Algebraic intermediate hyperbolicities

TL;DR

This paper extends the concept of algebraic hyperbolicity to intermediate hyperbolicity, linking it to the ampleness of cotangent bundle powers and deriving finiteness results for automorphisms and maps.

Contribution

It introduces a new notion of intermediate algebraic hyperbolicity and establishes its implications for automorphism groups and surjective maps.

Findings

Intermediate algebraic hyperbolicity holds when certain cotangent bundle exterior powers are ample.

Proves finiteness of the birational automorphism group under intermediate hyperbolicity.

Shows finiteness of surjective maps from a given projective variety.

Abstract

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.