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Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.

My question concerns specialisations of flag varieties over discrete valuation rings. Namely, let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$. Assume for simplicity that $K$ and $k$ are algebraically closed.

Let $X \to \textrm{Spec} R$ be a smooth projective scheme over $R$, and assume that the generic fibre is a flag variety. Is the special fibre also a flag variety?

Some remarks:

  • If the generic fibre is isomorphic to a projective space, then the answer to my question is yes. This is Theorem 2 of [1].
  • The converse to my question has a positive answer. Namely, if the special fibre is a flag variety, then the generic fibre is a flag variety. This is Proposition 5 of [2].

[1] Goren - Characterization and algebraic deformations of projective space.

[2] Démazure - Automorphismes et déformations des variétés de Borel.

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2 Answers 2

up vote 8 down vote accepted

There is a counter-example (with $k=\mathbb{C}$) due to Pasquier and Perrin, Math. Zeitschrift 265 (2010), 589-600. The generic fiber is an orthogonal grassmannian $\mathrm{Gr}_q(2,7)$, while the special fiber is a smooth projective non-homogeneous ("horospherical") variety with an action of $G_2$.

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In the positive direction, you should consult the work of Hwang-Mok. Regarding counterexamples, there are even easier examples than the one due to Pasquier and Perrin. Let $V$ be an even-dimensional vector space, and let $\omega$ be a symplectic pairing on $V$. Let $\mathbb{P}V$ be the associated projective space with tautological exact sequence, $$ 0 \to \mathcal{O}_{\mathbb{P}V}(-1) \to V\otimes_k \mathcal{O}_{\mathbb{P}V} \to Q \to 0.$$ The annihilator $A$ of the subbundle $\mathcal{O}_{\mathbb{P}V}(-1)$ with respect to $\omega$ is a subbundle of $V\otimes_k \mathcal{O}_{\mathbb{P}V}$ containing $\mathcal{O}_{\mathbb{P}V}(-1)$. The quotient $B=A/\mathcal{O}_{\mathbb{P}V}(-1)$ is a subbundle of $Q$, giving rise to a short exact sequence, $$ 0 \to B \to Q \to \mathcal{O}_{\mathbb{P}V}(1) \to 0.$$ In particular, this short exact sequence gives rise to a specialization of the extension $Q$ to the split extension $B\oplus \mathcal{O}_{\mathbb{P}V}(1)$. Thus, this also gives rise to a specialization of the partial flag variety $\text{Flag}(1,2;V) = \mathbb{P}_{\mathbb{P}V}Q$ to the inhomogeneous variety $\mathbb{P}_{\mathbb{P}V}(B\oplus \mathcal{O}_{\mathbb{P}V}(1))$.

Again, the fabulous positive progress is the work of Hwang-Mok. In positive characteristic, much of this has been generalized by Jan Gutt.

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