Derived equivalences of K3 surfaces and orientation

Huybrechts, Daniel; Macrì, Emanuele; Stellari, Paolo

doi:10.1215/00127094-2009-043

Cited by 60 publications

(112 citation statements)

References 32 publications

(79 reference statements)

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“…In this case one can show that the above product is just the product of the classical Atiyah and Kodaira-Spencer classes. This is the case, for instance, when X is flat over Spec(k[t]/t n+1 ) and X 0 is its base-change to Spec(k[t]/t n ): the situation studied in [5]. This explains why it was not necessary to work with (truncated) cotangent complexes in that paper.…”

Section: 5mentioning

confidence: 99%

“…The theory is not in its final form, however. The obstruction class is expected to be a product of Atiyah and Kodaira-Spencer classes, but this has only been proved in some situations (such as [5]). As a consequence the results are not strong enough to obtain virtual cycles [2,12], as discussed in [16,Sect.…”

Section: Introductionmentioning

confidence: 99%

“…2], for instance. This paper generalises [13,14] by working universally, using Fourier-Mukai kernels as in [5,6]. We put the discussion of [5, App.…”

Section: Introductionmentioning

confidence: 99%

“…While the classical Atiyah and Kodaira-Spencer classes were enough for the case treated in [5], Illusie's versions of these classes, defined in [7] using the cotangent complex, are necessary to go beyond the simplest cases. This was done for complexes of modules in [7] and for a single sheaf in [3].…”

Section: Introductionmentioning

confidence: 99%

“…But the explicit expression as the product of the truncated Atiyah class of E 0 and the truncated Kodaira-Spencer class of the thickening X 0 ⊂ X is crucial for some applications [5,16].…”

Section: Introductionmentioning

confidence: 99%

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Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

2009

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Abstract. We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira-Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.

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Section: 5mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…2], for instance. This paper generalises [13,14] by working universally, using Fourier-Mukai kernels as in [5,6]. We put the discussion of [5, App.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…But the explicit expression as the product of the truncated Atiyah class of E 0 and the truncated Kodaira-Spencer class of the thickening X 0 ⊂ X is crucial for some applications [5,16].…”

Section: Introductionmentioning

confidence: 99%

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Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

2009

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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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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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Derived equivalences of K3 surfaces and orientation

Cited by 60 publications

References 32 publications

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

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

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

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