Cyclic coverings: Deformation and Torelli theorem
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 \(Z={\mathbb{P}}_ n\), \(n\geq 3\).
scientific article; zbMATH DE number 3953234
scientific article