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Derived categories of Fano varieties of lines

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Pith's one-line read Generic cubic fourfolds whose Fano variety admits a rational Lagrangian fibration now satisfy the predicted derived equivalence to the Hilbert square of the Kuznetsov component.

desk verdict Clear, honest note proving Galkin's conjecture for generic Lagrangian-fibered Fano varieties and giving an all-cubic Hodge isometry, but the main theorem hinges on one unproved ingredient deferred to a first-author preprint. read the letter →

arxiv 2501.03534 v1 pith:7KVWCBEN submitted 2025-01-07 math.AG

Alessio Bottini , Daniel Huybrechts This is my paper · ORCID
classification math.AG MSC 14F0814J2814J4214C05
keywords derivedcategoriescubicfourfoldsFanovarietiesoflinesKuznetsovcomponentHilbertsquarehyperkählermanifoldsLagrangianfibrationsHodgestructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper targets a conjecture (Conjecture 0.1) asserting that for any smooth cubic fourfold $X$, the derived category of the Fano variety of lines is equivalent to the Hilbert square of the Kuznetsov component of $X$. It proves this equivalence for a Zariski-open family of cubics whose Fano variety admits a rational Lagrangian fibration — numerically, those lying in a Hassett divisor $C_d$ with $d/2$ a perfect square — and it proves the conjecture outright whenever the Fano variety is birational to a Hilbert scheme of a K3 surface. As a universal necessary condition, it also shows that the natural weight-two Hodge structures attached to the two sides are isometric for every smooth cubic fourfold, with no genericity assumption. The proof works by chaining derived equivalences through moduli spaces of twisted sheaves on K3 surfaces, compactified relative Jacobians, and Hilbert schemes.

What carries the argument

The argument is carried by a five-term chain of exact linear equivalences. Starting from the Fano variety, one moves to a fine moduli space $M_\alpha$ of $\alpha$-twisted sheaves supported on curves in a degree-two K3 surface (via a birational Mukai flop, a standard surgery between hyperkähler manifolds that preserves derived categories), then to a compactified relative Jacobian $M$ equipped with a Brauer class $\theta$, then to the Hilbert scheme of the K3 surface twisted by $\alpha^{[2]}$, then to the twisted derived category $D^b(S, \alpha)$ squared, and finally to the Hilbert square $A_X^{[2]}$ of the Kuznetsov component. Each arrow is an exact linear equivalence; the step from $M$ to $M_\alpha$ is the quoted comparison theorem that is the paper's main external input. On the Hodge side, the paper uses the decomposition of the Mukai lattice into transcendental and algebraic parts and a standard lattice-extension theorem to build the isometry between the weight-two structures without controlling the algebraic parts.

What would settle it

Check the deferred comparison theorem directly: for a generic degree-two K3 surface, compute the Fourier–Mukai transform associated with the twisted Poincaré bundle on the fibre product of the compactified relative Jacobian and the twisted moduli space, and verify that it is an equivalence; a single family of curves for which this fails would invalidate Theorem 2.1.

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Extended reading notes

Core claim

The central assertion is Theorem 2.1: if $d/2$ is a perfect square, then for a Zariski dense open subset of cubics $X$ in the Hassett divisor $C_d$ there exists an exact linear equivalence $D^b(F_X) \cong A_X^{[2]}$. This is the first verification of the conjecture for Fano varieties with a rational Lagrangian fibration. The paper also establishes Theorem 0.4, a Hodge isometry between the naive weight-two Mukai lattices of $F_X$ and $A_X^{[2]}$ for every smooth cubic fourfold, which is the Hodge-theoretic shadow that any derived equivalence would necessarily cast. In addition, Theorem 0.2 settles the conjecture for all cubics whose Fano variety is birational to a Hilbert scheme of a K3 surface. The two theorems together cover every known instance in which the conjecture has been verified: the paper records that no smooth cubic is known where the equivalence holds outside these families.

Load-bearing premise

The load-bearing premise is an unproved comparison theorem, quoted from a companion preprint, asserting that the compactified relative Jacobian of the universal curve family is derived equivalent to a certain moduli space of twisted sheaves; if that theorem has a gap, the chain of equivalences that proves the main result breaks.

