Singular surfaces and the rigidity of \({\mathbb{P}}_ 3\)
type: Item
instance of: scholarly article
title: Singular surfaces and the rigidity of \({\mathbb{P}}_ 3\)
zbMATH Open document ID: 0679.32022
DOI: 10.1007/BF01173702
author: Thomas Peternell
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155368
review text: The paper contains two proofs of the rigidity of \({\mathbb{P}}_ 3\) with respect to global deformations i.e. the following Theorem: Let X b a complex 4-dimensional manifold, \(\Delta\) \(\subset {\mathbb{C}}\) the unit disc, \(\pi\) : \(X\to \Delta\) a smooth proper map such that any fiber \(X_ t=\pi^{-1}(t)\cong {\mathbb{P}}_ 3\) for \(t\neq 0\). Then \(X_ 0\) is biholomorphic equivalent with \({\mathbb{P}}_ 3.\) The first proof which can be modified to the three-dimensional smooth quadric case uses the study of degenerations \(Y_ 0\subset X_ 0\) of the hyperplanes \({\mathbb{P}}_ 2\cong Y_ t\subset X_ t\) to exclude the case \(Y_ 0\) singular (the theorem is known if \(Y_ 0\) is non- singular). The second proof studies the degenerations of lines in \(Y_ t\approx {\mathbb{P}}_ 2\) to prove that \({\mathcal O}_{X_ 0}(1)\) is numerically effective (this is enough by a theorem due also to the author).
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 4113000
zbMATH Keywords: deformation of a complex structure
zbMATH Keywords: rigidity of the complex projective space
reviewed by: Nicolae Mihalache
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4113000