Invariant subvarieties of low codimension in the affine spaces

Let W be an irreducible subvariety of codimension r in a smooth affine vari ety X of dimension n defined over the complex field C. Suppose that W is le ft pointwise fixed by an automorphism of X of infinite order or by a one-di mensional algebraic torus action on X. In the present article, we consider whether or not X is then an affine space bundle over W of fiber dimension n - r. Our results concern the case r = 1 or the case r = 2 and n less than or equal to 3. As by-products, we obtain algebro-topological characterizati ons of the affine 3-space.