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 K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 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 non-empty, Zariski open subset in each divisor.

arXiv comment37 pages. Applications to algebraic cycles added, and other improvements following referees' suggestions. This is a slightly expanded version of the paper to appear in Duke Math J. v3: minor correction to section 7.3

References in corpus (10)

• Derived categories of cubic fourfolds
• Hochschild cohomology and Atiyah classes
• New derived symmetries of some hyperkähler varieties
• The Mukai pairing, I: the Hochschild structure
• Derived equivalences of K3 surfaces and orientation
• Orlov spectra: bounds and gaps
• Caldararu's conjecture and Tsygan's formality
• Infinitesimal Derived Torelli Theorem for K3 surfaces
• Cubic fourfolds containing a plane and a quintic del Pezzo surface
• A global Torelli theorem for rigid hyperholomorphic sheaves

Cited by in corpus (60)

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• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
• Cubic fourfolds fibered in sextic del Pezzo surfaces
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• Derived categories of cyclic covers and their branch divisors
• Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• Derived categories of K3 surfaces, O'Grady's filtration, and zero-cycles on holomorphic symplectic varieties
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• A note on the Kuznetsov component of the Veronese double cone
• The Brauer group is not a derived invariant
• Cycles, derived categories, and rationality
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• On the motive of O'Grady's ten-dimensional hyper-Kähler varieties
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• Derived equivalences of hyperkähler varieties
• Two-dimensional gauge dynamics and the topology of singular determinantal varieties
• On the double EPW sextic associated to a Gushel-Mukai fourfold
• Explicit rationality of some cubic fourfolds
• New examples of rational Gushel-Mukai fourfolds
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• The Voisin map via families of extensions
• ACM bundles on cubic fourfolds containing a plane
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• Rational cubic fourfolds in Hassett divisors
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• Lagrangian embeddings of cubic fourfolds containing a plane
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• Homological Projective Duality for Determinantal Varieties
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• Motives of moduli spaces on K3 surfaces and of special cubic fourfolds
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• Special cubic birational transformations of projective spaces
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• Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds
• New rational cubic fourfolds arising from Cremona transformations
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• Mirror symmetry and rigid structures of generalized K3 surfaces
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• On lattice polarizable cubic fourfolds
• Dg categories of cubic fourfolds
• Some Moduli of $n$-pointed Fano Fourfolds
• Hochschild cohomology and deformations of $\mathbb{P}$-functors
• The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties