A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold
type: Item
instance of: scholarly article
title: A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold
zbMATH Open document ID: 1409.14053
author: Radu Laza
author: Giulia Saccà
author: Claire Voisin
published in: Acta Mathematica
publication date: 2018-01-15T00:00:00Z
review text: The authors address the problem of construction and classification of compact hyper-Kähler manifolds; particularly, they are interested in an understanding the 10-dimensional example OG10 by \textit{K. G. O'Grady} [J. Reine Angew. Math. 512, 49--117 (1999; Zbl 0928.14029)] originally constructed as the desingularization of the moduli space of certain torsion-free sheaves of rank two on a projective \(K3\) surface. The main result of the paper is as follows: Let \(X\subset{\mathbb P}^5\) be a general cubic fourfold, \(U=({\mathbb P}^5)^\vee\setminus X^\vee\) the locus of smooth hyperplane sections, and \({\mathcal J}_U\to U\) the associated intermediate Jacobian fibration. Then there exists a smooth projective compactification \(\overline{\mathcal J}\to ({\mathbb P}^5)^\vee\), which carries a holomorphic symplectic form, with respect to which the fibration is Lagrangian. Furthermore, \(\overline{\mathcal J}\) is deformation equivalent to OG10. This settles in affirmative a conjecture of \textit{A. Kuznetsov} and \textit{D. Markushevich} [J. Geom. Phys. 59, No. 7, 843--860 (2009; Zbl 1181.14049)]. The construction of such a symplectic compactification \(\overline{\mathcal J}\) relies on the study of degenerations of intermediate Jacobians, which in turn uses the Mumford's idea to regard the intermediate Jacobian of a cubic threefold as the Prym variety \(\mathrm{Prym}(\widetilde C/C)\), where \(C\) is a quintic curve which is the discriminant locus of the conic bundle obtained by projecting the cubic threefold from a general line, and \(\widetilde C\) is the natural unramified double cover of \(C\). It is shown that this construction goes through mildly singular cubic threefolds.
reviewed by: Eugenii I. Shustin
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 53D99
zbMATH DE Number: 6826204
zbMATH Keywords: hyper-Kähler manifold
zbMATH Keywords: Lagrangian fibration
zbMATH Keywords: intermediate Jacobian
zbMATH Keywords: cubic fourfold
zbMATH Keywords: cubic threefold
MaRDI profile type: Publication
full work available at URL: acta.2017.v218.n1.a2
OpenAlex ID: W2963963601
DOI: 10.4310/ACTA.2017.V218.N1.A2
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description: scientific article; zbMATH DE number 6826204