Subvarieties of parallelizable manifolds
A compact complex manifold \(X\) is called parallelizable if the complex tangent bundle \(TX\) is holomorphically trivial. Equivalently, the complex Lie group of automorphisms \(G\) acts trivially and locally freely on \(X\), so that \(X \simeq G/ \Gamma\), where \(\Gamma\) is a discrete subgroup. The goal of the paper under review is to study subvarieties \(Z\) of complex parallelizable manifolds. The case of codimension one subvarieties being easy (and known), the authors study subvarieties of higher codimension. They consider a whole spectrum of related problems, such as existence of subvarieties, existence of nontrivial holomorphic vector bundles, number of closed orbits, etc. However their main interest is in studying the pluricanonical map of \(Z\) from a group theoretic viewpoint. An interesting application is that if \(G\) is simple, then \(\text{codim} Z \geq \sqrt {\text{dim}}G\), that is, the dimension of \(Z\) is bounded from above. Finally, the authors prove a version of Bloch's conjecture: if \(f : \mathbb{C} \to X\) is a holomorphic map, then \(\overline {f(\mathbb{C})}\) is an orbit of a complex Lie subgroup of \(G\).
scientific article; zbMATH DE number 537219
scientific article