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${\it\alpha}$ -INVARIANT AND THOMPSON’S CONJECTURE
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.
