Submanifolds with ample normal bundles and a conjecture of Hartshorne

TL;DR

This paper investigates Hartshorne's conjecture on the intersection of submanifolds with ample normal bundles in projective manifolds, confirming it in several cases and relating it to cycle cones.

Contribution

It provides new evidence for Hartshorne's conjecture under Griffiths positivity assumptions and explores its connections to cycle cone questions.

Findings

The conjecture holds generically under certain positivity conditions.

Verification of the conjecture in specific cases.

Relation of the conjecture to properties of cycle cones.

Abstract

The Hartshorne conjecture predicts that two submanifolds X and Y in a projective manifold Z with ample normal bundles meets as soon as dim X + dim Y is at least dim Z. We mostly assume slightly stronger that one of the normal bundles is positive in the sense of Griffiths. We observe that the conjecture holds generically, relate it to question on cones of cycles and verify it in various cases.