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$K3$ CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION

  • Junliang Shen (a1) and Qizheng Yin (a2)

Abstract

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov’s noncommutative $K3$ category associated to a nonsingular cubic 4-fold.

By introducing a filtration on the $\text{CH}_{1}$ -group of a cubic 4-fold $Y$ , we conjecture a sheaf/cycle correspondence for the associated $K3$ category  ${\mathcal{A}}_{Y}$ . This is a noncommutative analog of O’Grady’s conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.

Our method provides systematic constructions of (a) the Beauville–Voisin filtration on the $\text{CH}_{0}$ -group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in ${\mathcal{A}}_{Y}$ .

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1.Addington, N. and Lehn, M., On the symplectic eightfold associated to a Pfaffian cubic fourfold, J. Reine Angew. Math. 731 (2017), 129137.
2.Addington, N. and Thomas, R. P., Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163(10) (2014), 18851927.
3.Atiyah, M. F. and Hirzebruch, F., The Riemann–Roch theorem for analytic embeddings, Topology 1 (1962), 151166.
4.Bayer, A., Lahoz, M., Macrì, E., Nuer, H., Perry, A. and Stellari, P., Families of stability conditions, in preparation.
5.Bayer, A., Lahoz, M., Macrì, E. and Stellari, P., Stability conditions on Kuznetsov components. Appendix joint with X. Zhao, Preprint, 2017, arXiv:1703.10839.
6.Beauville, A., On the splitting of the Bloch–Beilinson filtration, in Algebraic Cycles and Motives, Vol. 2, London Math. Soc. Lecture Note Ser., Volume 344, pp. 3853 (Cambridge Univ. Press, Cambridge, 2007).
7.Beauville, A. and Donagi, R., La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301(14) (1985), 703706.
8.Beauville, A. and Voisin, C., On the Chow ring of a K3 surface, J. Algebraic Geom. 13(3) (2004), 417426.
9.Bloch, S. and Srinivas, V., Remarks on correspondences and algebraic cycles, Amer. J. Math. 105(5) (1983), 12351253.
10.Bridgeland, T., Stability conditions on triangulated categories, Ann. of Math. (2) 166(2) (2007), 317345.
11.Charles, F. and Pacienza, G., Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles, Preprint, 2014, arXiv:1401.4071v2.
12.Clemens, C. H. and Griffiths, P. A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281356.
13.Coskun, I. and Starr, J., Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. IMRN 2009(24) (2009), 46264641.
14.Fulton, W., Intersection theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, second edition, A Series of Modern Surveys in Mathematics, Volume 2 (Springer-Verlag, Berlin, 1998). xiv+470 pp.
15.Huybrechts, D., Chow groups of K3 surfaces and spherical objects, J. Eur. Math. Soc. 12(6) (2010), 15331551.
16.de Jong, A. J. and Starr, J., Cubic fourfolds and spaces of rational curves, Illinois J. Math. 48(2) (2004), 415450.
17.Kuznetsov, A., Derived categories of cubic fourfolds, in Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics, Volume 282, pp. 219243 (Birkhäuser Boston, Inc., Boston, MA, 2010).
18.Kuznetsov, A., Semiorthogonal decompositions in algebraic geometry, in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, pp. 635660 (Kyung Moon Sa, Seoul, 2014).
19.Kuznetsov, A., Derived categories view on rationality problems, in Rationality Problems in Algebraic Geometry, Lecture Notes in Mathematics, Volume 2172, pp. 67104 (Fond. CIME/CIME Found. Subser., Springer, Cham, 2016).
20.Kuznetsov, A. and Markushevich, D., Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys. 59(7) (2009), 843860.
21.Kuznetsov, A. and Perry, A., Derived categories of Gushel–Mukai varieties, Compos. Math. 154(7) (2018), 13621406.
22.Lahoz, M., Lehn, M., Macrì, E. and Stellari, P., Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects, J. Math. Pures Appl. (9) 114 (2018), 85117.
23.Laza, R., Sacca, G. and Voisin, C., A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold, Acta Math. 218(1) (2017), 55135.
24.Lehn, C., Lehn, M., Sorger, C. and van Straten, D., Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87128.
25.Li, C., Pertusi, L. and Zhao, X., Twisted cubics on cubic fourfolds and stability conditions, Preprint, 2018, arXiv:1802.01134.
26.Marian, A. and Zhao, X., On the group of zero-cycles of holomorphic symplectic varieties, Preprint, 2017, arXiv:1711.10045v2.
27.Markushevich, D. and Tikhomirov, A., The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom. 10(1) (2001), 3762.
28.Markushevich, D. and Tikhomirov, A., Symplectic structure on a moduli space of sheaves on a cubic fourfold, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), 131158.
29.Mumford, D., Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195204.
30.O’Grady, K. G., Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49117.
31.O’Grady, K. G., Moduli of sheaves and the Chow group of K3 surfaces, J. Math. Pures Appl. (9) 100(5) (2013), 701718.
32.Paranjape, K. H., Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139(3) (1994), 641660.
33.Rojtman, A. A., The torsion of the group of 0-cycles modulo rational equivalence, Ann. of Math. (2) 111(3) (1980), 553569.
34.Shen, J., Yin, Q. and Zhao, X., Derived categories of $K3$ surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties, Preprint, 2017, arXiv:1705.06953v2.
35.Shen, M., On relations among 1-cycles on cubic hypersurfaces, J. Algebraic Geom. 23(3) (2014), 539569.
36.Shen, M. and Vial, C., The Fourier transform for certain hyperkähler fourfolds, Mem. Amer. Math. Soc. 240(1139) (2016), vii+163 pp.
37.Voisin, C., Théorème de Torelli pour les cubiques de ℙ5, Invent. Math. 86(3) (1986), 577601.
38.Voisin, C., Intrinsic pseudo-volume forms and K-correspondences, in The Fano Conference, pp. 761792 (Univ. Torino, Turin, 2004).
39.Voisin, C., On the Chow ring of certain algebraic hyper-Kähler manifolds, Pure Appl. Math. Q. 4(3, part 2) (2008), 613649.
40.Voisin, C., Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady, in Recent Advances in Algebraic Geometry, London Mathematical Society Lecture Note Series, Volume 417, pp. 422436 (Cambridge Univ. Press, Cambridge, 2015).
41.Voisin, C., Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties, in K3 Surfaces and their Moduli, Progress in Mathematics, Volume 315, pp. 365399 (Birkhäuser/Springer, Cham, 2016).
42.Voisin, C., Hyper-Kähler compactification of the intermediate Jacobian fibration of a cubic fourfold: the twisted case, in Local and Global Methods in Algebraic Geometry, Contemporary Mathematics, Volume 712, pp. 341355 (American Mathematical Society, Providence, RI, 2018).
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$K3$ CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION

  • Junliang Shen (a1) and Qizheng Yin (a2)

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