Existence of Symplectic Surfaces

TL;DR

This paper proves that in high-dimensional symplectic manifolds, every positive homology class can be represented by an immersed symplectic surface, with embedded representatives possible in higher dimensions, and explores conditions for embeddings in four dimensions.

Contribution

It establishes the existence of immersed symplectic surfaces representing homology classes and provides conditions for embedded representatives in various dimensions.

Findings

Every degree 2 homology class with positive symplectic area is represented by an immersed symplectic surface.

Embedded symplectic surfaces exist in dimensions at least 6.

Conditions for embedded symplectic representatives in dimension 4 are analyzed.

Abstract

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4.