# GIT in the projective setting

Let us suppose that a complex reductive group ${\displaystyle G}$ acts on a projective variety ${\displaystyle X}$, with a linearization ${\displaystyle G\curvearrowright {\mathcal {O}}_{X}(1)}$ (the pullback of the hyperplane bundle). Since the quotient is supposed to be described by ${\displaystyle G}$-invariant functions on ${\displaystyle X}$, in the projective case it makes sense to set ${\displaystyle X//G=\operatorname {Proj} (\bigoplus _{l\geq 0}H^{0}(X,{\mathcal {O}}_{X}(l))^{G})}$, where ${\displaystyle H^{0}}$ is standard notation for global sections and ${\displaystyle {\mathcal {O}}_{X}(l)={\mathcal {O}}_{X}(1)^{\otimes l}}$.

Define a rational map ${\displaystyle X\rightarrow X//G}$ by ${\displaystyle x\mapsto \operatorname {ev} _{x}}$. Recall that

${\displaystyle \bigoplus _{l\geq 0}H^{0}(X,{\mathcal {O}}_{X}(l))^{G}}$
is finitely generated, in particular we may assume it is generated in some degree ${\displaystyle d}$ (the ${\displaystyle \operatorname {Proj} }$ construction is unchanged under taking the subring ${\displaystyle \bigoplus _{l\geq 0}R_{dl}}$ of ${\displaystyle \bigoplus _{l\geq 0}R_{l}}$); evaluate invariant sections of ${\displaystyle O_{X}(d)}$ at ${\displaystyle x}$ to obtain a map
${\displaystyle \operatorname {ev} _{x}\in \operatorname {Hom} (H^{0}(X,{\mathcal {O}}_{X}(d))^{G},\mathbb {C} _{x}).}$
This gives a rational map from ${\displaystyle X}$ to ${\displaystyle \mathbb {P} (H^{0}(X,{\mathcal {O}}_{X}(d))^{G})^{*}}$, which is not defined on points for which ${\displaystyle \operatorname {ev} _{x}=0}$. This motivates the following

Definition 7.5

A point ${\displaystyle x\in X}$ is called unstable if every ${\displaystyle s\in H^{0}(X,{\mathcal {O}}_{X}(d))^{G}}$ vanishes on ${\displaystyle x}$. It is called semi-stable if it is not unstable. The locus of semi-stable points in ${\displaystyle X}$ is denoted by ${\displaystyle X^{ss}}$.

Remark that this is a sensible definition because ${\displaystyle \bigoplus _{k\geq 0}H^{0}(X,{\mathcal {O}}_{X}(k))^{G}}$ is generated in degree ${\displaystyle d}$; if there is one section (of any degree) not vanishing at ${\displaystyle x}$, then there will also be a non-vanishing section of degree ${\displaystyle d}$. Furthermore, ${\displaystyle X^{ss}}$ is open in ${\displaystyle X}$ and we have a morphism ${\displaystyle X^{ss}\to X//G}$.

The next definition is motivated by the following: we want to pick those orbits which are separated from nearby ones by invariant functions; morally, we want to see around which points the space of orbits embeds into the GIT quotient.

Definition 7.6

A point ${\displaystyle x\in X}$ is stable if it is semi-stable, ${\displaystyle G}$ acts on ${\displaystyle X_{f}}$ with closed orbits (for some ${\displaystyle G}$-invariant function ${\displaystyle f}$ s.t. ${\displaystyle f(x)\neq 0}$) and the stabilizer of ${\displaystyle x}$ in ${\displaystyle G}$ is finite.

