The Mordell-Bombieri-Noguchi conjecture over function fields (Q1107594)

The author discusses the higher dimensional analogue of Mordell's finiteness conjecture over function fields. The algebraic fibre space whose general fibre a variety of general type is conjectured to be birationally trivial, i.e. birationally equivalent to the product of the base space and a general fibre, if there are an infinite number of cross sections whose union is Zariski dense. Many authors have proved this conjecture under some additional conditions. - In this paper the author proves it supposing that \({\mathcal O}(\alpha)\otimes p^*{\mathcal O}(-K_ X)\) is \(f\circ p\)-nef for some \(\alpha >0\), where p is the projection, \(p(\Omega^ s_ X)\to X\), where \(s=\dim (S)\) and \(f:\quad X\to S\) is a proper surjective map. - The main theorem follows from the authors previous result [see Math. Ann. 262, 101-123 (1983; Zbl 0438.14011)] and by a vanishing theorem due to \textit{J. Kollár} [Ann. Math., II. Ser. 123, 11-42 (1986; Zbl 0598.14015) and 124, 171-202 (1986; Zbl 0605.14014)].