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The goal of this work is to study positivity of subvarieties with nef normalbundle in the sense of intersection theory. After Ottem's work on amplesubschemes, we introduce the notion of a nef subscheme, which generalizes thenotion of a subvariety with nef normal bundle. We show that restriction of apseudoeffective (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 ofcycle classes of nef subvarieties via proper surjective maps. This conecontains the movable cone and shares similar intersection-theoretic propertieswith it, thanks to the aforementioned properties of nef subvarieties. On theother hand, we prove that if $Y\subset X$ is an ample subscheme of codimension$r$ and $D|_Y$ is $q$-ample, then $D$ is $(q+r)$-ample. This is analogous to aresult proved by Demailly-Peternell-Schneider. We use the theory of $q$-ampledivisors, as developed by Totaro, throughout the paper.

On nef subvarieties