The second part of this definition should mean that:

1. if we restrict to an affine open inside ${\displaystyle X_{f}}$ (and trivialize ${\displaystyle {\mathcal {O}}_{X}(1)}$), for any two orbits ${\displaystyle O_{1}}$ and ${\displaystyle O_{2}}$ we can find ${\displaystyle G}$-invariant functions ${\displaystyle s_{1},\ s_{2}}$ s.t. ${\displaystyle s_{i|O_{i}}=1}$ and ${\displaystyle s_{i|O_{2-i}}=0}$ for ${\displaystyle i=1,2}$;
2. this condition holds infinitesimally as well, i.e. for any vector ${\displaystyle 0\neq v\in T_{x}X/T_{x}(G.x)}$ we can find a ${\displaystyle G}$-invariant function ${\displaystyle s}$ that vanishes on ${\displaystyle G.x}$ but s.t. ${\displaystyle D_{v}(s)\neq 0}$.

Remark that also the locus of stable points is open. Unfortunately it might happen to be empty.

Theorem 7.1

Let ${\displaystyle G}$ be a reductive group acting on a projective variety ${\displaystyle X}$ with a linearization of the action.

1. The map ${\displaystyle \pi \colon X^{ss}\to Y=X//G}$ is a good quotient.
2. There exists an open subset ${\displaystyle Y^{s}\subseteq Y}$ s.t. ${\displaystyle \pi ^{-1}(Y^{s})=X^{s}}$ and the restriction of ${\displaystyle \pi }$ to ${\displaystyle X^{s}}$ is a geometric quotient.
3. For any ${\displaystyle x_{1},\ x_{2}\in X^{ss}}$, we have ${\displaystyle \pi (x_{1})=\pi (x_{2})}$ if and only if ${\displaystyle {\overline {G.x_{1}}}\cap {\overline {G.x_{2}}}\cap X^{ss}\neq \varnothing }$. A point ${\displaystyle x\in X^{ss}}$ is stable if and only if ${\displaystyle G.x}$ is closed in ${\displaystyle X^{ss}}$ and ${\displaystyle x}$ has finite stabilizer.

The following proposition asserts that we can distinguish (semi-)stable points by looking at the orbits of their liftings in ${\displaystyle \mathbb {C} ^{n+1}}$.

Proposition 7.1

Let ${\displaystyle G}$ act on ${\displaystyle X\subseteq \mathbb {P} ^{n}}$ as above. Let ${\displaystyle x\in X}$ and ${\displaystyle {\widetilde {x}}}$ be any lifting of ${\displaystyle x}$ to ${\displaystyle \mathbb {C} ^{n+1}}$. Then

• ${\displaystyle x}$ is semi-stable if and only if ${\displaystyle 0}$ lies in the closure of the ${\displaystyle G}$-orbit of ${\displaystyle {\widetilde {x}}}$.
• ${\displaystyle x}$ is stable if and only if it is semi-stable, ${\displaystyle G.{\widetilde {x}}}$ is closed in ${\displaystyle \mathbb {C} ^{n+1}}$ and ${\displaystyle {\widetilde {x}}}$ has finite stabilizer.

Observe that the action being linearized induces an action on ${\displaystyle {\mathcal {O}}_{X}(-1)\subseteq \mathbb {P} ^{n}\times \mathbb {C} ^{n+1}}$. A one-parameter subgroup (1PS) in ${\displaystyle G}$ is a homomorphism of algebraic groups ${\displaystyle \mathbb {G} _{m}=\mathbb {C} ^{*}\to G}$, i.e. a cocharacter. Any 1PS in ${\displaystyle G}$ induces an action of ${\displaystyle \mathbb {C} ^{*}}$ on ${\displaystyle X}$ and it is clear from the above that (semi-)stable points for ${\displaystyle G}$ are (semi-)stable for ${\displaystyle \mathbb {C} ^{*}}$ as well. A non-trivial and useful result is that the behaviour under all the 1PS's determines stability under the action of the whole group.

Theorem 7.2 (Hilbert-Mumford criterion)

Let ${\displaystyle G}$ and ${\displaystyle X}$ be as above. A point ${\displaystyle x\in X}$ is (semi-)stable under the action of ${\displaystyle G}$ if and only if it is (semi-)stable under the action of ${\displaystyle \mathbb {C} ^{*}}$ induced by every 1PS of ${\displaystyle G}$.

