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Cyclic coverings: Deformation and Torelli theorem. (English) Zbl 0593.32017
Main theorem. Let X be a compact complex manifold which can be represented as a k-fold cyclic covering \(f: X\to Z\) of a homogeneous rational manifold Z of rank one. Denote by \[ \begin{matrix} X &&\overset {g} \hookrightarrow && X \\ &f\searrow && \swarrow \pi \\ &&Z \end{matrix} \] the embedding of f into the total space \(Y={\mathbb{V}}(F^{-1})\) of the line bundle \(F={\mathcal O}_ Z(m)\in Pic(Z)\) whose dual \(F^{-1}\) generates the \({\mathcal O}_ Z\)-algebra \(f_*{\mathcal O}_ X\). Then for every complex germ S both of the maps \(Def(X/Y)(S)\to Def(X/Z)(S)\to Def(X)(S)\) are surjective, hence any deformation of X can be represented as a covering of Z which is embeddable into Y, in each of the following cases: i) dim \(Z\geq 3\) and \(K_ Z\otimes F\geq 0\) where \(K_ Z\) is the canonical bundle, ii) dim Z\(=2\) i.e. \(Z={\mathbb{P}}_ 2\), and X is not a K3-surface iii) \(Z={\mathbb{G}}(r,N)\) is a Grassmannian with \(2\leq r\leq N/2\) and (r,N)\(\neq (2,4)\), or \(Z={\mathbb{P}}_ n\), \(n\geq 3\).

MSC:
32G07 Deformations of special (e.g., CR) structures
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14E20 Coverings in algebraic geometry
32L20 Vanishing theorems
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
32H99 Holomorphic mappings and correspondences
32Q99 Complex manifolds
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