Entire holomorphic curves in surfaces
An aim of the work is the strengthen of the famous Bloch conjecture (1926): If a projective variety X has irregularity \(q>\dim (X)\), then every entire curve \(f: {\mathbb{C}}\mapsto X\) is algebraically degenerate, i.e. has image lying in a proper subvariety. It was proved for surfaces by \textit{T. Ochiai} [Invent. Math. 43, 83-96 (1977; Zbl 0374.32006)], in general case by \textit{M. Green} and \textit{P. Griffiths} [in Differential geometry, Proc. int. Chern Symp., Berkeley 1979, 41-74 (1980; Zbl 0508.32010)] and independently by \textit{Y. Kawamata} [Invent. Math. 57, 97- 100 (1980; Zbl 0569.32012)] and by \textit{P.-M. Wong} [Am. J. Math. 102, 493-502 (1980; Zbl 0439.32010)]. The main result of the paper states that this conjecture remains valid for projective surfaces of general type with irregularity \(q=2\) and with simple Albanese torus. Theorem (6.7 and 6.8): Let X be a normal algebraic surface of general type, A a simple 2-dimensional abelian variety, and \(\pi: X\mapsto A\) a surjective holomorphic map. Then \(\pi \circ f=const\quad for\) every entire curve \(f: {\mathbb{C}}\mapsto X.\) In particular, if \(\pi\) is a branched cover (i.e. \(\pi\) is finite-to-one), then there is no nonconstant entire curve in X [and so X is Kobayashi hyperbolic (the reviewer)]. In the case when A is not simple, the same method leads to the following conclusion: X is Kobayashi hyperbolic iff it contains no rational and elliptic curves.
scientific article; zbMATH DE number 3989541