Foliations modelling nonrational simplicial toric varieties

Details Foliations modelling nonrational simplicial toric varieties Foliations modelling nonrational simplicial toric varieties

2011-08-08

arxiv.org/abs/1108.1637v1

Mathematics

Complex Variables

20 pages, 4 figures

Scientific paper

We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. 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 can apply El Kacimi's basic version of the hard Lefschetz theorem. This allows us to reformulate Stanley's argument for the positivity of the $g$-vector: we give a proof that unifies rational and nonrational cases, and avoid singularities altogether.

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