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.

Criteria for complete intersections

“…A line bundle which is both q-ample and semi-ample-that is, some tensor power is globally generated-is called q-positive; the name is justified by the terminology in complex geometry. A summary of these notions can be found in [12,Appendix].…”

“…Proposition (cf. [12,Proposition 1.4]) Suppose W is smooth and Y is a local complete intersection (lci, for short). Then Y is q Y -ample if and only if the conditions below are satisfied:…”

“…This is due to the fact that his proof uses a technical G3-criterion (in the terminology of Hironaka-Matsumura) for subvarieties of projective spaces, still due to him. In here, this issue is hidden at the first step of the proof of Lemma 1.5, since the partial amplitude assumption yields the G3-property (see [12,Section 2.2] for details).…”

“…It's therefore natural to attempt bridging this difference. In this article we propose such a generalisation, which uses the author's work [12] on partially ample subvarieties.…”

Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

“…There are few sufficient geometric criteria for it, one of the best known is due to Speiser [14]. The author [12] introduced the notion of a partially ample subvariety and proved that the G3property holds for subvarieties satisfying the weakest such partial amplitude condition (named 1 ą0 , one-positive); the convenience of the partial amplitude property was shown by proving Fulton-Hansen-type connectedness results for pre-images. The 1 ą0 -property has, however, an unpractical aspect: it involves the cohomological dimension of the complement XzY which is not straightforward to control (see [10] for several methods to estimate it).…”