Editorial extensions

If this is right

  • For every cubic in the Zariski-open subset of $C_d$ with $d/2$ a perfect square, the Fano variety of lines is derived equivalent to the Hilbert square of the Kuznetsov component, so the conjecture holds on a dense set of the Lagrangian-fibered locus.
  • The Hodge isometry of Theorem 0.4 holds for all smooth cubic fourfolds, so any counterexample to the conjecture would have to be invisible to the weight-two Mukai lattice.
  • When $A_X$ is equivalent to the derived category of a K3 surface, the Hilbert-square equivalence reduces to the classical McKay correspondence for Hilbert schemes, so the genuinely new phenomenon is the twisted case.
  • The particular equivalence constructed here is shown not to deform to the generic cubic, so the general conjecture, if true, would need a different family of equivalences beyond the Lagrangian-fibered locus.
  • The paper identifies concrete smallest open cases (for instance $d=74$ in the untwisted K3 case and $d=24$ in the twisted case), giving test targets for any future approach.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A proof of the quoted comparison theorem in full generality would likely extend Theorem 2.1 from the Zariski-open subset to all smooth cubics in $C_d$ with $d/2$ a perfect square, since the remaining genericity assumptions are used to guarantee integral curves and fine moduli spaces.
  • The Hodge isometry, combined with the twisted Mukai-lattice formalism, suggests that the sign involution of the Hilbert square should induce a nontrivial autoequivalence of $D^b(F_X)$ on the Lagrangian-fibered locus; promoting this cohomological action to a categorical statement could yield new autoequivalences of hyperkähler fourfolds.
  • The same chain of moduli-space equivalences may be a template for other hyperkähler varieties of K3$^{[n]}$-type that are birational to moduli spaces of twisted sheaves, suggesting a general principle: a Hilbert-square construction on the associated K3 category matches the derived category exactly when the variety admits a Lagrangian fibration.
  • The paper's identification of the smallest open cases provides a concrete roadmap: the next decisive test of the conjecture lies in the untwisted K3 case $d=74$, where none of the current techniques apply.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies Galkin's conjecture that the derived category of the Fano variety of lines F_X of a smooth cubic fourfold X is equivalent to the Hilbert square A_X^[2] of the Kuznetsov component A_X. The main new result is Theorem 0.3 (Theorem 2.1): for generic cubics X in a Hassett divisor C_d with d/2 a perfect square, there is an exact linear equivalence D^b(F_X) ≅ A_X^[2]. The proof proceeds by a chain of derived equivalences involving a birational model M_α (a moduli space of twisted sheaves on a degree-two K3 surface), a compactified relative Jacobian M with a Brauer twist, and the Hilbert square S^[2] of a K3 surface. The paper also proves Theorem 0.4, a Hodge isometry between the naive weight-two Mukai structures of F_X and A_X^[2] for every smooth cubic, and contains a twisted version of the Bridgeland-King-Reid equivalence (Proposition 1.2) and a complete proof of the twisted Kawamata-Namikawa flop equivalence (Proposition 2.3). A comparison with the Beckmann-Taelman twist and a discussion of the sign involution are included in Section 3.

Significance. If the main theorem is correct, it establishes Galkin's conjecture for a previously inaccessible dense set of cubic fourfolds, namely those whose Fano variety admits a rational Lagrangian fibration. The Hodge-theoretic Theorem 0.4 is a meaningful necessary condition and provides evidence for the conjecture in full generality. The paper is well structured, and the parts that are proved in the text, in particular Propositions 1.2 and 2.3, are presented with reasonable detail. However, the central chain of equivalences in Section 2.5 relies on Proposition 2.6, whose proof is deferred entirely to an unpublished preprint by the first author; this makes the main theorem conditional as written. The manuscript would be strengthened substantially by a self-contained proof of that step or by a clear statement that the theorem depends on an external unrefereed result.

