# Specialisation of rigid varieties

Recall that a variety $X$ over a field $k$ is called rigid if $H^1(X, T_X) = 0$. I am interested in understanding this property under specialisation.

Let $R$ be a discrete valuation ring and let $\pi: X \to \mathrm{Spec }R$ be a smooth proper morphism with geometrically irreducible generic fibre. Assume that the generic fibre of $\pi$ is rigid. Then is the special fibre of $\pi$ also rigid?

Note that basic facts in deformation theory show that if the special fibre is rigid then the generic fibre is also rigid. I am interested in the converse.

• No, that is not true. The surface $\mathbb{P}^1\times \mathbb{P}^1$ is rigid, yet it specializes to a whole sequence of Hirzebruch surfaces that are not rigid. The traditional keyword for this question is "deformation in the large". Your question for the special case of projective homogeneous spaces is the subject of the "Hwang-Mok rigidity theorem". They prove that smooth projective specializations of irreducible compact Hermitian symmetric spaces are also irreducible compact Hermitian symmetric spaces. So that is one positive example for your question. – Jason Starr Jun 25 '15 at 10:24

I am just posting my comment above as an answer. Of course the only rigid curve is $\mathbb{P}^1$, and this specializes to $\mathbb{P}^1$. Building on Mori's work, Siu proved that every smooth, projective specialization of $\mathbb{P}^n$ is $\mathbb{P}^n$. The analogous result for quadric hypersurfaces in $\mathbb{P}^n$ with $n\geq 4$ was proved by Jun-Muk Hwang. There was a breakthrough with the following paper of Jun-Muk Hwang and Ngaiming Mok, a spectactular application of their notion of "variety of minimal rational tangents" (VMRT).

MR1608587 (99b:32027) Reviewed
Hwang, Jun-Muk(KR-SNU); Mok, Ngaiming(PRC-HK)
Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation.
Invent. Math. 131 (1998), no. 2, 393–418.
32G05 (32J27 32M15)

They proved that for every irreducible Hermitian symmetric space of compact type, i.e., for every projective homogeneous space of Picard number 1 that is "cominuscule", every smooth projective specialization is of the same type. Note, their proof makes essential use of complex differential geometry, particularly the Kobayashi-Ochiai theorem. Later, Jan Gutt found a purely algebro-geometric proof and extended the Hwang-Mok theorem to positive characteristic.

There are also many "non-examples". Of course the Hirzebruch surface $\mathbb{P}^1\times \mathbb{P}^1$ is rigid, yet specializes to non-rigid Hirzebruch surfaces. Perhaps more surprisingly, for every vector space $V$ of even dimension $n\geq 3$, the rigid Flag variety $\text{Flag}(1,2;V)$ specializes to a variety that is not rigid (and not homogeneous). I believe it remains an open problem to find the full list of projective homogeneous spaces such that all smooth, projective specializations are also homogeneous (although I think this is known in the case of Picard number 1, mainly through the work of Hwang-Mok).

Edit. Regarding Daniel Loughran's follow-up question -- "Is every smooth, projective Fano specialization of a rigid Fano manifold also rigid?" -- in his ICM 2006 report, Jun-Muk Hwang does conjecture that this is true for rational homogeneous spaces in characteristic $0$. I guess that means that we do not know rigidity for projective homogeneous spaces of Picard number $1$ that are not cominuscule (but Hwang conjectures that they are rigid).

Second Edit. I should have remembered this sooner, but actually, abx completely answered your original question and your follow-up question in his answer to your earlier question: Specialisations of flag varieties.

Just to repeat the answer by abx, Pasquier-Perrin found an example of a (non-rigid) horospherical variety $X_5$ for $G_2$ that has Picard number $1$ (hence is Fano), yet deforms to the orthogonal Grassmannian $\text{Gr}_q(2,\mathbb{C}^7)$, and this orthogonal Grassmannian is projective homogeneous. In particular, Conjectures 3.1 and 3.2 from Hwang's ICM 2006 report are false.

MR2644311 (2011d:14086) Reviewed
Pasquier, Boris(D-BONN-CM); Perrin, Nicolas(F-PARIS6-IMJ)
Local rigidity of quasi-regular varieties. (English summary)
Math. Z. 265 (2010), no. 3, 589–600.
14L30 (14B12 14F10 14M27)

• Hi Jason. Thanks for the answer! Your examples seem to suggest that things are a lot better behaved in the Fano case. Do you know of any counter-examples to my question if I moreover assume that the special fibre and generic fibre are both Fano? – Daniel Loughran Jun 25 '15 at 11:30
• I have a vague recollection that the Mukai-Umemura threefold can specialize to a non-rigid Fano threefold. I thought that was what Tian first used to proved reductivity is not enough for existence of a Kaehler-Einstein metric. I need to double-check this. – Jason Starr Jun 25 '15 at 11:45
• Is the Mukai-Umemura threefold really rigid? I thought that prime Fano threefolds of genus 12 had positive dimensional moduli. – Daniel Loughran Jun 25 '15 at 11:58
• I was wrong about the Mukai-Umemura example. It is an example where the automorphism group changes, but it is not rigid. – Jason Starr Jun 25 '15 at 12:01
• Ok. If you know of any counter-examples in the Fano case, I would be most interested. – Daniel Loughran Jun 25 '15 at 13:22

Since in Jason Starr's answer (and comments) the Fano case has been mentioned, maybe this is worth noting:

A K-polystable Fano manifold X cannot specialise (in a $\mathbb{Q}$-Gorenstein family with general fiber isomorphic to X) to a klt K-polystable Fano variety Y different from X itself.

This follows by the fact that $\mathbb{Q}$-Gorenstein smoothable K-polystable Fano varieties over $\mathbb{C}$ form separated (and compact) moduli spaces. See the arXiv pre-prints by Li-Wang-Xu, Spotti-Sun-Yao and Odaka. Note that the proof of such "algebraic" statement is, at present, highly transcendental, since it is based on the fact that smooth K-polystable Fanos admit (unique) Kahler-Einstein metrics, and vice versa.

An example (under the rigidity hypothesis): the klt $\mathbb{Q}$-Gorenstein degenerations of $\mathbb{P}^2$ (which is rigid and K-polystable since the Fubini-Study metrics is KE) have been classified by Hacking and Prokhorov http://arxiv.org/abs/math/0509529. Such degenerations are given by weighted projective planes $\mathbb{P}(a^2,b^2,c^2)$, where $a,b,c$ satisfies the Markov equation $a^2+b^2+c^2=3abc$ (and their partial $\mathbb{Q}$-Gorenstein smoothings). It is well-known that, beside the case of $\mathbb{P}^2$ itself, such (coarse) spaces do not admit singular (with isolated orbifold sing.) Kahler-Einstein metrics (hence they are not K-polystable).

• Yes, I asked around about this a few days ago. I was told that it is wide open to try to understand which Fano manifolds are K-polystable, even among Fano manifolds of Picard number 1, e.g., projective homogeneous spaces of Picard number 1 that are not compact Hermitian symmetric, Fano complete intersections in projective space, moduli spaces of stable rank r vector bundles on a genus g curve with fixed determinant of degree prime to r, etc. Via the example of Pasquier-Perrin, either $Gr_q(2,7)$ or the $G_2$-horospherical variety is not K-polystable; both have reductive automorphism group. – Jason Starr Jun 28 '15 at 17:58