# Quotients

We had a slogan that the goal of GIT is to construct quotients in algebraic geometry; one would like to consider the quotient topological space (which is the natural way to endow the set of orbits with a topology) and give it the structure of an algebraic variety. This is not always possible. On the other hand, at least in the affine setting, one is tempted to consider the spectrum of the algebra of invariants as a quotient, but this sometimes happen to record little information about the topological quotient. What we are going to do is to get rid of bad orbits and obtain a reasonable quotient space of the remaining ones. Let's now go through some definitions of reasonable quotients and the properties we would expect from them.

Definition 7.2

Let ${\displaystyle {\mathcal {C}}}$ be a category of algebro-geometric objects (we have in mind 1. (quasi-)affine algebraic varieties, 2. (quasi-)projective varieties, 3. schemes over some base, say ${\displaystyle \mathbb {C} }$). Let ${\displaystyle G}$ be a group (object in ${\displaystyle {\mathcal {C}}}$) acting on ${\displaystyle X\in \operatorname {ob} ({\mathcal {C}})}$. A morphism ${\displaystyle \pi \colon X\to Y}$ in ${\displaystyle {\mathcal {C}}}$ is called a categorical quotient if, for every ${\displaystyle G}$-invariant arrow ${\displaystyle f\colon X\to Z}$ (i.e. equivariant with respect to the trivial action of ${\displaystyle G}$ on ${\displaystyle Z}$) in ${\displaystyle {\mathcal {C}}}$, there exists a map ${\displaystyle Y\to Z}$ that factorizes ${\displaystyle f}$ through ${\displaystyle \pi }$.

Uniqueness follows from the universal property.

Exercise: let ${\displaystyle \varphi \colon G\times X\to X}$ be the action map. Write down the diagrams that this map is required to satisfy (i.e. identity, compatibility).

Definition 7.3

A map ${\displaystyle \pi \colon X\to Y}$ is called a good quotient if

1. ${\displaystyle \pi }$ is ${\displaystyle G}$-invariant (i.e. constant on orbits) and surjective;
2. ${\displaystyle \pi }$ is affine and, for every affine open ${\displaystyle U\subseteq Y}$, ${\displaystyle \pi ^{\sharp }}$ identifies ${\displaystyle \mathbb {C} [U]}$ as the subring of ${\displaystyle G}$-invariant functions inside ${\displaystyle \mathbb {C} [\pi ^{-1}(U)]}$;
3. if ${\displaystyle W}$ is closed and ${\displaystyle G}$-invariant in ${\displaystyle X}$, then ${\displaystyle \pi (X)}$ is closed (i.e. ${\displaystyle \pi }$submersive; it is sometimes required to be universally submersive);
4. if ${\displaystyle W_{1}}$ and ${\displaystyle W_{2}}$ are closed, ${\displaystyle G}$-invariants subsets of ${\displaystyle X}$ such that they do not intersect, then also ${\displaystyle \pi (W_{1})\cap \pi (W_{2})=\varnothing }$.

Definition 7.4

A map ${\displaystyle \pi \colon X\to Y}$ is called a geometric quotient if it is a good quotient and ${\displaystyle Y}$ is an orbit space.

Basically, a good quotient may fail to be a geometric quotient by grouping together a bunch of orbits. The basic results of GIT for reductive groups are the following:

• in the affine setting, one obtains a good quotient by taking ${\displaystyle X\to \operatorname {Spec} \mathbb {C} [X]^{G}}$;
• in the projective case, one can throw away some orbits and obtain a good quotient (that is also a projective variety) of the other ones (semistable orbits); the quotient map restricts to a geometric quotient on the set of stable orbits (the quotient is only quasi-projective).

Furthermore, it is possible to give a numerical characterization of (semi)-stable points.