Invariants for Lagrangian tori

Abstract

We define an simple invariant $\lambda \left(T\right)$ of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4–manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that $\lambda \left(T\right)$ is actually a $C^{\infty }$ invariant. In addition, this invariant is used to show that many symplectic 4–manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3–surface obtained from knot surgery using the trefoil knot.