TL;DR

This paper investigates the deformation and isotopy of pseudoholomorphic curves in symplectic geometry, establishing conditions for moduli space properties and proving that certain symplectically embedded surfaces are isotopic.

Contribution

It introduces a sufficient condition for the saddle point property of the moduli space and solves the local symplectic isotopy problem for embedded pseudoholomorphic curves.

Findings

Any two symplectically embedded surfaces of degree d ≤ 6 in CP^2 are symplectically isotopic.

A sufficient condition for the saddle point property of the moduli space is established.

The local symplectic isotopy problem is solved for embedded pseudoholomorphic curves.

Abstract

The deformation problem for pseudoholomorphic curves and related geometrical properties of the total moduli space of pseudoholomorphic curves are studied. A sufficient condition for the saddle point property of the total moduli space is established. The local symplectic isotopy problem is formulated and solved for the case of imbedded pseudoholomorphic curves. It is shown that any two symplectically imbedded surfaces Sigma_0, Sigma_1 in CP^2 of the same degree d\le 6 are symplectically isotopic.