The derived category of a non-generic cubic fourfold containing a plane

Moschetti, Riccardo

doi:10.4310/mrl.2018.v25.n5.a8

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…This shows the analogy with the untwisted case considered in the theorem above. We conclude by observing that a partial result in the case of cubics containing a plane is in [Mos16].…”

Section: Fano Varieties and Their Kuznetsov Components: Examplesmentioning

confidence: 73%

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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“…This shows the analogy with the untwisted case considered in the theorem above. We conclude by observing that a partial result in the case of cubics containing a plane is in [Mos16].…”

Section: Fano Varieties and Their Kuznetsov Components: Examplesmentioning

confidence: 73%

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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“…A version of the corollary also holds for K3 surfaces with a Brauer twist, completing a result by Huybrechts [Huy17, Theorem 1.4]; the corresponding Hodge-theoretic condition is the existence of a squarezero class in H Hdg (Ku(X), Z) (see Proposition 33.1). Partial results were also obtained in [Kuz10,Mos18].…”

Section: The Connected Component Stabmentioning

confidence: 83%

Stability conditions in families

et al. 2021

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We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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“…In [19] the equivalence of the Kuznetsov component to the twisted K3 surface is obtained by realizing X as a hyperplane section of a smooth cubic 5-fold containing the plane and applying the result of quadric surface bundles over threefolds in [14]. Section 5 in this paper provides a more direct proof.…”

Section: Example 62 ([19]mentioning

confidence: 99%

“…Acknowledgements. I would like to thank Arend Bayer, Qingyuan Jiang for numerous helpful conversations, and thank Alexander Kuznetsov for pointing out the reference [19]. The author is supported by the ERC Consolidator grant WallCrossAG, no.…”

Section: Introductionmentioning

confidence: 99%

Residual categories of quadric surface bundles

Xie¹

2022

Preprint

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We show the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme in two situations: (1) the quadric surface bundle has a smooth section; (2) the total space and the base of the quadric surface bundle are smooth and the base is a surface. We provide two proofs in situation (1) describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In situation (2) the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. As an application we apply the results to certain complete intersections of quadrics.

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The derived category of a non-generic cubic fourfold containing a plane

Cited by 4 publications

References 18 publications

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

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

Stability conditions in families

Residual categories of quadric surface bundles

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