Calabi-Yau threefolds with \(\rho > 13\)
type: Item
instance of: scholarly article
title: Calabi-Yau threefolds with \(\rho > 13\)
zbMATH Open document ID: 0759.14030
DOI: 10.1007/BF01934312
author: P. M. H. Wilson
author: D. R. Heath-Brown
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164988
review text: If a Calabi-Yau threefold \(X\) has Picard number \(\rho(X)>13\), then there is a birational morphism \(\varphi:X\to\overline X\) onto a normal projective threefold \(\overline X\) with \(\rho(\overline X)<\rho(X)\) and for which the exceptional locus of \(\varphi\) is covered by rational curves. This result improves one in an earlier paper of the second named author [Invent. Math. 98, No. 1, 139-155 (1989; Zbl 0688.14032)], where a similar result was proved under the assumption \(\rho(X)>19\). As in the earlier paper, the proof consists of combining classification methods from algebraic geometry with arguments of Davenport concerning diophantine approximation for integral cubic forms, the cubic form in question being the cup-product on the cohomology group \(H^ 2(X,\mathbb{Z})\). The improvement occurs because it is not actually necessary to search for integral zeros of the cubic form as was done in the previous paper, but instead for integral points at which the form is negative but with sufficiently small absolute value. This also has implications concerning the cone of Kähler classes in \(H^ 2(X,\mathbb{R})\), and its intersection being sparse in the cubic cone defined by the cup-product.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 126355
zbMATH Keywords: Calabi-Yau threefold
zbMATH Keywords: Picard number
zbMATH Keywords: classification
zbMATH Keywords: diophantine approximation for integral cubic forms
MaRDI profile type: Publication
cites work: Q5749404
cites work: Cubic forms in thirty-two variables
cites work: Non-Homogeneous Cubic Equations
cites work: Recent results in complex differential geometry
cites work: Calabi-Yau manifolds and a conjecture of Kobayashi
cites work: Calabi-Yau manifolds with large Picard number
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OpenAlex ID: W2036211641
description: scientific article; zbMATH DE number 126355