# GIT lecture 15-16

These lecture (GIT15-16) were given on 9/30/09 and 10/2/09.

### Excerpts

First this result was proved (it was only stated last lecture)

Prop. a torus of dim acts on of dim by and stabilizer of generic point is finite. Let with then is semi-stable iff and properly stable iff

- interior of

Cor. If is a torus, then a point is properly stable iff its properly stable with respect to all 1 – parameter subgroups.

Thm (Hilbert-Mumford numerical criterion) If is reductive then (and X quasi projetive) then is

- semi-stable iff is is semi-stable with repspect to any 1-parameter subroups.
- properly stable iff it is so for any 1-parameter subgroup.

The idea of the proof is simple. If a points fails to be semi-stable (or properly stable) then there is a 1-parameter subgroup that realizes this failure.

Some technical results are used to prove this. The notation is , .

Lemma: if then there exists (recall ) so this is effectively a matrix with coeff in , hence a matrix which has entries which are Laurent series in ) such that .

Thm(Iwahori). Let be a reductive algebraic group. Then each double coset contains a unique 1-parameter subgroup. In other words, given a point , then can write for and a 1-param. sub group.

This lemma is also useful.

Lemma: with the notation above, if exists (so then exists and .

This is proved checking everything component wise and using that for integers .

Here is part of the argument for the thm above. Suppose is not properly stable. This means either is not closed, or is closed by is infinite. Say is infinite. I forget at the moment, but for some reason is affine. It it was proper, then it would be finite, but the fibers are infinite, so its not proper. Now by the valuative criterion for properness there exist such that .

now use Iwahori to write . So by the lemma above

exists and , so . But is in the orbit of , because is closed, and all elements in the orbit have conjugate stabilizers, so contains a 1 parameter subgroup as well.

In the rest of the lecture there were some applications given, but I haven’t gone through it carefully.

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- Published:
- October 5, 2009 / 8:49 pm

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- GIT (course)

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