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The rational homology of toric varieties is not a combinatorial invariant. (English) Zbl 0671.52007

The author proves that for \(n\geq 3\) dimension the rational homology Betti numbers of a toric variety with singularities are not determined by the combinatorial type of the fan which determines it; i.e. by the poset formed by the cones in the fan.
The above result contrasts the positive result for nonsingular toric varieties.
Two combinatorially equivalent polytopes are given, for which the associated toric varieties have different Betti numbers.
Reviewer: L.A.Székely

MSC:

52Bxx Polytopes and polyhedra
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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