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 Symmetry.

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Let X be an algebraic variety and let S be a tropical variety associated to X. We study the tropicalization map from the moduli space of stable maps into X to the moduli space of tropical curves in S. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in X, the associated tropical curves in S deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.

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