Tutorial 3: Geometric Invariant Theory
summary
States that for reductive groups the invariant ring is finitely generated, so Spec of that ring gives a categorical quotient that is constant on orbits, separates closed orbits and has fibers equal to unions of orbits sharing the same closed orbit in their closure. Explains need to restrict to a semi-stable locus (projective cone points whose orbit closures avoid zero), gives the Hilbert–Mumford 1‑parameter subgroup test and Kempf’s optimal 1‑PS, applies these to determinant/permanent polystability, and extends finite‑generation via a transfer principle to some non‑reductive stabilizer cases.
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chapters (12)
- 0:00Introduction to GIT
- 1:48Finite generation of algebras
- 6:12Quotients and orbit spaces
- 10:50Categorical quotients
- 13:51Reynolds operator properties
- 20:25Fiber behavior and dimensions
- 26:47Projective setting definitions
- 34:22Stability and polystability
- 42:06Main theorem of GIT
- 53:33Hilbert-Mumford criterion
- 1:01:03Determinants and permanents
- 1:05:00Extensions to non-reductive groups
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