Hyperbolicity Related Problems for Complete Intersection Varieties

Damian Brotbek (1, 2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Jet differentials; ample cotangent bundle; hyperbolicity

Domaines: Géométrie algébrique [math.AG]

Damian Brotbek. Hyperbolicity Related Problems for Complete Intersection Varieties. Compositio Mathematica, 2014, 150 (3), pp.369-395. ⟨10.1112/S0010437X13007458⟩. ⟨hal-00556790⟩