Algebraic Aspects of Additive Group Actions on Complex Affine Space

Abstract

The automorphism group A n (ℂ) of the polynomial ring (ℂ[x 1, ..., x n ] in n variables over the complex field, equivalently the automorphism group of n-dimensional complex affine space, is known to have the structure of an infinite dimensional algebraic group [30]. Our concern in this paper is with embeddings of the additive group G a in A n (ℂ), in other words with algebraic (sometimes referred to as rational or polynomial) actions of G a on complex affine affine space. Throughout this report, all group actions on varieties are assumed to be algebraic (i.e. the orbit of any regular function spans a finite dimensional complex vector space).