Stability conditions on Kuznetsov components

Bayer, Arend; Lahoz, Martí; Macrì, Emanuele; Stellari, Paolo

doi:10.48550/arxiv.1703.10839

Cited by 38 publications

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“…Theorem 1.1 shows that the stability conditions constructed in [BLMS17] are Serre-invariant in the sense of Definition 4.1. As pointed out in Theorem 3.18, an analogous result holds more generally for certain Fano threefolds of Picard rank 1, index 1 and even genus.…”

Section: Introductionmentioning

confidence: 99%

“…Kuznetsov showed in [Kuz10] that for many cubic fourfolds (notably, in each case X was rational), KupXq is equivalent to the derived category of a K3 surface. Afterwards, it was shown in [BLMS17] that KupXq admits stability conditions, and that the corresponding moduli spaces of semistable complexes are smooth, projective hyperkahler varieties [BLM `21].…”

Section: Introductionmentioning

confidence: 99%

“…The richness of these results motivates the study of analogous situations. In particular, the derived categories of Gushel-Mukai (GM) varieties are known to contain exceptional collections [KP18], the Kuznetsov components of which have been shown to admit stability conditions in [BLMS17,PPZ19]. See Section 2.2 for a summary.…”

Section: Introductionmentioning

confidence: 99%

“…In [PY20, Proposition 5.7] it is shown that the stability conditions induced on the Kuznetsov component of a Fano threefold of Picard rank 1 and index 2 (e.g. a cubic threefold) with the method in [BLMS17] are Serre-invariant. Using this result, the authors further proved that non-empty moduli spaces of stable objects with respect to these stability conditions are smooth.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome this problem, we first induce stability conditions σps, qq on KupXq from (a double tilt of) tilt stability conditions defined on D b pXq, via the criterion given in Proposition 5.1 of [BLMS17], for pairs ps, qq in a region that is slightly larger than the one considered in [BLMS17], lying above the boundary given by Li in [Li19b] (see Propositions 3.1 and 3.2). Next, fixing one such stability condition, we study the action on it of the left mutation with respect to O X and then with respect to U X .…”

Section: Introductionmentioning

confidence: 99%

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Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

Pertusi¹,

Robinett²

2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

Pertusi¹,

Robinett²

2021

Preprint

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Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

Pertusi

Robinett

2023

Mathematische Nachrichten

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We show that the stability conditions on the Kuznetsov component of a Gushel–Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of GL2+(prefixdouble-struckR)$\text{GL}^+_2(\operatorname{\mathbb {R}})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel–Mukai fourfolds.

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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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Stability conditions on Kuznetsov components

Cited by 38 publications

References 50 publications

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

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

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