Additivity and Relative Kodaira Dimensions

Abstract

In this paper we introduce the notion of relative Kodaira dimension for a symplectic 4-manifold M with a possibly disconnected embedded symplectic surface F. One notable feature is that the sphere components of F have to be discarded, which resembles the definition of Thurston norm of 3-manifolds. To define it we need to establish several homological properties of embedded symplectic surfaces. We also formulate the notion of relative minimal model, and prove the existence and the somewhat surprising uniqueness.