On characterization of toric varieties

Abstract

We study the conjecture due to V.\,V. Shokurov on characterization of toric varieties. We also consider one generalization of this conjecture. It is shown that none of the characterizations holds true in dimension $\ge 3$. Some weaker versions of the conjecture(s) are verified.

14 pages, 1 figure; minor details to the proof of Lemma 4.4 are added; the surface Y in subsection A.1 is modified and the fact that Y is not the plane is proved by computation of its Makar-Limanov invariant (one more reference added)