Symplectic geometry on moduli spaces of J-holomorphic curves

TL;DR

This paper develops a symplectic form on the moduli space of J-holomorphic curves in a symplectic manifold, providing conditions for its non-degeneracy and applications to existence results.

Contribution

It introduces a new symplectic form on the space of immersed symplectic surfaces and establishes conditions for it to be symplectic on moduli spaces of J-holomorphic curves.

Findings

The 2-form on the space of immersed surfaces is closed and non-degenerate up to reparametrizations.

Conditions on J ensure the form restricts to a symplectic form on the moduli space.

Derived criteria for the existence of J-holomorphic curves in a given homology class.

Abstract

Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.