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.

中文翻译：

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-不变量和汤普森猜想

1981 年，汤普森证明，如果$n\geqslant 1$是任何整数并且$G$是的任何有限子群$\text{GL}_{n}(\mathbb{C})$， 然后$G$至多有一个半不变的度数$4n^{2}$. 他推测，事实上，存在一个普遍常数$加元这样对于任何$n\in \mathbb{N}$和任何有限子群$G<\text{GL}_{n}(\mathbb{C})$,$G$至多有一个半不变的度数$人民币$. 这个猜想意味着${\它\阿尔法}$-不变的${\it\alpha}_{G}(\mathbb{P}^{n-1})$，正如田在 1987 年介绍的那样，至多是$加元. 我们在本文中证明了 Thompson 猜想。