Non-kahler manifolds and GIT-quotients

Abstract

Bosio generalized the construction by Lopez de Medrano-Verjovsky- Meersseman (LVM) of a family of non-algebraic compact complex manifolds of any dimension. We describe how to construct the generalized family from certain Geometric Invariant Theory (GIT) quotients. We show that Bosio's generalization parallels exactly the extension from Mumford's GIT to the more general GIT developed by Bialynicki-Birula and Swiecicka. This point of view yields new results on the geometry of LVM and Bosio's manifolds. Lopez de Medrano and Verjovsky discovered in 1997 a way to construct many compact complex manifolds (cf. [ 14]). They start with a C-action on CPn induced by a diagonal linear vector field ( satisfying certain properties), and find an invariant open dense subset U. CPn where the action is free, proper and cocompact, so the quotient N = U/C is a compact complex manifold. Their construction was extended to Cm-actions by Meersseman in [ 15], yielding a vast family of non-Kahler compact manifolds, called LVM-manifolds. These manifolds lend themselves very well to various computations, and a thorough study of their properties is conducted in [ 15]. Furthermore, they are ( deformations of) a very natural generalization of Calabi-Eckmann manifolds. Finally, the topology of LVM-manifolds can be extraordinarily complicated: we refer to [ 5] for the most recent results about a study started off in [ 20] and [ 14].