Take the 2-minute tour ×  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?  share|improve this question Apart from
the topological motivation that for Hausdorff topological gp $G$
and subgp $H$ the quotient $G/H$ is Hausdorff precisely when $H$
is closed in $G$, perhaps (i) may sound better in view of the
following fact: if $f:H \rightarrow G$ is a homomorphism between
finite type gr