Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation
type: Item
instance of: scholarly article
title: Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation
zbMATH Open document ID: 1071.32022
author: Jun-Muk Hwang
author: Ngaiming Mok
published in: Inventiones Mathematicae
publication date: 2005-06-13T00:00:00Z
full work available at URL: http://hdl.handle.net/10722/48613
review text: The authors mainly prove the following theorem: Let \(\pi:X\to \Delta\) be a smooth and projective morphism from a complex manifold \(X\) to the unit disc \(\Delta\). Suppose for any \(t\in \Delta-\{0\}\), the fiber \(X_t=\pi^{-1}(t)\) is biholomorphic to a rational homogeneous space \(S\) of Picard number 1. Then the central fiber \(X_0\) is also biholomorphic to \(S\). Of independent interest are their results related to the following conjecture: Let \(X\) be a Fano manifold of Picard number 1. Then, at a general point \(x\) on \(X\), there does not exist any nonzero holomorphic vector field vanishing at \(x\) to the order \(\geq 3\). They prove the conjecture in the present article under the assumption that the variety of minimal rational tangents at a general point is nonsingular, irreducible and linearly nondegenerate.
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 2175703
zbMATH Keywords: rational homogeneous space
zbMATH Keywords: Picard number
zbMATH Keywords: Kähler deformation
MaRDI profile type: Publication
OpenAlex ID: W1998438394
cites work: Some theorems on actions of algebraic groups
cites work: Symmetric compact complex spaces
cites work: The classification of the irreducible complex algebras of infinite types
cites work: Uniruled projective manifolds with irreducible reductive G-structures.
cites work: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation
cites work: Q4949409
cites work: Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1.
cites work: Deformation rigidity of the rational homogeneous space associated to a long simple root
cites work: Rigidity of homogeneous contact manifolds under Fano deformation.
cites work: On the vanishing orders of vector fields on Fano varieties of Picard number 1
cites work: On Stability of Compact Submanifolds of Complex Manifolds
cites work: Families of singular rational curves
cites work: Q4859907
cites work: Q5517524
cites work: Q5547669
cites work: On Hartshorne's conjecture
cites work: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3
cites work: Geometry Associated with Semisimple Flat Homogeneous Spaces
cites work: Q3998356
cites work: Vanishing theorems on compact hermitian symmetric spaces
cites work: The infinite groups of Lie and Cartan. I: The transitive groups
cites work: Q4277782
cites work: A cohomological characterization of \({\mathbb{P}}^ n\)
cites work: Q4281152
cites work: Q3140224
DOI: 10.1007/S00222-004-0417-9
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description: scientific article; zbMATH DE number 2175703