Numerical Dimension and Locally Ample Curves

Abstract

In the paper [the author, “Fujita vanishing theorems for $q$-ample divisors and applications on subvarieties with nef normal bundle”, Preprint, arXiv:1609.07797], was shown that the restriction of a pseudoeffective divisor $D$ to a so-called nef subvariety $Y$ (e.g. $Y$ is lci in $X$ and has nef normal bundle) is pseudoeffective. Assuming the normal bundle is ample and that $D{|}_Y$ is not big, we prove that the numerical dimension of $D$ is bounded above by that of its restriction, i.e. ${\kappa }_{\sigma }(D)\le {\kappa }_{\sigma }(D{|}_Y)$. The main motivation is to study the cycle classes of “positive” curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movable curves. We do not impose any condition on the singularities on the curve or the ambient variety. For locally complete intersection curves in a smooth projective variety, this is the main result of J. C. Ottem [J. Eur. Math. Soc. (JEMS) 18, No. 11, 2459–2468 (2016; Zbl 1362.14007)]. The main tool in this paper is the theory of $q$-ample divisors.