On the existence of nontrivial 3-folds with vanishing Hodge cohomology
type: Item
instance of: scholarly article
title: On the existence of nontrivial 3-folds with vanishing Hodge cohomology
zbMATH Open document ID: 1117.14042
author: Jing Zhang
published in: Michigan Mathematical Journal
publication date: 2007-04-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 5140644
reviewed by: N. Mohan Kumar
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1156345606
OpenAlex ID: W2014130594
cites work: Q3952228
cites work: Q3659893
cites work: Q3714183
cites work: Q4941090
cites work: On Stability of Compact Submanifolds of Complex Manifolds
cites work: Higher direct images of dualizing sheaves. I
cites work: Higher direct images of dualizing sheaves. II
cites work: Q3496299
cites work: Global 2-forms on regular 3-folds of general type
cites work: Global holomorphic 2-forms and pluricanonical systems on threefolds
cites work: Q3015015
cites work: Introduction to the Mori program
cites work: Affine like Surfaces
cites work: Q5628266
cites work: Holomorphic convexity of spaces of analytic cycles
cites work: Q3973874
cites work: Q4661966
cites work: Analytic Sheaf Cohomology Groups of Dimension n of n-Dimensional Complex Spaces
cites work: Invariance of plurigenera
cites work: Fibres holomorphes à base et à fibre de Stein
cites work: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack
cites work: Q3658114
cites work: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface
cites work: Threefolds with vanishing Hodge cohomology
cites work: Q4364034
DOI: 10.1307/MMJ/1156345606
description: scientific article; zbMATH DE number 5140644