By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature of the 4-manifold $X$ is non-trivial.

Conjecture 1: $X$ and $B$ can be given complex structures such that the map $X \to B$ is holomorphic.

Conjecture 2: $X$ and $B$ can be given the structure of complex algebraic varieties such that the map $X \to B$ is algebraic.

Question: Are the above conjectures true? If not, can you provide a counter example?

Background: The bundle $X\to B$ induces a map $f:B \to M_g$ where $M_g$ is the moduli space of curves and $g$ is the genus of the fiber. The signature of $X$ is given by

$$\sigma(X) = 4\int _B f^*(\lambda)$$

where $\lambda$ is the first Chern class of the Hodge bundle. Since the signature of $X$ is non-zero, the map $f$ is non-trivial in homology and one might hope that an argument along the following lines is true: Find a representative $\tilde{f}$ in the same homotopy class as $f$ which has minimal energy. Using the fact that $M_g$ has a hyperbolic metric ($g>2$ since $\sigma(X)\neq 0$), prove this minimal energy map is holomorphic or even algebraic. $\tilde{f}$ provides $X$ with the desired structure.

This question is a variant on this recent question: A four-dimensional counterexample? the counter example given there has zero signature.