The Kodaira Dimension of Contact 3-Manifolds and Geography of Symplectic Fillings

We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. Contact 3-manifolds with distinct Kodaira dimensions behave differently when it comes to the geography of various kinds of symplectic fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2\chi +3\sigma $ for all of its minimal symplectic fillings. This is motivated by Stipsicz’s result in [38] for Stein fillings. Finally, we discuss various aspects of exact self-cobordisms of fillable contact 3-manifolds.