Effective bounds on ampleness of cotangent bundles

TL;DR

This paper establishes new degree bounds ensuring the ampleness of cotangent bundles for general complete intersections, significantly improving previous bounds and advancing understanding of their geometric properties.

Contribution

It provides explicit degree bounds for ampleness of cotangent bundles in complete intersections, improving upon prior super-exponential bounds in the dimension of the ambient projective space.

Findings

Ample cotangent bundles occur under specific degree and codimension conditions.

Degree bounds are polynomial in the dimension, not super-exponential.

Results do not cover cases where codimension is between n and 2n-2.

Abstract

We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and quadratic in $n$. This degree bound substantially improves the currently best-known super-exponential bound in $N$ by Deng, although our result does not address the case $n \leq c < 2n-2$.