Some remarks on the filling in problem for degenerations
type: Item
instance of: scholarly article
title: Some remarks on the filling in problem for degenerations
zbMATH Open document ID: 0583.32022
DOI: 10.1215/S0012-7094-85-05230-5
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32D99
Mathematics Subject Classification ID: 32H99
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3933444
zbMATH Keywords: degenerations
zbMATH Keywords: bimeromorphic map
zbMATH Keywords: Kähler manifold
zbMATH Keywords: birational map
zbMATH Keywords: irreducible curve
zbMATH Keywords: complex manifold
zbMATH Keywords: filling in problem
author: Robert Friedman
reviewed by: Michael Th. Rassias
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3933444