- (i) For each y in Y, the fiber is an orbit of G.
- (ii) The topology of Y is the quotient topology: a subset is open if and only if is open.
- (iii) For any open subset , 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 . In particular, if X is irreducible, then so is Y and : 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 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.
- The canonical map is a geometric quotient.
- If L is a linearlized line bundle on an algebraic G-variety X, then, writing for the set of stable points with respect to L, the quotient
- is a geometric quotient.
- Brion 2009, Definition 1.18
- M. Brion, "Introduction to actions of algebraic groups"