On Defining Functions and Cores for Unbounded Domains II

Abstract

We continue the study of the core $${\mathfrak {c}}(\Omega )$$ of a complex manifold $$\Omega $$ which was initiated in Harz et al. (Math Z, doi:10.1007/s00209-016-1792-9, 2016). The main focus lies on the investigation of pseudoconcavity and Liouville-type properties of the core. We construct examples of strictly pseudoconvex domains $$\Omega \subset {\mathbb {C}}^n$$ with nonempty core which show that, first, $${\mathfrak {c}}(\Omega )$$ may not contain any analytic variety of positive dimension, and, second, it can happen that $${\mathfrak {c}}(\Omega )$$ is connected but not every smooth and bounded from above plurisubharmonic function on $$\Omega $$ is constant on $${\mathfrak {c}}(\Omega )$$. However, we show that for each complex manifold of dimension two, there exists a decomposition of the core into disjoint connected sets which are 1-pseudoconcave in $$\Omega $$ and satisfy a Liouville-type theorem with respect to smooth plurisubharmonic functions on $$\Omega $$. We also introduce cores of higher order, by requiring not only failure of strict plurisubharmonicity of smooth and bounded from above plurisubharmonic functions $$\varphi $$ on $$\Omega $$, but also instead by prescribing an upper bound k for the rank of the complex Hessian of $$\varphi $$, with k possibly different from $$\dim _{\mathbb {C}}\Omega -1$$. We show that this sharpening of the definition does not lead to better results on pseudoconcavity of the core.