A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold
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.
scientific article; zbMATH DE number 6826204