TITLE: Specialisation of rigid varieties
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: 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).