## Tohoku Mathematical Journal 2000 March SECOND SERIES VOL. 52, NO. 1

 Tohoku Math. J. 52 (2000), 61-77

Title INVARIANT SUBVARIETIES OF LOW CODIMENSION IN THE AFFINE SPACES

Author Kayo Masuda and Masayoshi Miyanishi

(Received June 8, 1998, revised January 26, 1999)
Abstract. Let $W$ be an irreducible subvariety of codimension $r$ in a smooth affine variety $X$ of dimension $n$ defined over the complex field $\boldsymbol{C}$. Suppose that $W$ is left pointwise fixed by an automorphism of $X$ of infinite order or by a one-dimensional 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 $n\leq3$. As by-products, we obtain algebro-topological characterizations of the affine 3-space.

1991 Mathematics Subject Classification. Primary 14L30; Secondary 14F45.