The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings
arXiv:1610.06870
Abstract
We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of for all its minimal symplectic fillings. This is motivated by the bound of Stipsicz for Stein fillings. Finally, we discuss various aspects of exact self cobordisms of fillable 3-manifolds.