We study the non-klt locus of singularities of pairs. We show that
given a pair (X, B) and a projective morphism X → Z with connected fibres such
that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected
components near each fibre of X → 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