| Description |
In this dissertation, we take a cohomological approach to study positive properties of higher codimensional subvarieties. First, we prove 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. We also study the positivity of the cycle class of an ample curve or a curve with ample normal bundle. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme (resp. locally ample subscheme), which generalizes the notion of a subvariety with nef normal bundle (resp. ample 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, nefness and locally ampleness 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 $Y\subset X$ is an ample subscheme of codimension $r$ and $D|_Y$ is $q$-ample, then $D$ is $(q+r)$-ample. This is analogous to a result proved by Demailly-Peternell-Schneider. We also show that the cycle class of an ample curve (resp. locally ample curve) lies in the interior of the movable cone of curves (cone of curves). |
