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This is a question in complex geometry, but even for algebraic varieties I don't know the answer:

Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is an elliptic bundle, i.e there exists a morphism with connected fibres $\varphi: S \rightarrow C$ onto a smooth projective curve $C$ such that all the fibres are isomorphic to a fixed elliptic curve $E$.

I am looking for references that describe the automorphisms of $S$ over $C$, i.e. those automorphisms $f: S \rightarrow S$ that send $\varphi$-fibres onto $\varphi$-fibres. More precisely I would like to know:

  • Which groups can appear?

  • Given such an automorphism $f$, can you describe the locus of fixed points?

Remarks:

  • If $S$ is torus, there is a very detailed description by Fujiki, but I don't think his arguments apply to the case where $C$ has higher genus.

  • If one assumes that S becomes a product after finite base change (this always holds if $S$ is projective, see Jason's remark below) one can try to use descent arguments, but I don't think that this is representative of the general situation. I am looking for arguments that do not use the existence of a (multi-)section.

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    $\begingroup$ If all of the fibers are isomorphic to a fixed elliptic curve, then the family is isotrivial in a very strong sense. You stipulate that there is a Kaehler class. That makes things even more rigid. In the projective case, you quickly reduce the structure group to the finite group $\Gamma$ of automorphisms of a polarized fiber. Such a fiber bundle will be classified by a homomorphism $\rho$ from the fundamental group of $C$ to $\Gamma$. The induced automorphism $f_C$ of $C$ must be compatible with $\rho$. $\endgroup$ Mar 15, 2016 at 11:58
  • $\begingroup$ If I understand correctly the following holds in the projective case: since the group $\Gamma$ is finite, the monodromy representation $\rho$ becomes trivial after finite étale base change. So the surface $S$ becomes a product after finite base change ? $\endgroup$
    – roadrunner
    Mar 15, 2016 at 15:01
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    $\begingroup$ "So the surface $S$ becomes a product after finite base change." Yes. If there is an ample invertible sheaf, then there exists an ample invertible sheaf $\mathcal{L}$ whose restriction to geometric fibers of $\phi$ has degree $d\geq 3$. There exists a finite, etale morphism $C'\to C$ whose fiber over $t \in C$ parameterizes ordered triples $(x_0,x_1,x_2)$ of closed points of $S_t$ such that $\mathcal{L}\cong \mathcal{O}(d\cdot x_i)$, and such that $\mathcal{O}(x_1-x_0)$ and $\mathcal{O}(x_2-x_0)$ are a symplectic basis for the $d$-torsion. The base change to $C'$ is a product. $\endgroup$ Mar 15, 2016 at 15:16
  • $\begingroup$ Jason, I modified my second remark in the question to take into account your explanation. $\endgroup$
    – roadrunner
    Mar 15, 2016 at 19:35

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