I'm reading the work of Kodaira on the classification of elliptic surfaces, in particular On Compact Analytic Surfaces II.
What I retain are the following facts (up to confusion). Given a base $N$, functional $\mathcal{J}:N\longrightarrow \mathbb{H}/SL_2$ and homological invariant $\mathcal{G}$ (a sheaf over $N$):
There is a unique $B\overset{\pi_0}{\longrightarrow} N$ with these invariants and a global holomorphic section.
A $M'\overset{\pi'}{\longrightarrow} N$ with these invariants and no multiple singular fibers have locally sections near any singular point, Kodaira use this to prove that $M'$ is obtained by "gluing parts" of $B$ as in 1. through a cocyle of automorphisms of $B$.
Any $M\overset{\pi}{\longrightarrow} N$ is a logarithmic transformation of some $M'$ as in 2., so is in particular bimeromorphic to it.
Now, i'm confused because if we consider $\mathcal{J}$ constant (thus $\mathcal{G}$ constant sheaf), $B=N\times \mathbb{C}/\Lambda$ is the basic member (up to confusion). So point 2., if correct, would imply that any $M'$ with these invariant and no multiple fiber has no singular fiber and is a principal elliptic fiber bundle over $N$ ?
However, I think there are many counterexamples?