On a conjecture of Shokurov: Characterization of toric varieties

Abstract

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let $(X,D=\sum d_iD_i)$ be a three-dimensional log variety such that $K_X+D$ is numerically trivial and $(X,D)$ has only purely log terminal singularities. In this situation we prove the inequality \{center} $\sum d_i\le \rk\Weil(X)/(\operatorname{algebraic equivalence}) +\dim(X)$. \{center} We describe such pairs for which the equality holds and show that all of them are toric.