  On Kahler 4-manifolds, not necessarily compact or of finite topological type,
we obtain relationships between the fundamental group of compact embedded
Levi-flat or pseudoconvex submanifold and the fundamental group of the ambient
manifold $M^4$. When a Levi-flat submanifold $V^3$ has finite fundamental group
then $\pi_1(M^4)=\iota_*\pi_1(V^3)$; when a non-separating pseudoconvex
submanifold $V^3$ has finite fundamental group, then
$\pi_1(M^4)=\iota_*\pi_1(V^3)\rtimes\mathbb{Z}$. As applications, if a Kahler
manifold (compact or not) has an embedded holomorphic $\mathbb{P}^1$ of
positive self-intersection, it must intersect all other holomorphic $P^1$ of
non-negative self-intersection, the fundamental group of $M^4$ is trivial, and
no ALE or ALF ends exist. If a Levi-flat submanifold and an embedded
holomorphic $\mathbb{P}^1$ of positive self-intersection both exist, they
intersect. The total number of ALE plus ALF ends is zero or one regardless of
what other kinds of ends exist. We provide examples, such as a 2-ended
scalar-flat Kahler metric conformal to the Taub-NUT.
