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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.

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