THE ${\itα}$-INVARIANT AND THOMPSON'S CONJECTURE. (8th July 2016) - TIEP, PHAM HUU
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 ${\itα}$ -invariant ${\itα}_{G}(\mathbb{P}^{n-1})$, as introduced by Tian in 1987, is at most $C$ . We prove Thompson's conjecture in this paper.