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.



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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.



Moduli of flag varieties
Yes, there are only finitely many. One only needs to observe that for a given group, the variety $G/P$ has dimension at least the rank of the group $G$ (the dimension of a maximal torus). You can see this by inspecting cases by hand, but there's also a conceptual reason: the maximal torus of the adjoint group $G$ acts on $G/P$ faithfully with isolated fixed points, so it acts faithfully on the tangent space of some fixed point. Since a faithful module over a torus must have dimension $\geq$ that of the torus, this establishes the desired result. 


Thus, for a given dimension $n$, there can only be flag manifolds for groups with rank $\leq n$, there are finitely many of these, and only finitely many flag manifolds for each one. Q.E.D.


