On subvarieties with ample normal bundle (Q345184)

Different geometric properties of an algebraic variety \(X\) are reflected in its positively embedded subvarieties. In particular, smooth subvarieties \(Y \subset X\) with ample normal bundle are of interest.NEWLINENEWLINEFor instance, one can study the consequences on a divisor to be numerically trivial on such subvariety \(Y\). This is the content of Theorem 1 (over a field of characteristic zero): a pseudoeffective \(\mathbb{R}\)-divisor \(D\) such that its restriction to \(Y\) (of dimension bigger than \(0\)) is numerically trivial is far from being big, that is, its numerical dimension is \(0\). (Recall that the numerical dimension of \(D\) is measuring essentially the maximal order of growth in \(m\) of \(h^0(X, \mathcal{O}_X(\lfloor mD \rfloor +H))\), being \(H\) ample on \(X\), see Def. 6 for details.)NEWLINENEWLINEObserve that, in particular, if \(D\) is nef then it is numerically trivial. This result and duality statements between divisors and curves lead to Theorem 2 (over the complex numbers) where it is shown that a smooth curve with ample normal bundle is big (it is in the interior of the cone of curves). This seems not to be true in higher dimension (see the Introduction of this paper and references therein).NEWLINENEWLINEMoreover (see Theorem 3), these subvarieties with ample normal bundle intersect all but finitely many prime divisors on \(X\).