Abstract
In this paper, we first study generalizations of Fujita vanishing theorems for q-ample divisors. We then apply them to study positivity of subvarieties with nef normal bundle in the sense of intersection theory.After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties.We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. This cone contains the movable cone and shares similar intersection-theoretic properties with it, thanks to the aforementioned properties of nef subvarieties.On the other hand, we prove that if Y⊂X is an ample subscheme of codimension r and D|Y is q-ample, then D is (q+r)-ample. This is analogous to a result proved by Demailly-Peternell-Schneider and Küronya.
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