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

Pertusi, Laura; Robinett, Ethan

doi:10.48550/arxiv.2112.04769

Cited by 6 publications

(8 citation statements)

References 20 publications

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“…By this equivalence and the results for Y 3 , Y 4 , Y 5 , we deduce that Ku(X d ) has a unique orbit of Serre-invariant stability conditions for d = 14, 18, 22. Moreover, by [61,Theorem 4.25] (see also [115,Corollary 4.5]) this uniqueness result holds also for Ku(X 10 ). In the remaining cases, i.e.…”

Section: Cubic Threefolds and Beyondmentioning

confidence: 74%

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Categorical Torelli theorems: results and open problems

Pertusi¹,

Stellari²

2022

Preprint

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We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.Contents 24 6. Cubic fourfolds 40 References 47

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Section: Cubic Threefolds and Beyondmentioning

confidence: 74%

“…(1) By [115,Theorem 3.18] the stability conditions σ(s, q) on Ku(X d ) are Serre-invariant for every d = 10, 14, 18, 22. Among all, the most interesting case is d = 10, i.e.…”

Section: Cubic Threefolds and Beyondmentioning

confidence: 99%

Categorical Torelli theorems: results and open problems

Pertusi¹,

Stellari²

2022

Preprint

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“…When d = 4, 5, the heart A(α, β) has homological dimension 1. When d = 3, from [34] we know that these stability conditions are Serre-invariant. Thus by Lemma 3.10, to show ext 2 (E, E) = 0, we only need to show E ∈ A X is σ (α, β)-semistable.…”

Section: Proofmentioning

confidence: 99%

A note on Bridgeland moduli spaces and moduli spaces of sheaves on $$X_{14}$$ and $$Y_3$$

Liu

Zhang

2022

Math. Z.

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We study Bridgeland moduli spaces of semistable objects of $$(-1)$$ ( - 1 ) -classes and $$(-4)$$ ( - 4 ) -classes in the Kuznetsov components on index one prime Fano threefold $$X_{4d+2}$$ X 4 d + 2 of degree $$4d+2$$ 4 d + 2 and index two prime Fano threefold $$Y_d$$ Y d of degree d for $$d=3,4,5$$ d = 3 , 4 , 5 . For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{4d+2}$$ X 4 d + 2 and $$Y_d$$ Y d are isomorphic. We show that moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{14}$$ X 14 are realized by Fano surface $$\mathcal {C}(X)$$ C ( X ) of conics, moduli spaces of semistable sheaves $$M_X(2,1,6)$$ M X ( 2 , 1 , 6 ) and $$M_X(2,-1,6)$$ M X ( 2 , - 1 , 6 ) and the correspondent moduli spaces on cubic threefold $$Y_3$$ Y 3 are realized by moduli spaces of stable vector bundles $$M^b_Y(2,1,2)$$ M Y b ( 2 , 1 , 2 ) and $$M^b_Y(2,-1,2)$$ M Y b ( 2 , - 1 , 2 ) . We show that moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes on $$Y_{d}$$ Y d are isomorphic to the moduli spaces of instanton sheaves $$M^{inst}_Y$$ M Y inst when $$d\ne 1,2$$ d ≠ 1 , 2 , and show that there are open immersions of $$M^{inst}_Y$$ M Y inst into moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes when $$d=1,2$$ d = 1 , 2 . Finally, when $$d=3,4,5$$ d = 3 , 4 , 5 we show that these moduli spaces are all isomorphic to $$M^{ss}_X(2,0,4)$$ M X ss ( 2 , 0 , 4 ) .

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“…In the case of Gushel-Mukai varieties of odd dimension and quartic double solids, stability conditions are known to exist by [BLMS17]. We make use of [PR21] and [PY20] to control the homological dimension of the heart, and of a paper in preparation [PPZ22] where we show the density of the set of semistable objects. This provides the proof of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

Derived categories of hearts on Kuznetsov components

Li¹,

Pertusi²,

Zhang³

2022

Preprint

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We prove a general criterion which guarantees that an admissible subcategory K of the derived category of an abelian category is equivalent to the bounded derived category of the heart of a bounded t-structure. As a consequence, we show that K has a strongly unique dg enhancement, applying the recent results of Canonaco, Neeman and Stellari. We apply this criterion to the Kuznetsov component Ku(X) when X is a cubic fourfold, a Gushel-Mukai variety or a quartic double solid. In particular, we obtain that these Kuznetsov components have strongly unique dg enhancement and that exact equivalences of the form Ku(X) ∼ − → Ku(X ′ ) are of Fourier-Mukai type when X, X ′ belong to these classes of varieties, as predicted by a conjecture of Kuznetsov.

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

Cited by 6 publications

References 20 publications

Categorical Torelli theorems: results and open problems

Categorical Torelli theorems: results and open problems

A note on Bridgeland moduli spaces and moduli spaces of sheaves on $$X_{14}$$ and $$Y_3$$

Derived categories of hearts on Kuznetsov components

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