This question arose by reading the paper **[1]**, in particular, the remark at p. 737:



> 
> As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective surface to a smooth
> projective curve. Suppose that there exists a $y \in Y$ such that the
> Kodaira-Spencer map $$\rho\_{f, \, y} \colon T\_{Y, \, y} \to H^1(X\_y, \,
> T\_{X\_y})$$ is zero. Then $\Omega\_X|\_{X\_y}=\mathcal{O}\_{X\_y} \oplus
> \Omega\_{X\_y}$, so $\Omega\_X|\_{X\_y}$ is not ample.
> 
> 
> 


Smooth, non-isotrivial fibrations from a surface to a curve are called *Kodaira fibrations* (see **[2]** for a good introduction to the subject) and the usual way to construct them is taking suitable ramified covers of a product of two curves.


Now, I do not see how to perform the construction in such a way that the above condition on the Kodaira-Spencer map is satisfied. In the paper **[3]** Kodaira shows, by means of a local computation, that in all his original examples the Kodaira-Spencer map is everywhere non-vanishing (p. 212-213). On the other hand, I am not aware of any example in which it vanishes at some point (in fact, I am not aware of any other example of Kodaira fibration for which the Kodaira-Spencer map has been explicitly computed).


So let me ask the



> 
> **Question.** What are examples of non-isotrivial smooth projective morphisms $f \colon X \to Y$ from a smooth projective surface to a
> smooth projective curve such that the Kodaira-Spencer map vanishes at
> some $y \in Y$?
> 
> 
> 


**References.**


**[1]** *Jabbusch, Kelly*, [**Positivity of cotangent bundles**](http://dx.doi.org/10.1307/mmj/1260475697), Mich. Math. J. 58, No. 3, 723-744 (2009). [ZBL1186.14037](https://zbmath.org/?q=an:1186.14037).


**[2]** *Catanese, Fabrizio*, [**Kodaira fibrations and beyond: methods for moduli theory**](http://dx.doi.org/10.1007/s11537-017-1569-x), Jpn. J. Math. (3) 12, No. 2, 91-174 (2017). [ZBL1410.14010](https://zbmath.org/?q=an:1410.14010).


**[3]** *Kodaira, Kunihiko*, [**A certain type of irregular algebraic surfaces**](http://dx.doi.org/10.1007/BF02788717), J. Anal. Math. 19, 207-215 (1967). [ZBL0172.37901](https://zbmath.org/?q=an:0172.37901).



Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point
Any ramified base change of a smooth, non-isotrivial fibration should give an example.


You can give a simple proof using one of the various characterizations of the Kodaira spencer map:
<https://en.wikipedia.org/wiki/Kodaira%E2%80%93Spencer_map#In_scheme_theory>


Namely, if $f:X\to Y$ is a smooth fibration over a curve $Y$, then the Kodaira spencer map at a point $\operatorname{Spec} k = y\in Y$ is obtained by restricting $f$ to a first order thickening $\overline{f}:\mathfrak{X}\_{y}\to \operatorname{Spec} k[\varepsilon]$ of $y$ and considering the associated sequence of differentials
$$0\to \overline{f}^\*\Omega\_{k[\varepsilon]/k}\to \Omega\_{\mathfrak{X}\_y}\to \Omega\_{\mathfrak{X}\_y/k[\varepsilon]}\to 0 $$
By smoothness of the morphism, this sequence is exact and the last nonzero term is locally free as an $\mathcal{O}\_{\mathfrak{X}\_y}$-module. In particular, the sequence remains exact when you restrict to the reduced fiber $X\_y$. By the base change properties of differentials, you get an exact sequence
$$0\to f^\*T\_{Y,y}^{\vee}\to \Omega\_{\mathfrak{X}\_y}|\_{X\_y} \to \Omega\_{X\_y}\to 0$$


The associated element in $T\_{Y,y}^{\vee}\otimes \operatorname{Ext}^1(\Omega\_{X\_y},\mathcal{O}\_X)$ is equivalent to the data of a map $T\_{Y,y}\to H^1(X\_y,T\_{X\_y})$ which is the Kodaira spencer map.


Now if $h:Y'\to Y $ is a morphism of smooth curves that is ramified at $y'\in Y'$ such that $h(y')=y$, we get a diagram
$$ \begin{array}{ccc} \mathfrak{X}\_{y'} & \rightarrow & \mathfrak{X}\_y \\
\downarrow & & \downarrow \\
\operatorname{Spec}k[\varepsilon] & \rightarrow & \operatorname{Spec}k[\varepsilon] \end{array}$$
in which the lower horizontal map is given by the dual map on algebras with $\varepsilon\mapsto 0$.
By functoriality of differentials, it follows that we have a morphism of exact sequences
$$ \begin{array}{ccccccccc} 0 & \rightarrow & f\_y^\*T\_{Y,y}^{\vee} & \rightarrow & \Omega\_{\mathfrak{X}\_y}|\_{X\_y}&\rightarrow &\Omega\_{X\_y}& \rightarrow & 0\\
& &\downarrow & & \downarrow& &\downarrow & & \\0 & \rightarrow & h^\*f\_{y'}^\*T\_{Y',y'}^{\vee} & \rightarrow & \Omega\_{\mathfrak{X}\_y}|\_{X\_y}&\rightarrow &\Omega\_{X\_y}& \rightarrow & 0 \end{array}$$
Here, the first vertical map is the differential of $h$, which vanishes when we have ramification, while the last vertical map is the identity.
Finally, it follows that we have a commutative diagram of Kodaira-Spencer maps
$$ \begin{array}{ccc} T\_{Y',y'} & \rightarrow & H^1(X\_{y'},T\_{X\_{y'}}) \\
\downarrow & & \downarrow \\T\_{Y,y} & \rightarrow & H^1(X\_y,T\_{X\_y})\end{array}$$
where the first vertical map is zero, and the last vertical map an isomorphism. This implies that the upper horizontal map is zero.