major comments (2)
  1. [§2.2, Proposition 2.6] Proposition 2.6, which supplies the equivalence D^b(M,θ) ≅ D^b(M_α) in Eq. (2.8), is load-bearing for Theorem 2.1: in the chain in §2.5, it is the only step linking the twisted Jacobian compactification to the twisted moduli space M_α, and without it the derived equivalence D^b(F_X) ≅ A_X^[2] is not established. The proof given in the text is a single sentence that attributes the statement to Arinkin [4], Addington-Donovan-Meachan [2], and then to [10, Thm. 3.3 & Rem. 3.4]. Since [10] is an arXiv preprint by one of the authors and is not verified anywhere in this manuscript, the theorem remains conditional. The authors should either include a complete proof of Proposition 2.6 or explicitly state, with a precise match of hypotheses (the chosen fine component M_α from Proposition 2.8, the compactification M, and the Brauer class θ), that the result is taken from an external preprint. As written, the most exposed link of the main chain is not justified within the paper.
  2. [§2.4, Proposition 2.8 and §2.5] The proof of Proposition 2.8 invokes the birational correspondence F_X ≈ M_α from [27, Thm. 1.2] and then uses the condition (v,w)=1 to assert that M_α is fine. The paper does not explain how the hypotheses of [27, Thm. 1.2] are satisfied for the specific generic cubics considered, nor how the birational correspondence is obtained as a composition of Mukai flops so that Proposition 2.3 applies repeatedly in §2.5. While these are likely standard consequences of the cited results, the reader is left to assemble the argument; the main theorem would be clearer if the genericity assumptions and the flop decomposition were stated explicitly in the proof of Theorem 2.1.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'generic Fano varieties admitting a rational Lagrangian fibration' in the abstract is imprecise: the theorem is about a Zariski open subset of cubics X in C_d, and the Lagrangian fibration is a property of F_X. Please align the terminology between the abstract, Theorem 0.3, and Theorem 2.1.
  2. [§1.2, Proposition 1.2] The proof of Proposition 1.2 would benefit from a short explanation of why the Fourier–Mukai kernel O_{Z_n} is well defined with the stated twisting, in particular why the restriction of (α^{[n]})^{-1} ⊠ α^{n} to Z_n is trivial; the current argument is compressed.
  3. [§3.1, proof of Theorem 0.4] The notation in the displayed chain of Hodge isometries is dense and the role of the Nikulin extension step (3.1) is not fully explained; in particular, it would help to explicitly state which lattice is the orthogonal complement in each occurrence and why the extension sends λ_1 to γ.
  4. [Global] The manuscript contains several typographical issues (e.g., 'F ano' in the title, inconsistent spacing around 'BrpZq'), and some references listed in the bibliography, such as [35] and [39], are not cited in the body of the text. A careful proofreading pass is recommended.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 2.1 rests on Proposition 2.6, whose proof is deferred to the first author's unpublished preprint [10]; the key twisted equivalence (2.8) is load-bearing but not established in this paper.

  1. self citation load bearing [Section 2.2, Proposition 2.6 (proof of (2.8))]
    "The result goes back to Arinkin [4, Thm. C & § 7] who proved that for an integral curve with planar singularities the Poincaré bundle extends to a sheaf on the square of the compactification of the Jacobian which taken as a Fourier–Mukai kernel induces an auto-equivalence. For relative compactifications of Picard schemes of any degree the result was proved by Addington, Donovan, and Meachan [2]. The result claimed here is [10, Thm. 3.3 & Rem. 3.4]."

    The proof of Theorem 2.1 in Section 2.5 uses the chain D^b(F_X) ≅ D^b(M_α) ≅ D^b(M,θ) ≅ D^b(S^[2],α^[2]) ≅ A_X^[2]. The middle equality is (2.8), supplied by Proposition 2.6. Instead of proving (2.8), the paper explicitly cites [10, Thm. 3.3 & Rem. 3.4], an arXiv preprint by the first author, and gives no independent verification that its hypotheses match the chosen component M_α and compactified relative Jacobian M. Thus the central positive claim reduces, at this link, to an un-audited self-citation. The statement is not definitionally forced by an assumption of the conjecture itself, but it is load-bearing: if this unproved equivalence fails, the chain proving Theorem 2.1 breaks.

