Partially ample subvarieties of projective varieties

Abstract

The goal of this article is to define partially ample subvarieties of projective varieties, generalizing Ottem’s work on ample subvarieties, and also to show their ubiquity. As an application, we obtain a connectedness result for pre-images of subvarieties by morphisms, reminiscent to a problem posed by Fulton–Hansen in the late 1970s. Similar criteria are not available in the literature.

Appendix A: Background material

1.1 A.1 Cohomological q-ampleness

This notion was introduced by Arapura and Totaro.

Definition A.1

Let Y be a projective scheme, an ample line bundle.

• (i) ([24, Theorem 7.1]) An invertible sheaf ${\mathscr {L}}$ on Y is q-ample if, for any coherent sheaf ${\mathscr {G}}$ on X, holds: there exists $\mathop {\mathrm{ct}}^{{\mathscr {G}}}$ such that for all $a\geqslant \mathop {\mathrm{ct}}^{{\mathscr {G}}}$ and all $t>q$, .

  It is enough to verify the property for

  

  , $k\geqslant 1$ (cf. [24, Theorem 6.3, 7.1]).

• (ii) ([2, Lemmas 2.1, 2.3]) A locally free sheaf ${\mathscr {E}}$ on Y is q-ample if ${\mathscr {O}}_{\mathbb {P}({\mathscr {E}}^\vee )}(1)$ on is q-ample. It is equivalent saying that, for any coherent sheaf ${\mathscr {G}}$ on Y, there is such that:

  

  , for all $t>q$ and all $a\geqslant \mathop {\mathrm{ct}}^{{\mathscr {G}}}$.

  The q-amplitude of${\mathscr {E}}$, denoted by $q^{\mathscr {E}}$, is the smallest integer q with this property. Note that ${\mathscr {E}}$ is q-ample if and only if so is ${\mathscr {E}}_{Y_{\mathrm{red}}}$ (cf. [24, Corollary 7.2]). Also, any locally free quotient ${\mathscr {F}}$ of ${\mathscr {E}}$ is still q-ample; indeed, .

1.2 A.2 q-Positivity

Proposition A.2

([23, Proposition 1.7]) For a globally generated, locally free sheaf ${\mathscr {E}}$ on Y, the following statements are equivalent:

• (i) ${\mathscr {E}}$ is q-ample (cf. Definition A.1);

• (ii) The fibres of the morphism $\mathbb {P}({\mathscr {E}}^\vee )\rightarrow |{\mathscr {O}}_{\mathbb {P}({\mathscr {E}}^\vee )}(1)|$ are at most q-dimensional.

We say that ${\mathscr {E}}$ is Sommese-q-ample if it satisfies any of these conditions.

Definition A.3

(cf. [1]) Suppose X is a smooth, complex projective variety. A line bundle ${\mathscr {L}}$ on X is q-positive, if it admits a Hermitian metric whose curvature is positive definite on a subspace of ${\mathscr {T}}_{X,x}$ of dimension at least $\dim X-q$, for all $x\in X$; equivalently, the curvature has at each point $x\in X$ at most q negative or zero eigenvalues.

Theorem A.4

• (i) ([1, Proposition 28]) q-positive line bundles are q-ample.

• (ii) ([19, Theorem 1.4]) Assume ${\mathscr {E}}$ is globally generated. Then it holds:

  $$\begin{aligned} {\mathscr {E}}\text { is Sommese-}q\text {-ample}\;\;\Longleftrightarrow \;\; {\mathscr {O}}_{\mathbb {P}({\mathscr {E}}^\vee )}(1)\text { is }q\text {-positive.} \end{aligned}$$

• (iii) ([3, 21]) Let be q-positive and $Y\in |{\mathscr {L}}|$ a smooth divisor. Then it holds: