# Global algebraic linear differential operators

## TL;DR

This paper studies the existence and growth of global algebraic differential operators on projective schemes, especially projective space, and explores the non-existence of algebraic elliptic operators on most smooth varieties.

## Contribution

It characterizes when global differential operators exist, proves their dimension grows polynomially with order, and introduces the concept of algebraic elliptic operators with non-existence results.

## Key findings

- Global differential operators exist for large N on projective space.
- Dimension of global operators grows polynomially with degree 2n.
- Algebraic elliptic operators generally do not exist on most smooth varieties.

## Abstract

In this note, we want to investigate the question, given a projective algebraic scheme X/k and coherent sheaves F, E on X, when do global differential operators of order N greater than zero, between E and F exist. We investigate in particular the case of projective n-space and prove that for large N always global differential operators between E and F exist. We also show that the dimension of global operators of order N grows like a polynomial in N of degree 2n. Finally, we define algebraic elliptic operators in the classical sense and show that for "most" algebraic smooth complete varieties, they do not exist on locally free sheaves.