Singular surfaces and the rigidity of \({\mathbb{P}}_ 3\)
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).
rigidity of the complex projective space
scientific article; zbMATH DE number 4113000
scientific article