full rationale

The paper contains no fitted parameters and does not assume Galkin's Conjecture 0.1, so the gross forms of circularity (prediction from fitted data, renaming a known result, or defining the target into existence) are absent. The Hodge-isometry result Theorem 0.4 is proved internally by lattice arguments and does not depend on the main categorical chain. The strongest positive theorem, Theorem 2.1, however, passes through Proposition 2.6, whose essential content (2.8) is deferred to the first author's preprint [10]. This is a load-bearing self-citation: the paper does not prove the equivalence, does not verify [10]'s hypotheses in the present setting, and gives no machine-checked, code-reproduced, or externally falsifiable substitute. I therefore cannot call the derivation self-contained. Secondary dependencies in Proposition 2.8 on [27, Thm. 1.2] and [23, Thm. 1.4] also come from the second author's earlier results, but these are separate published/posted theorems rather than the target statement. The authors themselves acknowledge the generic-only nature of the proof, which is a limitation and not a circularity. Overall, the central claim has substantial independent content and is not forced by construction; the main defect is that one key step is a same-author preprint cited as a black box. Score 4.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data, and no constants depend on tunable inputs. The axioms are standard mathematical background plus several external theorems used as black boxes; the most fragile is [10, Thm. 3.3 & Rem. 3.4], an unpublished preprint by one of the authors. No new objects requiring independent evidence are introduced beyond the definitions of A_X^[2] and the naive Hodge structure.

assumptions (7)
  • standard math Semiorthogonal decomposition D^b(X)=⟨A_X,O_X,O_X(1),O_X(2)⟩ and admissibility of A_X
    Starting point for defining A_X and its Hilbert square; cited to Kuznetsov [31] and Huybrechts [24, Ch. 7].
  • standard math Mukai flops induce derived equivalences, including the twisted version in Proposition 2.3
    Used to compare D^b(F_X), D^b(M_α), and D^b(M,θ); the untwisted result is due to Kawamata and Namikawa and a proof is included in Section 2.1.
  • domain assumption The Arinkin-type equivalence D^b(M,θ) ≅ D^b(M_α) of Proposition 2.6
    Deferred to [10, Thm. 3.3 & Rem. 3.4], an arXiv preprint by the first author; this is the least independently verified input in the proof of Theorem 2.1.
  • domain assumption Huybrechts-Mattei [27, Thm. 1.2]: for generic non-special X, F_X is birational to a moduli space M_α of twisted sheaves on a K3 surface
    Used in Proposition 2.8 to identify the birational model of F_X; stated as an external theorem.
  • domain assumption Huybrechts [23, Thm. 1.4]: a Hodge isometry rH(A_X,Z) ≅ rH(S,α,Z) implies A_X ≅ D^b(S,α) for generic cubics
    Upgrades the lattice-theoretic statement in Proposition 2.8 to an actual derived equivalence.
  • standard math Twisted BKR equivalence D^b(S^[n], α^[n]) ≅ D^b(S,α)^[n] (Proposition 1.2)
    Needed to identify D^b(S^[2],α^[2]) with A_X^[2]; proof given using Haiman's isospectral Hilbert scheme and [8, 18].
  • standard math Nikulin's lattice extension theorem for Hodge isometries of transcendental lattices
    Used in the proof of Theorem 0.4 to extend the isometry between transcendental lattices to the full Mukai lattices.

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Pith. "Pith review of Derived categories of Fano varieties of lines." pith.science (2026). https://pith.science/paper/7KVWCBEN

@misc{pith2026250103534,
  author       = {Pith},
  title        = {Pith review of: Derived categories of Fano varieties of lines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KVWCBEN}},
  note         = {Machine review of arXiv:2501.03534}
}
read the original abstract

We gather evidence for a conjecture of Galkin predicting the derived category of the Fano variety of lines contained in a smooth cubic fourfold to be equivalent to the Hilbert square of the Kuznetsov component of the derived category of the cubic. We prove the conjecture for generic Fano varieties admitting a rational Lagrangian fibration and show that the natural Hodge structures of weight two associated with the Fano variety and the Hilbert square are isometric.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A twisted derived category of hyper-K\"ahler varieties of $K3^{[n]}$-type

    math.AG 2025-02 accept novelty 6.0 of 10

    A twisted derived category of K3^[n]-type hyper-Kähler varieties is governed by the oriented Markman-Mukai lattice, proven under a primitivity condition and yielding derived equivalences between fine K3 moduli spaces ...

