Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers. 


Assume that both $Y$ and the general fiber of $f$ admit a non-trivial first order infinitesimal deformation, that is $Ext^1(\Omega\_Y,\mathcal{O}\_Y)\neq 0$, and $Ext^1(\Omega\_{f^{-1}(y)},\mathcal{O}\_{f^{-1}(y)})\neq 0$ for $y\in Y$ general. 


Does this imply that $X$ admits a non-trivial first order infinitesimal deformation as well?



Infinitesimal deformations of a fibration
A candidate for a counterexample is the *Cartwright-Steger surface*, see <http://arxiv.org/abs/1412.4137>.


It is complex surface $X$ of general type with $$p\_g(X)=q(X)=1, \quad K\_X^2=9,$$ hence $K\_X^2=9 \chi(\mathcal{O}\_X)$. Since equality is attained in the Miyaoka-Bogomolov-Yau inequality, $X$ is a ball quotient, i.e. a quotient of $\mathbb{B}^2\_{\mathbb{C}}$ by a cocompact subgroup of $\textrm{PSU}(2, \, 1)$. 


By Mostow rigidity theorem, $X$ is biholomorphically rigid, i.e. its Kuranishi deformation space consists of exactly one point, in other words there are no deformations of $X$ *in the large*.


On the other hand, since $q=1$ and $X$ is of general type, the Albanese map of $X$ is a fibration $$a \colon X \to E,$$ whose base $E$ is an elliptic curve (actually, $E = \mathbb{C}/(\mathbb Z \oplus \omega \mathbb{Z})$, where $\omega$ is a third root of unity) and whose general fibre $F$ of $a$ is a smooth curve of genus at least $2$ (actually, $g(F)=19$). Therefore both $E$ and $F$ admit non-trivial first-order deformations.


Note that one should check that $H^1(X, \, T\_X) =0$ in order to be sure that that $X$ has no *obstructed* first-order deformations. 


**ADDENDUM.** As explained [here](https://mathoverflow.net/questions/211064/infinitesimal-deformations-of-fake-projective-planes-or-ball-quotients?noredirect=1#comment523071_211064), one actually has $H^1(X, \, T\_X)=0$ by a result of Calabi and Vesentini, so $X$ really provides a counterexample. 


