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**, Mich. Math. J. 58, No. 3, 723-744 (2009). ZBL1186.14037.

**[2]** *Catanese, Fabrizio*, **Kodaira fibrations and beyond: methods for moduli theory**, Jpn. J. Math. (3) 12, No. 2, 91-174 (2017). ZBL1410.14010.

**[3]** *Kodaira, Kunihiko*, **A certain type of irregular algebraic surfaces**, J. Anal. Math. 19, 207-215 (1967). ZBL0172.37901.