Abstract. We obtain a global version and a twisted version (in the sense of [BP05]) of the main theorem of [BKR01].

We prove a numerical characterization of n for varieties with at worst isolated local complete intersection quotient singularities. In dimension 3, we prove such a numerical characterization of 3 for normal -Gorenstein projective varieties.

We present a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.

Abstract. We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne-Mumford stacks. We also obtain a sharper bound on −(KX + D)-degree of (KX + D)-negative extremal rays for projective Q-factorial log terminal threefold pair (X, D).