A geometric characterization of toric singularities

Abstract

Given a projective contraction $\pi :X\to Z$ and a log canonical pair $(X,B)$ such that $-(K_X+B)$ is nef over a neighborhood of a closed point $z\in Z$, one can define an invariant, the complexity of $(X,B)$ over $z\in Z$, comparing the dimension of X and the relative Picard number of $X/Z$ with the sum of the coefficients of those components of B intersecting the fiber over z. We prove that, in the hypotheses above, the complexity of the log pair $(X,B)$ over $z\in Z$ is non-negative and that when it is zero then $(X,\lfloor B\rfloor )\to Z$ is formally isomorphic to a morphism of toric varieties around $z\in Z$. In particular, considering the case when π is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities, thus resolving a conjecture due to Shokurov.