THE -INVARIANT AND THOMPSON’S CONJECTURE

Abstract

In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$, then $G$ has a semi-invariant of degree at most $4n^{2}$. He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G<\text{GL}_{n}(\mathbb{C})$, $G$ has a semi-invariant of degree at most $Cn$. This conjecture would imply that the ${\it\alpha}$-invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$, as introduced by Tian in 1987, is at most $C$. We prove Thompson’s conjecture in this paper.