I work over an algebraically closed field $k$ of characteristic zero.

Recall that a *flag variety* is a projective variety which is a homogeneous space for some semisimple algebraic group. Every flag variety is of the form $G/P$, where $G$ is a semisimple algebraic group and $P$ is a parabolic subgroup. It seems to me that this data is "discrete", so I expect flag varieties to have discrete moduli. Moreover flag varieties are Fano varieties, and it is known that there are only finitely many deformation types of Fano varieties of fixed dimension. This leads to my question.

For each $n \in \mathbb{N}$, are there are only finitely flag varieties of dimension $n$, up to isomorphism?

A proof/disproof or a reference for this would be much appreciated. By "up to isomorphism" I mean up to isomorphism as an algebraic variety.

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