Global algebraic linear differential operators

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.

14 pages