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Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.

What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true that there are only finitely many such complex surfaces (i.e. with a given $\pi_1$ and $\kappa(S) = 1$) up to diffeomorphism?

Clearly, such a statement doesn't hold when such $S$ is a simply connected complex surface with $\kappa(S) = 1$ since elliptic surfaces $E(n; p, q)$ with two logarithmic transformations provide such infinite family of diffemorphism types, where $(p, q) = 1$. Probably, a similar infiniteness result holds when $\pi_1(S) = \mathbb{Z}_{k}$, but I am not sure when $b_{1}(S) \geq 2$. Any references would be greatly appreciated.

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Maybe the book by Friedman and Morgan "Smooth four-manifolds and complex surfaces" could be of some help (chapter 2 is devoted to the study of smooth structures on elliptic surfaces)? –  Benoit Oct 24 '13 at 18:34

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Any such surface is an elliptic surface over a curve. Over any curve, you can get the topological Euler characteristic arbitrarily large. Simply pull back your favorite elliptic surface on $\mathbb P^1$ back by a typical map of large degree from your curve to $\mathbb P^1$. And as long as the Euler characteristic is at least $36-12g$, it will certainly have Kodaira dimension $1$.

Fixing the Euler characteristic, you should be fine. You will have a bound on the degree of the $j$ invariant map, and the amount of ramifications, so you can describe it up to change of complex structure with a finite amount of topological data . Then you only need to note how the special fibers are quadratically twisted, which is also finite information.

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Dear Will, Thanks for your answer. Why your argument doesn't apply when $S$ is simply connected? I did not understand the statement "then you only need to note how the special fibers are quadratically twisted, which is also finite information"? –  guest2014 Oct 8 '13 at 1:14
    
Yes, sorry, this only works for Jacobian elliptic surfaces. So even with bounded Euler characteristic, I think logarithmic transformations will give you infinitely many kodaira dimension 1 surfaces. I don't see how the fundamental group could help: just take your infintely many surfaces, and pull back the family from $\mathbb P^1$ to some higher genus curve. –  Will Sawin Oct 8 '13 at 5:07

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