THE α-INVARIANT AND THOMPSON'S CONJECTURE. (8th July 2016) - TIEP, PHAM HUU
Abstract : In 1981, Thompson proved that, if n1 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 nN and any finite subgroup G<GL_n(C), G has a semi-invariant of degree at most Cn . This conjecture would imply that the α -invariant α_G(P^n-1), as introduced by Tian in 1987, is at most C . We prove Thompson's conjecture in this paper.