THE α-INVARIANT AND THOMPSON'S CONJECTURE

In 1981, Thompson proved that, if n >= 1 is any integer and G is any finite subgroup of GL(n) (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 is an element of N and any finite subgroup G < GL(n) (C), G has a semi-invariant of degree at most Cn. This conjecture would imply that the alpha-invariant alpha(G)(Pn-1), as introduced by Tian in 1987, is at most C. We prove Thompson's conjecture in this paper.