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Automorphisms of elliptic surfaces, inducing the identity in cohomology. (English) Zbl 1200.14068

Let \(S\) be a nonsingular complex surface of Kodaira dimension one. Then \(S\) admits at least one elliptic fibration \(f : S \to C\). Assume that the Jacobian fibration of \(f\) is not a product. The author considers the question whether \(S\) admits an automorphism \(\sigma\) that acts trivially on \(H^2(S,\mathbb{Q})\). More precisely, let \(G\) be the kernel of the map \(\mathrm{Aut}(S) \to \mathrm{Aut}(H^2(S,\mathbb{Q}))\). For each \(\sigma\in G\) there exists an automorphism \(\tilde{\sigma} \in \mathrm{Aut}(C)\) such that \(f\circ \sigma=\tilde{\sigma}\circ f\). This defines a map \(G\to \mathrm{Aut}(C)\), let \(H\) be the kernel of this map, and let \(Q := G/H\). The main result of the paper is that if \(G\) is nontrivial one of the following cases occurs:
(a) \(\chi(\mathcal{O}_S)\) = 1. If moreover \(p_g(S) > 0\) holds then we are in the following situation: \(H\) is trivial; \(Q\) has order 2 and \(C/Q = \mathbb{P}^1\), i.e., \(C\) is hyperelliptic. Moreover, in this case the fibration \(f : S \to C\) is the minimal desingularization of the base change of an elliptic surface \(g : X \to \mathbb{P}^1\), such that at the ramification point of \(C \to \mathbb{P}^1\) the fibers are as follows:
(1) \(2g + 1\) fibers of type \(I^*_0\) and one \(mI_4\) fiber (possible with multiplicity).
(2) \(2g + 1\) fibers of type \(I^*_0\) and one \(IV\) fiber.
(3) \(2g\) fibers of type \(I^*_0\), one fiber of type \(IV\) and one of type \(IV^*\).
(4) \(2g + 1\) fibers of type \(I^*_0\), one fiber of type \(IV\).
In the first and second case it turns out that there are 2 fibers of type \(mI_1\) and no further singular fibers (except possibly for fibers of type \(mI_0\)); in the third case all other singular fibers are of type \(mI_0\) and in the fourth case there is a fiber of type \(II\) and no further singular fibers except for fibers of type \(mI_0\). Conversely, if X, S,C are as above, then \(|Q| = 2\).
(b) \(\chi(\mathcal{O}_S) = 0\). If moreover \(p_g(S) > 2\) holds then we are in the following situation. Note that the \(j\)-function is constant. If \(j\) is not 0 or 1728 then \(H = (\mathbb{Z}/2\mathbb{Z})^2\), if \(j = 0\) then \(H\) is trivial, \(\mathbb{Z}/3\mathbb{Z}\) or \((\mathbb{Z}/2\mathbb{Z})^2\) and if \(j = 1728\) then \(H = \mathbb{Z}/2\mathbb{Z}\) or \((\mathbb{Z}/2\mathbb{Z})^2\). The group \(Q\) is trivial or of order 2. If \(Q\) is nontrivial then \(C/Q\) is an elliptic curve or \(\mathbb{P}^1\), and \(S\) is the base change of an elliptic fibration over \(\mathbb{P}^1\) (resp. elliptic curve) such that over each ramification point the fibers is of type \(I_0^*\) .
From this it follows that possible \(G\) are \((\mathbb{Z}/n\mathbb{Z})\) for \(n \in \{1, 2, 3\}\), \((\mathbb{Z}/2\mathbb{Z})^k\), \(k \in \{1, 2, 3\}\) and \(\mathrm{Sym}(3)\).

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
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