# GIT, wedging, and moment map

So this stuff didn’t come from a GIT lecture, but it is stuff I spent time thinking about because of a GIT lecture.

### Wedging

I should say this is just a discussion about finite dimensional vector spaces (maybe even finite rank free modules?)

I was always told that if then elements of can be thought of as alternating multilinear maps , where is the ground field.

So if are functionals, then should give rise to an alternating map . The answer is (or should be)

the only question is, how do you determine that this is the answer?

Roughly an answer (Due to Theo)

By the universal property of wedging, the vector space of alternating bilinear forms is naturally isomorphic to . The nontrivial step is to explain why, at least if is finite dimensional, then .

Recall: If are finite dim, then

pf: because of fin dim can choose basis . Then and are determined by and

The map in one direction takes to the bilinear map . In the reverse direction, choose such that for some . Set ; then . Then set . QED.

To simplify stuff set

- is the subset of that give rise to alternating bilinear maps, i.e. .

Note is defined as a quotient of , so there is map in that direction. There is also an averaging map (analogous to the Renolds operator) . For example,

So we have

Let be any lift, for example . Then produces a multilinear function:

In the example at hand, this will take to some multiple of .

Its possible that requiring will pin down uniquely, but in any case, as long as the characteristic isn’t 2, this should produce an isomorphism which roughly explains that

and if the characteristic is 0, then you’ll get similar isomorphism for higher wedge products.

### The Moment Map

This material is taken directly from ‘Cohomology of Quotients in Symplectic and Algebraic Geometry’ by Kirwan.

This works for any symplectic manifold , but at least now I usually just worry about being a projective variety. So there is a closed 2-form . Get iso,

is a lie group with lie alg. ; acts on and preserves the symplectic form.

Preliminaries.

###1: For and , setting gives an action of on . There might also be some transpose business to work out. Recall .

###2:Given a map where the latter is the dual of a vector space, then for every we get a map on tangent spaces and for every , the composition

where is evaluation at , is a 1-form.

###3: Thinking of the lie algebra as the tangent space at the identity, any gives a vector field on . Let be a 1-param. subgroup of with tan. vec. . Then over the point , assing the tan. vec. .

A moment map is a map s.t.

- (-equivariance) for and , holds.
- , is a 1-form and is another 1-form. Its required that

Claim: If acts on by matrix multiplication on column vectors, then a moment map is determined by

where , , is the natural pairing of vector space and its dual, and is the standard Hermitian form on (apparently called the Study-Fubini form. The constant .

Now

This shows 1, and transitivity of the action means 2 can be checked just at an easy point. An easy point is for example .

###4 Evidently if is a lie algebra element, it acts on as multiplication by a matrix, , and is just the first column of this matrix. Also which means , so in particular , this will be used later.

###5 For the point , local coordinate are . Now is a function that depends on :

You can multiply it out, differentiate it via and evaluate at the point . Then the only terms that don’t disappear are exactly the terms of the form . This produces

where the remark in ###4 is used; note this is a 1-form.

### Finally…

There’s another way of getting a 1-form from . Namely, look at . To compute this its required to understand what the Study-Fubini form will do to the tangent vector . Write this out in real coordinates: set and then

###6: Recall . Which is to say . Using this as motivation, should be some linear combination of and such that . This analysis yields

So and similarly for . Doing this change of basis yeilds

.

Then it becomes clear, that when you apply the 1-form you get agrees with the previous computation.

## About this entry

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- Published:
- November 5, 2009 / 10:59 pm

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- GIT (course)

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