We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMBmanifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.
Abstract. We give a proof of the Hard Lefschetz Theorem for orbifolds that does not involve intersection homology. We use a foliated version of the Hard Lefschetz Theorem due to El Kacimi.
Bosio generalized the construction by López de Medrano-VerjovskyMeersseman (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 Białynicki-Birula andŚwiȩcicka. This point of view yields new results on the geometry of LVM and Bosio's manifolds.López de Medrano and Verjovsky discovered in 1997 a way to construct many compact complex manifolds (cf. [14]). They start with a C-action on CP n induced by a diagonal linear vector field (satisfying certain properties), and find an invariant open dense subset U ⊂ CP n where the action is free, proper and cocompact, so the quotient N = U/C is a compact complex manifold. Their construction was extended to C m -actions by Meersseman in [15], yielding a vast family of non-Käh-ler 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].
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