Hodge theory and derived categories of cubic fourfolds

By Nicolas Addington, Richard P. Thomas. Duke mathematical journal, 2012

178 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5611d1a0d942bc0511a79bce5b3ec02ab1b58cbf

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Citations in the corpus

Lectures on K3 Surfaces (2016, methodology): “is in fact obtained by composing α ∈ HomDb(X)(E,E[1]) with β ∈ HomDb(X)(E,E[1]) ' HomDb(X)(E[1], E[2]) followed by the trace giving tr(β ◦ α) ∈ H2(X,O) ' k.”
Derived Categories View on Rationality Problems (2015, background): “bic fourfolds was suggested by B. Hassett in [Has00] in terms of Hodge structures, see also a discussion in [Add15]. The relation between the Conjecture of Hassett and Conjecture 4.2 was discussed in [AT14]. A general discussion of the category AX for a cubic fourfold can be found in [H15]. Finally,”
On two rationality conjectures for cubic fourfolds (2014, background): “In [1] this was called a weight-two Hodge structure, but it should really be called weight-zero.”
Stability conditions on Kuznetsov components (2017, background): “Moreover, the paper benefited from many useful discussions with Asher Auel, Agnieszka Bodzenta, François Charles, Igor Dolgachev, Frédéric Han, Daniel Huybrechts, Alexander Kuznetsov, Chunyi Li, Richard Thomas, Yukinobu Toda, and Song Yang.”
On the universal $CH_0$ group of cubic hypersurfaces (2014, unknown): “A G ] 1 9 Se p 20 15”
Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, methodology): “to the self-intersection H2 of a hyperplane class in W. These special cubic fourfolds organize themselves in divisors of Noether-Lefschetz type. Theorem 3.7 was ﬁrst proved by Addington and Thomas in [AT14] generically on these divisors. The completion of their result is in [BLM+19]. Both results th”
Derived categories of Gushel–Mukai varieties (2016, background): “HH•(S) =    k[2]⊕ k22 ⊕ k[−2] if S is K3, k12[0] if S is Enriques, k[2]⊕ k4[1]⊕ k6 ⊕ k4[−1]⊕ k[−2] if S is abelian, k2[1] ⊕ k4 ⊕ k2[−1] if S is bielliptic.”
Derived categories of cyclic covers and their branch divisors (2014, background): “(f∗(f ∗(ι kF))) = πk(ιk(F) ⊗RY ) = πk((F⊗χk)⊗RY ) = F. Similarly, to prove jk∗ is fully faithful we show j∗ k ◦ jk∗ = id. For this we take F∈ Db(Z)µn and consider the distinguished triangle F⊗L−1 Z ⊗χ[1] → j∗(j∗F) → F, 10 ALEXANDERKUZNETSOVANDALEXANDERPERRY see (3.7). If F = ιk(G) for G ∈ Db(Z), the”
The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, methodology): “Recently, CY2 categories have attracted a great deal of attention due to their connections to birational geometry, Hodge theory, and the construction of hyperkähler varieties [1, 39, 40, 7, 61, 6, 57, 76].”