The action of ${\displaystyle \mathbb {C} ^{*}}$ on ${\displaystyle \mathbb {C} ^{n+1}}$ can be diagonalized in some basis ${\displaystyle \{e_{0},\ldots ,e_{n}\}}$; suppose that ${\displaystyle \mathbb {C} ^{*}}$ acts with weight ${\displaystyle p_{i}}$ on each ${\displaystyle e_{i}}$. We may take a lifting of ${\displaystyle x}$ and express it in this basis, ${\displaystyle {\widetilde {x}}=\sum x_{i}e_{i}}$; then ${\displaystyle \lambda .{\widetilde {x}}=\sum \lambda ^{p_{i}}x_{i}e_{i}}$. Define ${\displaystyle \mu ^{+}(x)=\max\{-p_{i}:x_{i}\neq 0\}}$ and ${\displaystyle \mu ^{-}(x)=\max\{p_{i}:x_{i}\neq 0\}}$. Then ${\displaystyle \mu ^{+}(x)>0\Leftrightarrow \lim _{\lambda \to 0}\lambda .x}$ does not exist and ${\displaystyle \mu ^{+}(x)=0\Leftrightarrow \lim _{\lambda \to 0}\lambda .x}$ exists and is not ${\displaystyle 0}$. Analogous statements hold with ${\displaystyle \mu ^{-}}$ and ${\displaystyle \lim _{\lambda \to \infty }}$. Therefore

Lemma 7.2

A point ${\displaystyle x\in X}$ is semi-stable for the action of ${\displaystyle \mathbb {C} ^{*}}$ if and only if ${\displaystyle \mu ^{+}(x)\geq 0}$ and ${\displaystyle \mu ^{-}(x)\geq 0}$. It is stable iff strict inequalities hold.

It may be useful to notice that, for a 1PS ${\displaystyle \varphi \colon \mathbb {C} ^{*}\to G}$, ${\displaystyle x\in X}$ and ${\displaystyle g\in G}$, one has ${\displaystyle \mu ^{\pm }(g.x,\varphi )=\mu ^{\pm }(x,g^{-1}\varphi g)}$. Therefore one can change the 1PS by conjugation in ${\displaystyle G}$ whenever this makes the computations easier; in particular any 1PS can be conjugated into a chosen maximal subtorus of ${\displaystyle G}$ (e.g. diagonal matrices will work for the special linear group).

From the above it follows that the Hilbert-Mumford criterion may be expressed in the following numerical form.

Theorem 7.3

Let ${\displaystyle x\in X}$. For any 1PS ${\displaystyle \varphi \colon \mathbb {C} ^{*}\to G}$, set ${\displaystyle x_{0}=\lim _{\lambda \to 0}\varphi (\lambda ).x}$. Then ${\displaystyle x_{0}}$ is a fixed point for the 1PS, therefore ${\displaystyle \mathbb {C} ^{*}}$ acts on ${\displaystyle {\mathcal {O}}(-1)_{x_{0}}}$ with a certain weight, say ${\displaystyle p}$. Then ${\displaystyle x}$ is stable iff ${\displaystyle p<0}$ for any 1PS and semi-stable iff ${\displaystyle p\leq 0}$.

Example 7.3

Let us go back to the example of ${\displaystyle \mathbb {C} ^{*}}$ acting on ${\displaystyle \mathbb {C} ^{n}}$ via ${\displaystyle \lambda \mapsto \lambda {\textit {Id}}}$. One possibility is to consider ${\displaystyle \mathbb {C} ^{*}}$ as acting on the trivial line bundle over ${\displaystyle \mathbb {C} ^{n}}$ but with a non-trivial character, say with weight ${\displaystyle -p}$. Then it is an exercise to check that ${\displaystyle H^{0}(\mathbb {C} ^{n},L^{k})^{\mathbb {C} ^{*}}}$ are homogeneous polynomials of degree ${\displaystyle pk}$; one concludes that ${\displaystyle \mathbb {C} ^{n}//\mathbb {C} ^{*}=\mathbb {P} ^{n-1}}$ with the line bundle ${\displaystyle {\mathcal {O}}(p)}$ - this is obvious for ${\displaystyle p=1}$, for ${\displaystyle p>1}$ prove that homogeneous primes of a graded ring ${\displaystyle R=\bigoplus _{k\geq 0}R_{k}}$ correspond one-to-one to homogeneous primes of ${\displaystyle S=\bigoplus _{k\geq 0}R_{kd}}$ for any fixed integer ${\displaystyle d}$.

