I wonder what could be a Kähler surface, which is a total space of principal elliptic bundle over a curve. I believe that there is a classification and that it must be pretty simple, but I cannot find it in the literature.


Fo example, though there exist a full and complicated classification of elliptic surfaces, I didn't manage to find the list of all elliptic surfaces with no singular fibers.


Is it true, for example, that any of such surface is covered by the product of the base and the fiber?
In [Höfer, Thomas, Remarks on torus principal bundles, J. Math. Kyoto Univ.
33 (1993), no. 1, 227–259.] it is proved, that such a fibration must be flat (in the following sence: it is covered by a flat principal $\mathbb{C}^\*$-bundle), so it is enough to show that it has finite monodromy...



Principal elliptic bundles over curve with Kähler total space
A (partial) answer was given by [Donu Arapura](https://mathoverflow.net/users/4144/donu-arapura) at [Are most Kähler manifolds non-projective?](https://mathoverflow.net/questions/257147/are-most-k%C3%A4hler-manifolds-non-projective)



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> Let $C$ be a smooth projective curve of genus $g>0$, $\Gamma =\pi\_1(C)$, and $\tilde{C}$ the universal cover. Choose an elliptic curve $E$ and a group homomorphism $h\colon \Gamma \to E$. Define an action of $\Gamma$ on $\tilde{C} \times E$ by $\gamma(x,y)=(\gamma x,y+h(\gamma))$, and let $S$ be the quotient.
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> $S$ is Kähler. If hh has infinite image, then $S$ is not algebraic.
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> Proof. $S$ is Kähler because $\tilde{C}\times E$ has an invariant Kähler metric. For the second statement, assume that $h$ has infinite image. Projection on the first factor gives a holomorphic map $f \colon S \to C$. The fibres of $f$ can be identified with $E$. Restricting a meromorphic function $F$ on $S$ to a fibre gives a meromorphic function on $E$ which is constant on the orbits $\{y+h(\gamma)\}$ and therefore constant. Therefore $F$ comes from $C$. This shows that transcendence degree of the field of meromorphic functions on $S$ is 1.
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