an:05653828
Zbl 1192.14030
Imayoshi, Yoichi; Komori, Yohei; Nogi, Toshihiro
Holomorphic sections of a holomorphic family of Riemann surfaces induced by a certain Kodaira surface
EN
Kodai Math. J. 32, No. 3, 450-470 (2009).
00256783
2009
j
14J15 14J27
holomorphic families of Riemann surfaces; elliptic surfaces; Kodaira surfaces
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 \textit{holomorphic family of closed Riemann surfaces} of genus \(g_0\) over \(B\).
A holomorphic mapping \(s : B \to M\) is said to be a \textit{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 \textit{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)], \textit{H. Grauert} [Publ. Math., Inst. Hautes ??tud. Sci. 25, 363--381 (1965; Zbl 0137.40503)], \textit{Y. Imayoshi} and \textit{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)], \textit{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 \textit{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.
Vasile Br??nz??nescu (Bucure??ti)
Zbl 0178.55102; Zbl 0137.40503; Zbl 0696.30044; Zbl 0583.32061; Zbl 0361.32014