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 $B \to X$?
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asked Jul 20, 2011 at 10:45
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1$\begingroup$ I have no intuition on complex manifolds, so this may be entirely silly: I don't understand your heuristic at all. I mean, of course you can equip each fiber with a group structure, but how is this group structure determined purely in terms of $f$? In other words, I don't understand how $b \mapsto 0_b$ is supposed to be defined in the first place. $\endgroup$– t.b.Jul 20, 2011 at 11:35
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$\begingroup$ As an example of what I have in mind: consider the unit tangent bundle on a $2$-dimensional Riemannian manifold. This certainly is an $S^1$-bundle, but how do you equip the fibers with a group structure here? There must be some magic going on in the complex setting. $\endgroup$– t.b.Jul 20, 2011 at 11:37
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$\begingroup$ No, no magic, you're quite right, I should have thought about this more before asking. One can also fabricate examples of such fibrations where no holomorphic section exists (so one can't get a "holomorphic" variation of group structures), by taking a 2-dimensional torus that doesn't split as a product and quotienting by a sub-torus. $\endgroup$– Gunnar Þór MagnússonJul 20, 2011 at 12:10
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$\begingroup$ That makes sense, yes. Thank you for clarifying. $\endgroup$– t.b.Jul 20, 2011 at 12:17
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$\begingroup$ The problem is that the group structure of the complex torus is not canonical. You need to specify beforehand a point on the torus (which is going to be the identity of the group). So, if this zero-point is chosen "randomly", the zero section does not need to be even continuous. $\endgroup$– Andrea MoriJul 20, 2011 at 13:04
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1 Answer
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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 $S^3\times S^1$.
