Geometric quotient

In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties ${\displaystyle \pi :X\to Y}$ such that^[1]

(i) The map ${\displaystyle \pi }$ is surjective, and its fibers are exactly the G-orbits in X.

(ii) The topology of Y is the quotient topology: a subset ${\displaystyle U\subset Y}$ is open if and only if ${\displaystyle \pi ^{-1}(U)}$ is open.

(iii) For any open subset ${\displaystyle U\subset Y}$, ${\displaystyle \pi ^{\#}:k[U]\to k[\pi ^{-1}(U)]^{G}}$ is an isomorphism. (Here, k is the base field.)

The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii) may also be phrased as an isomorphism of sheaves ${\displaystyle {\mathcal {O}}_{Y}\simeq \pi _{*}({\mathcal {O}}_{X}^{G})}$. In particular, if X is irreducible, then so is Y and ${\displaystyle k(Y)=k(X)^{G}}$: rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).

For example, if H is a closed subgroup of G, then ${\displaystyle G/H}$ is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same).

Relation to other quotients

A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory.

A geometric quotient is precisely a good quotient whose fibers are orbits of the group.

Examples

• The canonical map ${\displaystyle \mathbb {A} ^{n+1}\setminus 0\to \mathbb {P} ^{n}}$ is a geometric quotient.
• If L is a linearized line bundle on an algebraic G-variety X, then, writing ${\displaystyle X_{(0)}^{s}}$ for the set of stable points with respect to L, the quotient

${\displaystyle X_{(0)}^{s}\to X_{(0)}^{s}/G}$  

is a geometric quotient.

References

1. Brion, M. "Introduction to actions of algebraic groups" (PDF). Definition 1.18.