FAMILIES OF GROUP ACTIONS, GENERIC ISOTRIVIALITY, AND LINEARIZATION

Abstract

Our base field is the field ℂ of complex numbers. We study families of reductive group actions on $$ {\mathbb A} $$ 2 parametrized by curves and show that every faithful action of a non-finite reductive group on $$ {\mathbb A} $$ 3 is linearizable, i.e., G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine morphisms 𝑝: S ⟶ Y and q : Τ ⟶ Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant étale base change φ: U ⟶ Y . A special case is the following result. Call a morphism φ: X ⟶ Y a fibration with fiber F if φ is at and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an étale dominant morphism μ: U ⟶ Y such that the pull-back is a trivial fiber bundle: U × Y X ≅ U × F. As an application we give short proofs of the following two (known) results: (a) Every affine A1-_bration over a normal variety is locally trivial in the Zariskitopology (see [KW85]). (b) Every affine A2-_bration over a smooth curve is locally trivial in the ZariskiTopology (see [KZ01]).