Hyperbolicity Related Problems for Complete Intersection Varieties

TL;DR

This paper investigates hyperbolicity, jet differentials, and cotangent bundle positivity for high-degree complete intersection varieties in complex projective spaces, extending known results and addressing conjectures in algebraic geometry.

Contribution

It demonstrates hyperbolicity for generic high-degree complete intersections, generalizes jet differential existence theorems, and provides results supporting Debarre's conjecture on cotangent bundle positivity.

Findings

Hyperbolicity for generic high multidegree complete intersections.

Existence of jet differentials generalizing Diverio's theorem.

Generic high-degree complete intersection surfaces have ample cotangent bundles.

Abstract

In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and high codimension from the known results on hypersurfaces. Then we prove a existence theorem for jet differentials that generalizes a theorem of S. Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has ample cotangent bundle.