On the existence of nontrivial 3-folds with vanishing Hodge cohomology
The author continues her study of algebraic threefolds with vanishing Hodge cohomology. The aim is to study smooth algebraic varieties \(Y\) over complex numbers of dimension \(n\) with \(H^i(Y,\Omega^j_Y)=0\) for all \(i>0\) and all \(j\). A question of \textit{J.-P. Serre} [in: Centre Belge Rech. math., Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles du 11 au 14 mars, 57--68 (1953; Zbl 0053.05302)] asks whether such a \(Y\) is Stein. If \(n=1\), clearly \(Y\) is affine and the case of surfaces in the above situation was completely determined by the reviewer [J. Algebr. Geom. 2, No. 4, 689--703 (1993; Zbl 0810.14015)]. In the paper under review, the author continues her study of such \(Y\) when \(n=3\) with the extra hypothesis that \(Y\) has non-constant regular functions. In particular it is shown that the fibres of the morphism induced by any of these nonconstant functions must be of similar type of surfaces from the above classification of surfaces.
scientific article; zbMATH DE number 5140644
scientific article