Some remarks on the filling in problem for degenerations
Suppose \(\pi^*: X^*\to \Delta^*\) is a proper smooth map with connected fibers from a complex manifold \(X^*\) to the punctured disk \(\Delta^*\). The filling in problem divides into two parts: (1) Existence: Does there exist an irreducible X containing \(X^*\) as a dense open subset and a commutative diagram \[ \begin{tikzcd}[column sep=tiny] X^\ast \ar[d,"\pi^\ast" '] \ar[r, phantom, "\subseteq" description] & X \ar[d,"\pi"] \\ \Delta^\ast \ar[r, phantom, "\subseteq" description] & \Delta \end{tikzcd} \] where \(\pi\) is proper and flat ? (2) Uniqueness: Up to bimeromorphic equivalence, how many different \(X\) are there ? The author gives some (not very deep, as the author intends) answers to (1) and (2) and illustrates the problems involved with several examples and further problems.
scientific article; zbMATH DE number 3933444