CHARACTERIZATION OF FERMAT HYPERSURFACES BY ABELIAN GROUPS

Abstract

We set $n\ge 2$ and $d\ge 2$. Let $X$ be the Fermat hypersurface of degree $d$ in the projective space ${\mathbb{P}}^n$. It is known that the automorphism group of $X$ contains a subgroup which is isomorphic to $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$. Let $G$ be a subgroup of the projective linear group $\text{PGL}(n+1,ℂ)$ such that $G$ is isomorphic to $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$. We show that $G$ can be lifted into a subgroup of the general linear group $\text{GL}(n+1,ℂ)$ except for the cases where $(n,d)=(2,3)$ and $(3,2)$. As a result, we show that for a smooth hypersurface $Y$ in ${\mathbb{P}}^n$ of degree $d$, if $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$ acts faithfully on $Y$, then $Y$ is the Fermat hypersurface unless $(n,d)=(2,3)$ and $(3,4)$.