Is there a “zero section” of any fibration of tori?

Question

I'm sure this is a silly question, but suppose we have a proper holomorphic fibration $f : X \to B$ of complex manifolds where the fibers of $f$ are complex tori. Each torus $X_b = f^{-1}(b)$ has a group structure, and thus an identity element $0_b$. Does the function $b \mapsto 0_b$ give a holomorphic section $B \to X$?

Answer 1

The answer is no. There are counterexamples already in dimension $2$, for instance Hopf surfaces which are elliptic fibrations over $\mathbb P^1$ or Kodaira surfaces which are elliptic fibrations over some elliptic curve. Both are described in the book of Barth, Hulek, Peters and Van de Ven. One such Hopf surface $M$ is obtained by taking the quotient of $\mathbb C^2\setminus\{0\}$ under the action of the cyclic group generated by a contraction $z\mapsto \frac{z}{2}$. It is clear that the canonical projection $\mathbb C^2\setminus\{0\}\to \mathbb P^1$ induces the desired elliptic fibration $M\to \mathbb P^1$ with fiber $E$. The existence of a section would imply a product structure $M\cong \mathbb P^1\times E$ but on the ther side one checks that $M$ is diffeomorphic to $S^3\times S^1$.