On connectedness of non-klt loci of singularities of pairs

Abstract

We study the non‑klt locus of singularities of pairs. We show that given a pair $(X,B)$ and a projective morphism $X \to Z$ with connected fibres such that $-(K_X + B)$ is nef over $Z$, the non‑klt locus of $(X,B)$ has at most two connected components near each fibre of $X \to Z$. This was conjectured by Hacon and Han.

In a different direction we answer a question of Mark Gross on connectedness of the non‑klt loci of certain pairs. This is motivated by constructions in Mirror Symmetry.