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Works this paper leans on

42 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [2]

    Addington, W

    N. Addington, W. Donovan, C. Meachan Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences. J. LMS 93 (2016), 846–865. 8, 9, 12

    work page 2016

  2. [10]

    Bottini O’Grady’s tenfolds from stable bundles on hyper-Kähler fou rfolds

    A. Bottini O’Grady’s tenfolds from stable bundles on hyper-Kähler fou rfolds. arXiv:2411.18528 8, 12

    arXiv

  3. [27]

    Huybrechts, D

    D. Huybrechts, D. Mattei The special Brauer group and twisted Picard varieties. arXiv:2310.04032. 10, 11, 13

    arXiv

  4. [4]

    Arinkin Autoduality of compactified Jacobians for curves with plane singularities

    D. Arinkin Autoduality of compactified Jacobians for curves with plane singularities. JAG 22 (2013), 363–

    work page 2013

  5. [1]

    Addington On two rationality conjectures for cubic fourfolds

    N. Addington On two rationality conjectures for cubic fourfolds. Math. Res. Lett. 23 (2016), 1–13. 2, 13

    work page 2016

  6. [3]

    Addington, R

    N. Addington, R. Thomas Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163 (2014), 1885–1927. 1, 2

    work page 2014

  7. [5]

    Bayer, M

    A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry, P. Stellar i Stability conditions in families. Publ. Math. IHES 133 (2021), 157–325. 2, 3, 14

    work page 2021

  8. [6]

    Beauville, R

    A. Beauville, R. Donagi La variété des droites d’une hypersurface cubique de dimens ion 4. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703–706. 16, 17

    work page 1985

Show all 42 references
  1. [7]

    Beckmann Derived categories of hyper-Kähler manifolds: extended Mu kai vector and integral structure

    T. Beckmann Derived categories of hyper-Kähler manifolds: extended Mu kai vector and integral structure. Comp. Math. 159 (2023), 109–152. 3, 7, 15, 16, 18

    2023

  2. [8]

    Beckmann, G

    T. Beckmann, G. Oberdieck On equivariant derived categories. Eur. J. Math. 9 (2023), no. 2, Paper No. 36, 39 pp. 5, 6

    2023

  3. [9]

    Belmans, L

    P. Belmans, L. Fu, and T. Raedschelders Derived categories of flips and cubic hypersurfaces. Proc. LMS 125 (2022), 1452–1482. 2

    2022

  4. [11]

    Bridgeland Stability conditions on K3 surfaces

    T. Bridgeland Stability conditions on K3 surfaces. Duke Math. J. 141 (2008), 241–291. 1 19

    2008

  5. [12]

    Bridgeland, A

    T. Bridgeland, A. King, M. Reid The McKay correspondence as an equivalence of derived categ ories. JAMS 14 (2001), 535–554. 1, 2, 5

    2001

  6. [13]

    Colliot-Thélène, A

    J.-L. Colliot-Thélène, A. Skorobogatov The Brauer–Grothendieck group. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, volume 71. 2021. 8

    2021

  7. [14]

    Elagin On equivariant triangulated categories

    A. Elagin On equivariant triangulated categories. arXiv:1403.7027 4, 5, 18

    arXiv

  8. [15]

    Galkin, E

    S. Galkin, E. Shinder The Fano variety of lines and rationality problem for a cubic hypersurface. arXiv:1405.5154. 2

    arXiv

  9. [16]

    Ganter, M

    N. Ganter, M. Kapranov Symmetric and exterior powers of categories. Transform. Groups 19 (2014), 57–103. 5

    2014

  10. [17]

    Grothendieck Le groupe de Brauer III

    A. Grothendieck Le groupe de Brauer III. Exemples et compléments. in: Dix exposés sur la cohomologie des schémas. volume 3 of Adv. Stud. in Pure Maths. North Holland A msterdam (1968), 88–188. 8

    1968

  11. [18]

    Haiman Hilbert schemes, polygraphs and the Macdonald positivity c onjecture

    M. Haiman Hilbert schemes, polygraphs and the Macdonald positivity c onjecture. JAMS 14 (2001), 941–1006. 6

    2001

  12. [19]

    Hassett Special cubic fourfolds

    B. Hassett Special cubic fourfolds. Comp. Math. 120 (200), 1–23. 2, 4

  13. [20]

    Huybrechts Compact hyperkähler manifolds: Basic results

    D. Huybrechts Compact hyperkähler manifolds: Basic results. Invent. Math. 135 (1999), 63–113. 8

    1999

  14. [21]

    Huybrechts Fourier–Mukai Transforms in Algebraic Geometry

    D. Huybrechts Fourier–Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. 2006. 2, 6, 8, 9

    2006

  15. [22]

    Huybrechts Lectures on K3 Surfaces

    D. Huybrechts Lectures on K3 Surfaces. volume 158 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 4, 15

    2016

  16. [23]

