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THE ${\it\alpha}$-INVARIANT AND THOMPSON’S CONJECTURE

  • PHAM HUU TIEP (a1)
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.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Forum of Mathematics, Pi
  • ISSN: -
  • EISSN: 2050-5086
  • URL: /core/journals/forum-of-mathematics-pi
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