THE ${\it\alpha}$ -INVARIANT AND THOMPSON’S CONJECTURE
In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of...
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2016-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508616000032/type/journal_article |
| Summary: | 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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| ISSN: | 2050-5086 |