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 such that[1]

(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.

Examples

• 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.

References

1. Brion 2009, Definition 1.18
• M. Brion, "Introduction to actions of algebraic groups"
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