    Huybrechts The K3 category of a cubic fourfold

    D. Huybrechts The K3 category of a cubic fourfold. Comp. Math. 153 (2017), 586–620. 1, 3, 7, 13, 14

    2017

  17. [24]

    Huybrechts The Geometry of Cubic Hypersurfaces

    D. Huybrechts The Geometry of Cubic Hypersurfaces. volume 206 of Cambridge Studies in Advanced Math- ematics. Cambridge University Press, Cambridge, 2023. 2, 3, 4, 5, 13, 16, 17

    2023

  18. [25]

    Huybrechts The period-index problem for hyperkähler manifolds

    D. Huybrechts The period-index problem for hyperkähler manifolds. arXiv:2411.17604. 6, 11

    arXiv

  19. [26]

    Huybrechts, M

    D. Huybrechts, M. Lehn The Geometry of Moduli Spaces of Sheaves. Cambridge Mathematical Library. Cambridge University Press. (2010) 14

    2010

  20. [28]

    Kawamata D-equivalence and K-equivalence

    Y. Kawamata D-equivalence and K-equivalence. JDG 61 (2002), 147–171. 2, 8, 9

    2002

  21. [29]

    Krashen, M

    D. Krashen, M. Lieblich Index reduction for Brauer classes via stable sheaves. IMRN 2008 (2008), 31 pp. 11, 12

    2008

  22. [30]

    Krug Extension groups of tautological sheaves on Hilbert scheme s

    A. Krug Extension groups of tautological sheaves on Hilbert scheme s. JAG 23 (2014), 571–598. arXiv:1111.4263. 6

    2014 arXiv

  23. [31]

    Kuznetsov Base change for semiorthogonal decompositions

    A. Kuznetsov Base change for semiorthogonal decompositions. Comp. Math. 147 (2011), 852–876. 1, 5

    2011

  24. [32]

    C. Lehn, M. Lehn, C. Sorger, D. van Straten Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731 (2017), 87–128. 5

    2017

  25. [33]

    C. Li, L. Pertusi, X. Zhao Derived categories of hearts on Kuznetsov components. J. LMS 108 (2023), 2146–2174. 7

    2023

  26. [34]

    Maulik, J

    D. Maulik, J. Shen, Q. Yin, R. Zhang The D-equivalence conjecture for hyper-Kähler varieties v ia hyper- holomorphic bundles. arXiv:2408.14775. 2

    arXiv

  27. [35]

    Milne Jacobian varieties

    J. Milne Jacobian varieties. In: Arithmetic geometry. Springer-Verlag, New York (1986) , 167–212

    1986

  28. [36]

    Mukai On the moduli space of bundles on K3 surfaces

    S. Mukai On the moduli space of bundles on K3 surfaces. I. In: Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math. 11 (1987), 341–413. 1, 3

    1987

  29. [37]

    Namikawa Mukai flops and derived categories

    Y. Namikawa Mukai flops and derived categories. J. Reine Angew. Math. 560 (2003), 65–76. 2, 8, 9

    2003

  30. [38]

    Orlov Equivalences of derived categories and K3 surfaces

    D. Orlov Equivalences of derived categories and K3 surfaces. J. Math. Sci. 84 (1997), 1361–1381. 1, 3 20 A. BOTTINI & D. HUYBRECHTS

    1997

  31. [39]

    Polishchuk Abelian Varieties, Theta Functions and the Fourier Transfo rm

    A. Polishchuk Abelian Varieties, Theta Functions and the Fourier Transfo rm. Cambridge University Press, 2003

    2003

  32. [40]

    Popov Twisted cubics and quadruples of points on cubic surfaces

    P. Popov Twisted cubics and quadruples of points on cubic surfaces. arXiv:1810.04563. 2, 5

    arXiv

  33. [41]

    Taelman Derived equivalences of hyperkähler varieties

    L. Taelman Derived equivalences of hyperkähler varieties. Geom. Top. 27 (2023), 2649–2693. 3, 7, 15, 16, 18

    2023

  34. [42]

    Yoshioka Moduli spaces of twisted sheaves on a projective variety

    K. Yoshioka Moduli spaces of twisted sheaves on a projective variety. In: Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., 45 (2006), 1–30. 4, 10, 13 Max-Planck Institute for Ma thema tics, Viv a tsgasse 7, 53111 Bonn, Germany & Ma thema tical Institute and Hausdo...

    2006

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