A criterion for flatness of Hodge bundles over curves and geometric applications
Since Iitaka proposed a scheme for classifying higher dimensional varieties, it has become important to understand ''fibrations'', i.e. surjective morphisms between varieties with connected fibres. Especially the so-called \(C_{n,m}\)-conjecture about Kodaira dimensions for such fibrations has been stimulating much research during the past decade [cf. \textit{H. Esnault}, Sémin Bourbaki 1980/81, Exposé 568, Lect. Notes Math. 901, 111-131 (1981; Zbl 0481.14013)]. It has been T. Fujita's idea to use the theory of variations of Hodge structure, especially the curvature properties of those. Subsequently Y. Kawamata added to the tools the use of Schmid's asymptotic analysis, but only in a peripheral way. The main novel idea in the paper under review is to exploit the asymptotic analysis more systematically in order to obtain information when previously established inequalities stemming from the semi- positivity of certain Hodge bundles become equalities. This information yields the criterion for flatness in the title of the paper. Of course, in order to extract geometrical information from this, the fibres of the fibration should satisfy a certain Torelli property. The result is a certain generalisation of Arakelov's theorem, which says that if \(f:\quad X\to C\) is a fibration with curves as fibres and base (C is complete) and if \(\deg ree (f_*\omega_{X/C})=0,\) then all smooth fibres are isomorphic. There are some technical difficulties, since unlike the situation of Arakelov's theorem, one is working in the setting of Kähler geometry. As a bonus, however, one obtains a certain extension of the \(C_{n,1}\)-conjecture (as proved by Kawamata) valid for Kähler varieties. This extension is crucial for classification purposes.
scientific article; zbMATH DE number 3873478
scientific article