Here we mean Kodaira fibration $f: X \rightarrow C$ where $f$ is a holomorphic submersion with maximal rank everywhere, but not a complex fiber bundle map. Such a surface has been constructed by Kodaira (J. Anal. Math. 1967, 207-215) and also Barth-Hulek-Peters-Van de Ven (p220-223) for more details.

My interest here is its connection to negative holomorphic bisectional curvature in K\"ahler geometry. By a result of To-Yeung (Bull. London. Math. Soc. 2011). They show any holomorphic fibration $f: Y \rightarrow S$ with $\operatorname{dim} Y=n$ and $\operatorname{dim} S=n-1$ (surjective and maximal rank along the base), if the Kodira-Spencer map $T_s S \rightarrow H^1(Y_s, TY_s)$ is injective for each $s \in S$ (i.e. effectively parametrized), then $Y$ admits a Kahler metric of negative bisectional curvature. in particular any Kodaira surface satisfies this assumption.

Kodaira surfaces has base curve genus $\geq 2$ and the fiber $g \geq 3$. The intuition of To-Yeung's result is that the varying fiber will be a curve in the moduli space of genus-$g$ Riemann surface $\mathcal{M}_g$, then one can define a map from Y to the moduli space of genus-$g$ one-punctured Riemann surface $\mathcal{M}_{g,1}$, Then an earlier result of Wolpert shows that the Weil-Petersson metric on Teichm\"uller space of $\mathcal{M}_g$ (they need to generalize it to the case of $\mathcal{M}_{g,1}$) has negative bisectional curvature. The induced metric on Kodaira surfaces will have the desired negativity of curvature.

Here, my question is to learn more examples with high-dimensional base from To-Yeung's result. It boils down to if we know $\mathcal{M}_g$ could contain high-dimensional subvariety, I knew nothing about this and only recently learned that one way to see high-dimensional subvariety is to take orbit closure of complex geodesics, those will give algebraic varieties by recent works of Eskin-Mirzakhani-Mohammadi and Filip. However, I have no idea how these general existence results lead to some concrete examples.

**Question 1:** Can one give some interesting subvariety in
$\mathcal{M}_g$? Or besides taking orbit closure of complex geodesic
are there other methods to produce subvariety in $\mathcal{M}_g$?

**Question 2:** What do we know about fundamental group of
Kodaira fibration surfaces? Recall that an old result of Griffiths
states the universal cover of such a surface is a bounded (but not symmetric) domain in $\mathbb{C}^2$.