# 2024-07-24-10-03-07 (Hoskins III)

## Reductive GIT

:::{.remark}
Last time: reductive $G\actson X \subseteq \PP^n$ linearly, via a representation $\rho: G\to \GL_{n+1}$.
We saw the Hilbert-Mumford criterion for the semistable locus.
Fix $T \leq G$ a maximal torus and choose a Weyl-invariant norm on the cocharacter space $X_*(T)_\RR$.
Then for any 1PS $\lambda: \GG_m\to G$, the conjugate $g \lambda g ^{-1}: \GG_m\to T$ is well-defined.

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:::{.theorem title="Kempf, Hesselink, Kirwan, Ness"}
For reductive $G\actson X = \PP^n$ linearly and a choice of norm $X_*(G)/G$ (modding out by conjugation), there is a finite stratification $X = \disjoint_{\beta\in B} S_\beta$ into $G\dash$invariant locally closed subvarieties with a partial order on $B$ satisfying

1. The lowest stratum is the semistable set, $S_0 = X^\ss$.
    Higher strata are more unstable.
2. $\bar S_ \beta \subseteq \disjoint_{\beta' \geq \beta} S_{ \beta'}$, i.e. closures of strata are contained in higher strata.
3. $B$ is determined combinatorially from the $T\dash$weights for $T$ a maximal torus, as is the norm.
4. $S_ \beta$ are determined by simpler limit sets which in turn are GIT semistable loci for smaller groups.

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:::{.remark}
These stratifications have been used for smooth varieties in various ways:

- (Kirwan) to compute cohomology of GIT quotients with $\QQ$ coefficients,
- (Dolgachev-Hu, Thaddeus) to describe birational transformations in VGIT, i.e. wall-crossing,
- (Ballard-Favero-Katzarkov-Halpern-Leistnes) to describe semiorthogonal decompositions in $D^b( [X^\ss/G] )$.

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:::{.remark}
See the Kempf-Ness theorem.
Over $\CC$, the GIT stratification will coincide with a symplectic stratification for $\norm{\mu}^2$ where $\mu$ is a moment map.
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:::{.remark}
For $x\in \PP^n$, define $\mu(x) \da \inf\ts{ {\mu(x, \lambda) \over \norm{\lambda} } \st \lambda: \GG_m \to G }$. 
Note that $x$ is semistable iff $\mu(x) \geq 0$.
We say \( \lambda \) is adapted to $x$ if it achieves this $\inf$.
Write the set of all primitive 1PS adapted to $x$ as \( \Lambda(x) \); Kempf shows this is nonempty.
To get a stratification, stratify the unstable locus by pairs $\beta = ([ \lambda], d)$ where the first term is a conjugacy class of 1PS of $G$ and $d$ is a value of $M$.
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:::{.definition title="Parabolic subgroups associated to 1PS"}
Let $P_ \lambda = \ts{ g\in G \st \lim_{t\to 0} \lambda(t) g \lambda(t)\inv \text{ exists in } G }$ and let $L_{\lambda} \da \ts{g\in G \st \lambda(t) g \lambda(t)\inv = g}$. The former is a parabolic, the latter is a Levi.
There is a map $q_ \lambda: P_ \lambda\to L_ \lambda$, and $U_ \lambda \da \ker q_ \lambda$ is unipotent.
There is a decomposition $P_ \lambda = U_ \lambda\semidirect L_ \lambda$.


![](figures/2024-07-24_10-40-20.png)

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:::{.remark}
For $\beta = ( [ \lambda] , d)$, we define $S_ \beta = \ts{x\in X \st [\lambda] \intersect \Lambda(x) \neq \emptyset \land M( \lambda) = d}$.
Fix \( \lambda \in [\lambda] \), define $Z_\lambda = \ts{ x\in X^{\lambda(\GG_m)} \st  {\mu(x, \lambda) \over \norm{\lambda} } = d }$.
This is a limit set.
Define a blade by $Y_ \lambda = \ts{ x \in X \st \lim_{t\to 0} \lambda(t) x \in Z_ \lambda }$.
Note that $\exists p_ \lambda: Y_ \lambda\to X_ \lambda$.
Similarly define semistable versions of these.
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## Non-reductive group actions


:::{.remark}
Non-reductive groups appear naturally in many moduli problems, e.g. 

- Moduli of unstable objects, e.g. VBs of a fixed HN type.
    Parabolic subgroup actions appear here.
- Moduli of hypersurfaces in weighted projective spaces or toric varieties.
    The automorphism groups are now generally non-reductive.
- Moduli of $k$-jets, i.e. germs of holomorphic maps truncated at order $k$.

Let $G\actson \spec A$ be non-reductive.
Even if $A^G$ is finitely generated, other issues arise:

- $X\to X\modmod G = \spec A^G$ is not surjective, and the image is generally only constructible.
- $G\dash$invariants don't separate closed orbits.
- $G\dash$invariants don't extend to ambient affines.

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