## Duke Mathematical Journal

### Hodge theory and derived categories of cubic fourfolds

#### Abstract

Cubic fourfolds behave in many ways like $K3$ surfaces. Certain cubics—conjecturally, the ones that are rational—have specific $K3$ surfaces associated to them geometrically. Hassett has studied cubics with $K3$ surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with $K3$ surfaces associated to them at the level of derived categories.

These two notions of having an associated $K3$ surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.

#### Article information

Source
Duke Math. J., Volume 163, Number 10 (2014), 1885-1927.

Dates
First available in Project Euclid: 8 July 2014

https://projecteuclid.org/euclid.dmj/1404824304

Digital Object Identifier
doi:10.1215/00127094-2738639

Mathematical Reviews number (MathSciNet)
MR3229044

Zentralblatt MATH identifier
1309.14014

#### Citation

Addington, Nicolas; Thomas, Richard. Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163 (2014), no. 10, 1885--1927. doi:10.1215/00127094-2738639. https://projecteuclid.org/euclid.dmj/1404824304

#### References

• [1] D. Arinkin, J. Block, and T. Pantev, ∗-Quantizations of Fourier-Mukai transforms, Geom. Funct. Anal. 23 (2013), 1403–1482.
• [2] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math. 3, Amer. Math. Soc., Providence, 1961, 7–38.
• [3] M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151–166.
• [4] A. Auel, M. Bernardara, M. Bolognesi, and A. Várilly-Alvarado, Cubic fourfolds containing a plane and a quintic del Pezzo surface, Algebraic Geom. 2 (2014), 181–193.
• [5] W. A. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528.
• [6] A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge Univ. Press, Cambridge, 1984.
• [7] M. Ballard, D. Favero, and L. Katzarkov, Orlov spectra: Bounds and gaps, Invent. Math. 189 (2012), 359–430.
• [8] A. Beauville, J.-P. Bourguignon, and M. Demazure, Géométrie des surfaces K3: Modules et périodes, Astérisque 126, Soc. Math. France, Paris, 1985.
• [9] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703–706.
• [10] S. Bloch, Semi-regularity and de Rham cohomology, Invent. Math. 17 (1972), 51–66.
• [11] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1–36, 258.
• [12] R.-O. Buchweitz and H. Flenner, The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character, Adv. Math. 217 (2008), 243–281.
• [13] D. Calaque, C. Rossi, and M. Van den Bergh, Căldăraru’s conjecture and Tsygan’s formality, Ann. of Math. (2) 176 (2012), 865–923.
• [14] D. Calaque and M. van den Bergh, Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010), 1839–1889.
• [15] A. Căldăraru, The Mukai pairing, I: The Hochschild structure, preprint, arXiv:math/0308079.
• [16] A. Căldăraru, The Mukai pairing, II: The Hochschild-Kostant-Rosenberg isomorphism, Adv. Math. 194 (2005), 34–66.
• [17] C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.
• [18] D. A. Cox, Primes of the Form $x^{2}+ny^{2}$, Wiley, New York, 1989.
• [19] B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), 103–114.
• [20] B. Hassett, Special cubic fourfolds, Compositio Math. 120 (2000), 1–23.
• [21] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006.
• [22] D. Huybrechts, Complex and real multiplication for K3 surfaces, lecture notes, GAeL–Géométrie Algébrique en Liberté, Aranjuez, Spain, 2008, www.math.uni-bonn.de/people/huybrech/Transcent.pdf.
• [23] D. Huybrechts, Lectures on K3 surfaces, preprint, www.math.uni-bonn.de/people/huybrech/K3Global.pdf (accessed 28 April 2014).
• [24] D. Huybrechts, E. Macrì, and P. Stellari, Derived equivalences of K3 surfaces and orientation, Duke Math. J. 149 (2009), 461–507.
• [25] D. Huybrechts and R. P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346 (2010), 545–569.
• [26] Y. Kawamata, Unobstructed deformations: A remark on a paper of Z. Ran, “Deformations of manifolds with torsion or negative canonical bundle,” J. Algebraic Geom. 1 (1992), 183–190.
• [27] A. Kuznetsov, “Derived categories of cubic fourfolds” in Cohomological and Geometric Approaches to Rationality Problems, Progr. Math. 282, Birkhäuser, Boston, 2010, 219–243.
• [28] R. Laza, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2) 172 (2010), 673–711.
• [29] M. Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), 175–206.
• [30] E. Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009), 213–233.
• [31] E. Macrì and P. Stellari, Infinitesimal derived Torelli theorem for K3 surfaces, with appendix by S. Mehrotra, Int. Math. Res. Not. IMRN 2009, no. 17, 3190–3220.
• [32] E. Macrì and P. Stellari, Fano varieties of cubic fourfolds containing a plane, Math. Ann. 354 (2012), 1147–1176.
• [33] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/1964), 347–361.
• [34] S. Mukai, “On the moduli space of bundles on $K3$ surfaces, I” in Vector Bundles on Algebraic Varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Mumbai, 1987, 341–413.
• [35] V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238.
• [36] Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), 279–291.
• [37] M. Rapoport, Complément à l’article de P. Deligne “La conjecture de Weil pour les surfaces K3,” Invent. Math. 15 (1972), 227–236.
• [38] Y. Toda, Deformations and Fourier-Mukai transforms, J. Differential Geom. 81 (2009), 197–224.
• [39] C. Voisin, Théorème de Torelli pour les cubiques de $\mathbb{P}^{5}$, Invent. Math. 86 (1986), 577–601.
• [40] C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge Stud. Adv. Math. 76, Cambridge Univ. Press, Cambridge, 2002.
• [41] C. Voisin, Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), 261–296.
• [42] S. Zucker, The Hodge conjecture for cubic fourfolds, Compositio Math. 34 (1977), 199–209.