A geometric characterization of toric singularities

Abstract

Given a projective contraction π:X→Z and a log canonical pair (X,B) such that −(KX+B) is nef over a neighborhood of a closed point z∈Z, one can define an invariant, the complexity of (X,B) over z∈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∈Z is non-negative and that when it is zero then (X,⌊B⌋)→Z is formally isomorphic to a morphism of toric varieties around z∈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.