GIT lecture 14

This lecture (GIT14) was given on 9/28/09.

Excerpts

The setup is G \subset GL(V) is reductive and it acts on an invariant subvariety X \subset \mathbb{P}(V)X = \mbox{Proj} R, want X//G = \mbox{Proj} R^G.

If X is projective then x \in X is semi-stable if x \in X_f for some homogeneous f \in R^G of positive degree.  It is stable if action on \mathbb{P}(V)_f is closed; the notion of stability depends on the embedding into projective space.  x \in X is unstable if it is not semi stable.

If X is just quasi projective the definitions of semi-stable and stable are strengthened: you require X_f to be affine (for semistability) and the action on X_f needs to be closed (for stable).

For x \in X \subset \mathbb{P}(V), denote the affine cone over X by X_a \subset V and x_a = kv - 0 be all the representatives of x.

Prop. 14.5 x is semistable iff 0 \not \in \overline{G\cdot v}, and x is stable iff G\cdot v is closed and x \in X is regular.

1/2 proof: Say x \in X is stable.  Then by a result from lecture 11 we know there is x \in X_f with f homogeneous and invariant such that the action of G is closed on X_f.  Now f(v) \ne 0 and f is constant on orbits, so f(G \cdot v) \ne 0, so G\cdot v \subset X_f, and the action of G is closed on X_f so orbits are closed so G\cdot v is closed.  Also, in the affine case showed X^{stable}_f = X_f - \phi^{-1}(Z) where \phi is the map to the affine quotient, and Z is the image of the irregular points.  So x wasn’t regular, it couldn’t be stable.

Conversely, the seperation lemma (which applies since G\cdot v is closed) gives a homogeneous invariant f such that x \in X_f.  It remains to show x \in X_f - \phi^{-1}(Z).  But if x \in \phi^{-1}(Z), then its in the fiber of an irregular point, so its orbit is in the closure of another orbit, so G \cdot v wouldn’t be closed (hmm think).

To construct a quotient you want to locally construct a quotient \mbox{Spec} R_f^G and then glue, but you can do this around unstable points. So just work with semi stable points: X//G := X^{ss}//G.

Another setup. (note projective stuff isn’t explicitly mentioned here) properly stable means stable with finite stabilizer. T is an alg. torus of dimension n that acts on V of dim N.  Associated to T are T^\vee = \{\chi \colon T \to k^*\} and P = \{ \lambda \colon k^* \to T\}.  Composing gives a map k^* \to k^* of the form t \mapsto t^m, for an integer m.  So there is a pairing P \times T^\vee \to \mathbb{Z}; it is non degenerate.

It can be shown the action of T looks like t.(x_1, ..., x_N) = (\chi_1(t)x_1, ...., \chi_N(t)x_N) with \cap_i \ker \chi_i finite (we assume the stabilizer of a generic point is finite). In any case, the support supp(x) = \{\chi_i | x = (x_1, ..., x_N) \mbox{ and } x_i \ne 0\}.

Prop 14.8 With the above setup, x \in V with supp(x) = \{\chi_1, ..., \chi_k\} then x is semi-stable iff 0 \in C(\chi_1, ..., \chi_k) and properly stable iff

  1. \dim span\{\chi_1, ..., \chi_k\} = n
  2. 0 \in interior of C(\chi_1, ..., \chi_k)

Note the reversed role of 0 being an element of something for checking semi stability.

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