A rigidity theorem for homogeneous rational manifolds of rank 1. (English) Zbl 0588.14005
The authors prove the following rigidity theorem for homogeneous-rational manifolds of rank 1 under algebraic deformations: Let \(f:\quad X\to Y\) be a proper morphism of complex algebraic varieties, where Y is regular and all fibers of f are smooth algebraic varieties of constant dimension. Assume that for a closed point \(z\in Y\) the fiber \(V:=f^{-1}(z)\) is a homogeneous-rational manifold of rank 1. Then \(f^{-1}(y)\cong V\) for all closed points \(y\in Y\). Furthermore there is an integer \(r>0\) such that \({\mathcal O}_{f^{-1}(z)}(r)\) extends to an invertible sheaf H on X and \(X\hookrightarrow {\mathbb{P}}(f_*H)\).
Reviewer: K.Oeljeklaus
MSC:
14D20 | Algebraic moduli problems, moduli of vector bundles |
14M20 | Rational and unirational varieties |
32G05 | Deformations of complex structures |
14M17 | Homogeneous spaces and generalizations |
14D15 | Formal methods and deformations in algebraic geometry |
Keywords:
algebraic deformation; rigidity theorem for homogeneous-rational manifolds; morphism of complex algebraic varietiesReferences:
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