# Kodaira fibration and moduli space of Riemann surfaces

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 $\mathbb{C}^2$.

• For every $g\geq 2$, there is a dense, Zariski open subset $U\subset M_g$ such that every point of $U$ is contained in a proper curve $B$ inside $M_g$ (even inside $U$). This follows from the existence of the Satake compactification of $M_g$. Now "iterate" to produce high-dimensional families. Let $\pi_B:Y_B\to B$ be the family of curves. For every integer $n$, there is a cover $\widetilde{Y}_{B,n}\to Y_B$ whose restriction to each fiber of $\pi$ is a normal cover $\widetilde{Y}\to Y$ with Abelian group of deck transformations isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{2g}$ . . . – Jason Starr May 11 '16 at 0:03
• . . . Fix an integer $m$ that divides $n^{2g}$. Set $2\widetilde{g} - 2 = n^{2g}(2g-2)$. There is a cover $Y'\to Y$ of degree $m^{2\widetilde{g}}$ whose fiber over $y\in Y$ parameterizes a choice of invertible sheaf $\mathcal{L}$ on $\widetilde{Y}$ such that $\mathcal{L}^{\otimes m}$ is isomorphic to the pullback to $\widetilde{Y}$ of $\mathcal{O}_Y(\underline{y})$. Now form the branched cover of $\widetilde{Y}$ branched over the preimage of $y$ associated to $\mathcal{L}$. So $Y'$ is a complete $2$-dimensional variety and the branched covers have maximal variation, etc. – Jason Starr May 11 '16 at 0:08