Holomorphic sections of a holomorphic family of Riemann surfaces induced by a certain Kodaira surface.

*(English)*Zbl 1192.14030Let \(M\) 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.

(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.

Reviewer: Vasile Brînzănescu (Bucureşti)