Questions Suggested by the Parabolic Subgroup Definition - MathOverflow

Questions Suggested by the Parabolic Subgroup Definition

2010-02-08T08:23:15Z

Take the following definition: "A parabolic subgroup of a linear algebraic group defined over $\mathbb{C}$ is a subgroup, closed in the Zariski topology, for which the quotient space is a projective algebraic variety."

My questions are:

(i) Why include closed in the definition?

(ii) What is an example of a projective algebraic variety that is the quotient of a linear algebraic group by a non-Zariski-closed subgroup?

(iii) What is an example of a quotient by a parabolic subgroup that is not a flag manifold?

(iv) Elliptic curves cannot be described as quotients of linear algebraic groups. What are other examples of families of varieties that cannot be expressed in this form?

Answer by Mike Skirvin for Questions Suggested by the Parabolic Subgroup Definition

2010-02-08T16:28:08Z

i)-ii) If the subgroup isn't closed, there's no reason in general for the quotient space to even be an algebraic variety. So for the definition to even make sense, there has to be some guarantee that $G/P$ is a variety. A standard result says that if $H \subset G$ is a closed subgroup of a linear algebraic group, then the quotient $G/H$ is a quasi-projective variety. You can find these results, for example, in Borel's text on linear algebraic groups. I don't know of an example of a non-closed subgroups whose quotient both exists and is not complete.

(iii) Pretty much by definition, $G/P$ is what's known as a partial flag variety. For the classical groups, you can directly verify that you get flag-like objects in this way, and for more general groups this is taken as the definition. However, if by flag variety you mean the full flag variety, then just pick any parabolic subgroup which is not a Borel subgroup. The simplest example occurs for $G = GL_3$ where you have parabolic subgroups corresponding the variety of lines in $\mathbb{C}^3$ (i.e., $\mathbb{P}^2$) and its dual, the variety of planes in $\mathbb{C}^3.$

(iv) I'm not sure exactly what you're asking for here. Just examples of varieties that aren't quotients of linear algebraic groups?

Answer by Victor Ostrik for Questions Suggested by the Parabolic Subgroup Definition

2010-02-08T16:42:19Z

If $G={\mathbb C}$ is the additive group and $L\subset G$ is a lattice, then $L$ is not Zariski closed, $G/L$ exists (in a sense of analytic geometry) and is projective. But of course $L$ is not a parabolic subgroup of $G$.

Answer by Jim Humphreys for Questions Suggested by the Parabolic Subgroup Definition

2010-05-30T13:33:54Z

Mike Skirvin has sorted out some of the main issues here, but the language used in the original question tends to confuse matters. The framework is the Borel-Chevalley structure theory of linear algebraic groups over an arbitrary algebraically closed field. Nothing special here about $\mathbb{C}$ . Starting with $G$ (which can be assumed connected), define a parabolic subgroup to be a closed subgroup $H$ for which the quotient variety $G/H$ is projective. By analogy with some classical examples, this variety is usually referred to as a "partial flag variety". Note that the construction of a quotient variety in this setting involves a morphism $G \rightarrow G/H$ whose fibers are $H$ and its other left cosets. Thus $H$ must be closed in order for $G/H$ to make sense as a variety. The language of manifolds doesn't come in unless this picture is compared with the Lie group picture for $\mathbb{C}$ , where fortunately everything translates well for reductive groups.

In the structure theory, it is shown that $H$ is parabolic precisely when it contains some Borel subgroup, then that $H$ is connected and self-normalizing, etc. All of this is most interesting when $G$ is reductive (and has a nontrivial semisimple derived group).

Answer by S. Carnahan for Questions Suggested by the Parabolic Subgroup Definition

2010-05-30T18:05:11Z

(iv) Theorem (Borel-Hirzebruch 1958, maybe?) A projective variety in characteristic zero with a transitive action of a connected linear algebraic group is smooth and Fano, i.e., its anticanonical bundle is ample.

This means "most" varieties don't arise as quotients by parabolic subgroups. In particular, curves of positive genus, and smooth hypersurfaces in $\mathbb{P}^n$ of degree at least $n+1$ make good examples.