Symbols at infinity and Lie algebras of vector fields

Abstract

If $X$ is smooth projective complex variety, $D$ is a simple normal crossings divisor and $A$ an ample divisor with support $D$, then there is an associated filtration of the Lie algebra of on the affine variety $U=X\setminus D$ induced from the subspaces $T_X(\log D)(nA)\subset T_U.$ We show that the associated graded Lie algebra is finitely generated in a strong sense. We deduce as a consequence that vector fields on any smooth affine variety form a finitely generated Lie algebra.