The K3 category of a cubic fourfold

Huybrechts, Daniel

doi:10.1112/s0010437x16008137

Cited by 62 publications

(113 citation statements)

References 80 publications

(286 reference statements)

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“…The key to link S ∼ X, (S, L), and (S, α) ∼ X to the properties ( * * ) and ( * * ′ ) is the following result in [AT14] generalized to the twisted case in [Hu17]. For (ii), again one direction is easy, as H 1,1 (S, Z) contains the B-field shift of (H 0 ⊕H 4 )(S, Z), cf.…”

Section: The Mukai Vectormentioning

confidence: 99%

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Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories

Huybrechts

2019

Lecture Notes of the Unione Matematica Italiana

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These are notes of lectures given at the school 'Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.The primitive Hodge structure of a smooth cubic fourfold X ⊂ P 5 is concentrated in degree four and it is of a very particular type. Once a Tate twist is applied and the sign of the intersection form is changed, it reveals its true nature. It very much looks like the Hodge structure of a K3 surface. In his thesis Hassett [Ha00] studied this curious relation and the intricate lattice theory behind it in greater detail. He established a transcendental correspondence between polarized K3 surfaces of certain degrees and special cubic fourfolds, some aspects of which are reminiscent of the Kuga-Satake construction. The geometric nature of the Hassett correspondence is still not completely understood but it seems that derived categories are central for its understanding. Work of Addington and Thomas [AT14] represents an important step in this direction, combining Hassett's Hodge theory with Kuznetsov's categorical approach to hypersurfaces.

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Section: The Mukai Vectormentioning

confidence: 99%

“…The goal of[AT14] was to compare Hassett's condition ( * * ) with the condition A X ≃ D b (S). Building upon[AT14], the twisted version was later dealt with in[Hu17].…”

mentioning

confidence: 99%

Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories

Huybrechts

2019

Lecture Notes of the Unione Matematica Italiana

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“…We argue similarly as using the global Torelli theorem . Let us use the notation from [, Proposition 4.1]. We have to show that there is an embedding

H^{2} false(X, double-struckZ false) ↪ \tilde{H} false(S, α_{3} false)

(into the Hodge structure of the twisted

K 3

‐surface see [, Definition 2.5]) that is compatible with the Hodge structure.…”

Section: The Third Construction‐moduli Space Of Twisted Sheavesmentioning

confidence: 99%

“…Let us use the notation from [, Proposition 4.1]. We have to show that there is an embedding

H^{2} false(X, double-struckZ false) ↪ \tilde{H} false(S, α_{3} false)

(into the Hodge structure of the twisted

K 3

‐surface see [, Definition 2.5]) that is compatible with the Hodge structure. Given the embedding, we find a vector

v \in \tilde{H} false(S, α_{3} false)

in the orthogonal complement of the image of

H^{2} false(X, double-struckZ false)

having

(v, v) = 2

.…”

Section: The Third Construction‐moduli Space Of Twisted Sheavesmentioning

confidence: 99%

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Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

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Abstract. We show that a Hilbert scheme of conics on a Fano fourfold double cover of P 2 × P 2 ramified along a divisor of bidegree (2, 2) admits a P 1 -fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of K3 surfaces of degree 2, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler 4-folds of type K3 [2] . As a consequence we present also three constructions of quartic Kummer surfaces in P 3 : as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over P 1 .

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Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

Macrì

Stellari

2019

Lecture Notes of the Unione Matematica Italiana

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We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds.These notes originated from the lecture series by the first author at the school on BirationalFrom the homological point of view, it was observed by Kuznetsov [Kuz10] that the derived category D b (W ) of a cubic fourfold W contains an admissible subcategory K u(W ) which is the right orthogonal of the category generated by the three line bundles O W , O W (H) and O W (2H). We will refer to K u(W ) as the Kuznetsov component of W . The category K u(W ) has the same homological properties of D b (S), for S a K3 surface: it is an indecomposable category with Serre functor which is the shift by 2 and the same Hochschild homology as D b (S). But, for W very general, there cannot be a K3 surface S with an equivalence K u(W ) ∼ = D b (S). This is the reason why we should think of Kuznetsov components as non-commutative K3 surfaces.The study of non-commutative varieties was started more than thirty years ago. Artin and Zhang [AZ84] investigated the case of non-commutative projective spaces (see also the book in preparation [Yek18]). In these notes we will follow closely the approach developed by Kuznetsov [Kuz10] and Huybrechts [Huy17]. One important feature is that the Kuznetzov component comes with a naturally associated lattice H(K u(W ), Z) with a weight-2 Hodge Non-commutative Calabi-Yau varietiesIn this section, we follow closely the presentation and main results in [Kuz15]; foundational references are also [BO95, BvdB03, Kuz07, Kuz14, Perr18a]. We start with a very short review on semiorthogonal decompositions and exceptional collections in Section 2.1. We then introduce the notion of non-commutative variety in Section 2.2 and study basic facts about Serre functors and Hochschild (co)homology. Finally, in Section 2.3 we sketch the

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The K3 category of a cubic fourfold

Cited by 62 publications

References 80 publications

Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories

Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories

Hyper-Kähler fourfolds and Kummer surfaces

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

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