10
$\begingroup$

I believe that the following is true:

Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point.

Where can I find a proof of this statement? Is it really true that no one ever wrote down this statement in such a form?

Improve this question
$\endgroup$
3
  • 4
    $\begingroup$ The "only if" follows from the Borel fixed point theorem. The "if" follows from the structure theorems for group schemes over a field, usually attributed to Chevalley. The identity component of the automorphism group scheme of your manifold is an extension of an Abelian variety by a linear algebraic group. You want to prove that the induced homomorphism from your $\mathbb{C}^\times$ to the Abelian variety is a constant homomorphism. $\endgroup$
    – Jason Starr
    Commented Jun 4, 2019 at 11:47
  • $\begingroup$ Thanks for this comment Jason! I wonder if there is a pedestrian reference for "if" part (one that uses the complex geometry language and doesn't use the word "scheme"). Also where can I find this result of Chevalley? (I haven't succeeded so far with google). $\endgroup$
    – aglearner
    Commented Jun 4, 2019 at 12:15
  • 5
    $\begingroup$ Here is a link to an article of Brian Conrad proving Chevalley's theorem in modern language: math.stanford.edu/~conrad/papers/chev.pdf $\endgroup$
    – Jason Starr
    Commented Jun 4, 2019 at 14:58

0

Reset to default

You must log in to answer this question.