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Holomorphic sections of a holomorphic family of Riemann surfaces induced by a certain Kodaira surface. (English) Zbl 1192.14030
Let \(M\) be a two-dimensional complex manifold and \(B\) be a Riemann surface. One assumes that a proper holomorphic mapping \(\pi : M \to B\) satisfies the following two conditions:
(i) The Jacobi matrix of \(\pi\) has rank one at every point of \(M\).
(ii) The fiber \(S_b = \pi^{-1}(b)\) over each point \(b\) of \(B\) is a closed Riemann surface of genus \(g_0\).
One calls such a triple \((M, \pi, B)\) a holomorphic family of closed Riemann surfaces of genus \(g_0\) over \(B\).
A holomorphic mapping \(s : B \to M\) is said to be a holomorphic section of a holomorphic family \((M, \pi, B)\) of Riemann surfaces if \(\pi \circ s\) is the identity mapping on \(B\).
Let \(\mathcal{S}\) be the set of all holomorphic sections of \((M, \pi, B)\). Denote by \(\# \mathcal{S}\) the number of all holomorphic sections of \(\mathcal{S}\). Next result is called Mordell conjecture in the functional field case.
By Yu. Manin, [Am. Math. Soc., Transl., II. Ser. 50, 189–234 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395–1440 (1963; Zbl 0178.55102)], H. Grauert [Publ. Math., Inst. Hautes Étud. Sci. 25, 363–381 (1965; Zbl 0137.40503)], Y. Imayoshi and H. Shiga [in: Holomorphic functions and moduli II, Proc. Workshop, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 11, 207–219 (1988; Zbl 0696.30044)], J. Noguchi [Publ. Res. Inst. Math. Sci. 21, 27–46 (1985; Zbl 0583.32061)], one has:
Theorem 1.1. The number of all holomorphic sections of \(\mathcal{S}\) is finite.
In this paper, the authors consider a holomorphic family of closed Riemann surfaces of genus two which is constructed by G. Riera [Duke Math. J. 44, 291–304 (1977; Zbl 0361.32014)]. The goal of this paper is to estimate the number of holomorphic sections of this family.

14J15 Moduli, classification: analytic theory; relations with modular forms
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
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