Symplectic surfaces in a fixed homology class

Abstract

The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there are many instances when K can be representated by infinitely many non-isotopic symplectic tori.