This can also be seen as follows: we can embed ${\displaystyle \mathbb {C} ^{n}}$ as an affine chart of ${\displaystyle \mathbb {P} ^{n}}$, mapping ${\displaystyle {\underline {z}}}$ to ${\displaystyle [{\underline {z}}:1]}$. The action of ${\displaystyle \mathbb {C} ^{*}}$ can be extended to ${\displaystyle \lambda .[{\underline {z}}:w]=[\lambda {\underline {z}}:\lambda ^{-n}w]}$. Show that unstable points are ${\displaystyle [0:\ldots :0:1]}$ and all those of the form ${\displaystyle [{\underline {z}}:0]}$ for ${\displaystyle {\underline {z}}\in \mathbb {C} ^{n}\setminus \{0\}}$; all other points are stable and the quotient is ${\displaystyle \mathbb {P} ^{n-1}}$.

Example 7.4 (Configurations of $n$ points in $\mathbb P^1$)

A length ${\displaystyle n}$ subscheme of ${\displaystyle \mathbb {P} ^{1}}$ is (semi-)stable for the natural action of ${\displaystyle SL_{2}}$ iff each point has multiplicity ${\displaystyle (${\displaystyle \leq }$).

Consider a torus ${\displaystyle \mathbb {C} ^{*}\to SL_{2}}$ and fix a basis in which it is diagonal ${\displaystyle \lambda \mapsto {\bigl (}{\begin{smallmatrix}\lambda ^{p}&0\\0&\lambda ^{-p}\end{smallmatrix}}{\bigr )}}$. Length ${\displaystyle n}$ subschemes are represented by degree ${\displaystyle n}$ homogeneous polynomials in two variables, i.e. ${\displaystyle \mathbb {P} H^{0}(\mathbb {P} ^{1},{\mathcal {O}}(n))}$. Write ${\displaystyle f(x,y)=a_{0}x^{n}+\ldots +a_{n}y^{n}}$ in the chosen basis. Observe that ${\displaystyle \lambda .(x^{i}y^{n-1})=\lambda ^{p(2i-n)}x^{i}y^{n-i}}$, so ${\displaystyle \lambda .f\to \infty }$ for ${\displaystyle \lambda \to 0}$ unless ${\displaystyle a_{0}=\ldots =a_{\lceil n/2\rceil }=0}$, i.e. unless ${\displaystyle [1:0]}$ has multiplicity ${\displaystyle >n/2}$.

Alternatively, we could have said that ${\displaystyle f}$ tends to ${\displaystyle a_{i}x^{n-i}y^{i}}$ for the smallest ${\displaystyle i}$ s.t. ${\displaystyle a_{i}\neq 0}$; and ${\displaystyle \mathbb {C} ^{*}}$ acts on the fiber over ${\displaystyle x^{n-i}y^{i}}$ with weight ${\displaystyle 2i-n}$.

Exercise Consider the case of ${\displaystyle S^{k}\mathbb {P} ^{n}}$, configurations of ${\displaystyle k}$ points in ${\displaystyle \mathbb {P} ^{n}}$. We can see them as union of ${\displaystyle k}$ hyperplanes in the dual projective space.

Exercise Consider the action of ${\displaystyle SL_{r}}$ on ${\displaystyle \operatorname {Hom} (\mathbb {C} ^{r},\mathbb {C} ^{n})}$ for ${\displaystyle r. Show that semi-stable points correspond to injective homomorphisms. There are no strictly semi-stable points. The GIT quotient is ${\displaystyle \operatorname {Gr} (r,n)}$.