Seminar Algebra and Geometry: Jérôme Tambour (Université de Bourgogne)

LVMB manifolds and triangulations of spheres

It is not easy to construct examples of non k ̈ahler compact complex manifold. For instance, every algebraic variety, and every Riemann surface, is ka ̈hler. The classical examples of such varieties are Hopf manifolds (1948) and Calabi-Eckmann manifolds (1953) which are complex structures on product of spheres Sp × Sq (with p and q odd). Santiago Lopez de Medrano, Alberto Verjovsky and Laurent Meersseman gave a generalization of this construction. The interest of theirs manifolds, known as LVM manifolds, stand in the fact that it is practical to compute some of their topological invariants (homology and cohomology for example). They are also endowed with a very nice action of a compact torus.

The talk will mainly deal with a generalization due to Bosio of the LVM manifolds, emphazing the combinatorial aspect of the LVM manifolds. These new manifolds are known as LVMB manifolds. In particular, our aim will be to show the very strong connection between LVMB manifolds, toric varieties and triangulations of spheres.

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