Deformation families of open Calabi-Yau manifolds and Steinness

Abstract: This paper is to construct deformation families of open Calabi-Yau manifolds together with some discussions on Steinness. Firstly, we construct a nontrivial compactifiable deformation of open Calabi-Yau manifolds. Secondly, we construct a compactifiable deformation family admitting a holomorphic involution and claim the non-Steinness for all the compactifiable deformation families constructed. Finally, we give some discussions on the non-Steinness of the quasi-projective variety from the blow-up of CP2 at nine points corresponding to a conjecture by Macro Brunella and present deformation families whose fibers are Stein.