THE ${\it\alpha}$-INVARIANT AND THOMPSON'S CONJECTURE. (8th July 2016)

• Record Type: Journal Article
• Authors: TIEP, PHAM HUU
• Is Part Of: Forum of mathematics. Volume 4(2016)
• Publication Date: 2016-07-08
• Subjects: 20C15 (primary), -- 13A50, -- 14B05, -- 20C33 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2016.3
• Languages: English
• ISSNs: 2050-5086

Abstract

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.