Calabi-Yau hypersurfaces in products of semi-ample surfaces
type: Item
instance of: scholarly article
title: Calabi-Yau hypersurfaces in products of semi-ample surfaces
zbMATH Open document ID: 0658.14020
DOI: 10.1007/BF01466771
author: T. Hübsch
author: Paul S. Green
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: This paper continues the quest of string theorists for a Calabi-Yau 3- manifold with small Euler characteristic. One of the features of these manifolds is the fact that the Euler characteristic determines the difference \(b_{1,1}-b_{1,2}\) of Hodge numbers but the individual numbers themselves are not so easy to compute and yet have great geometrical significance. In this case, where \(c_ 1=0\), \(b_{1,2}\) is isomorphic to the infinitesimal deformation number \(\dim (H^ 1(X,\theta))\) and physicists have already encountered the fact that \textit{K. Kodaira}, too, noticed thirty years ago [see ``Complex manifolds and deformation of complex structures'' (1986; Zbl 0581.32012)] namely that the number of moduli is usually easier to compute than the number \(b_{1,2}.\) Here, the authors aim to produce Calabi-Yau manifolds as anti-canonical divisors in the product of two surfaces with ample (or semi-ample) anti- canonical bundle. Representing such surfaces as blow-ups of \({\mathbb{C}}P^ 2\), they obtain information about moduli and finally calculate Euler characteristics. In fact, there are certain features which escape their attention. One (which may be of physical interest) is the relationship between the intersection form on the orthogonal complement of the anti-canonical divisor of a surface with ample anti-canonical bundle and the Dynkin diagrams of type \(E_ k\). - The second one is the basic geometric fact that all cubics which pass through eight points in the plane pass through a ninth and consequently an anti-canonical divisor for the product of two planes blown up eight times is singular. - Some small Euler characteristics which seem to be attainable from their formula (3.1) do not therefore exist.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 81T99
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 4075269
zbMATH Keywords: Calabi-Yau 3-manifold with small Euler characteristic
zbMATH Keywords: Hodge numbers
zbMATH Keywords: infinitesimal deformation number
zbMATH Keywords: number of moduli
zbMATH Keywords: anti-canonical bundle
zbMATH Keywords: intersection form
reviewed by: Nigel Hitchin
MaRDI profile type: Publication
cites work: Q3756080
cites work: Calabi's conjecture and some new results in algebraic geometry
cites work: Q3785565
cites work: Calabi-Yau manifolds - motivations and constructions
cites work: Calabi-Yau manifolds as complete intersections in products of complex projective spaces
cites work: On a theorem of Lefschetz
cites work: Q3704179
cites work: Polynomial deformations and cohomology of Calabi-Yau manifolds
cites work: Q4195061
cites work: Q4154669
cites work: On fiber products of rational elliptic surfaces with section
full work available at URL: https://doi.org/10.1007/bf01466771
OpenAlex ID: W1968278666
Wikidata QID: Q125766850
description: scientific article; zbMATH DE number 4075269