Complex structures on four-manifolds with symplectic two-torus actions

We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a 4-manifold with a symplectic 2-torus action admits an invariant complex structure and give an identification of those that do not admit a K¨ahler structure with Kodaira’s class of complex surfaces which admit a nowhere vanishing holomorphic (2, 0)-form, but are not a torus or a K3 surface