# Ampleness of Schur powers of cotangent bundles and k-hyperbolicity

## TL;DR

This paper proves the ampleness of Schur powers of cotangent bundles for generic complete intersections, linking positivity properties to hyperbolicity, with explicit codimension bounds and applications to complex hyperbolic geometry.

## Contribution

It establishes the ampleness of Schur powers of cotangent bundles for generic complete intersections with explicit codimension bounds, advancing understanding of positivity and hyperbolicity.

## Key findings

- Ampleness of Schur powers of cotangent bundles is achieved for generic complete intersections.
- Explicit codimension and multi-degree bounds are provided.
- Ampleness implies intermediate hyperbolic properties on complex manifolds.

## Abstract

In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.