Derived Categories of Cubic Fourfolds

Kuznetsov, Alexander

doi:10.1007/978-0-8176-4934-0_9

Cited by 145 publications

(152 citation statements)

References 7 publications

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“…In the case where X contains a plane, Kuznetsov constructed in [29] a semi-orthogonal decomposition a smooth complex hypersurface of degree 3 in P 5 .…”

Section: Cubic Fourfolds Containing a Planementioning

confidence: 99%

Noncommutative motives of Azumaya algebras

Tabuada

Bergh

2014

J. Inst. Math. Jussieu

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Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then computer the R-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional k-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element of the Grothendieck group of X which has rank r becomes invertible in the R-linearized Grothendieck group, and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.Comment: 22 pages; revised versio

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“…In the case where X contains a plane, Kuznetsov constructed in [29] a semi-orthogonal decomposition a smooth complex hypersurface of degree 3 in P 5 .…”

Section: Cubic Fourfolds Containing a Planementioning

confidence: 99%

Noncommutative motives of Azumaya algebras

Tabuada

Bergh

2014

J. Inst. Math. Jussieu

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“…In their address to the 2002 International Congress of Mathematicians in Beijing, Bondal and Orlov suggest that the bounded derived category

{sans-serifD}^{normalb} false(X false)

of coherent sheaves on a smooth projective variety

X

could provide new tools to explore the birational geometry of

X

, in particular via semiorthogonal decompositions. The work of many authors now provides evidence for the usefulness of semiorthogonal decompositions in the birational study of complex projective varieties of dimension at most 4. A survey can be found in .…”

Section: Introductionmentioning

confidence: 96%

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Auel

Bernardara

2018

Proc. London Math. Soc.

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We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero‐dimensional components. When S has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

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“…The most famous of these cases is that of a cubic fourfold (see [Kuz10]), when the category A X can be thought of as a noncommutative K3 surface. The case of a cubic hypersurface of dimension 7 (when A X is a 3-Calabi-Yau category) was discussed in [IM11].…”

Section: Combining This We Obtainmentioning

confidence: 99%

“…Moreover, for special cubic fourfolds the category A X 3 is equivalent to D b (S) for a K3 surface S (see [Kuz10]) and the same is expected to be true for some Gushel-Mukai sixfolds (see [KP15]). It is also expected that the same is true for special Debarre-Voisin varieties.…”

Section: Combining This We Obtainmentioning

confidence: 99%

Calabi–Yau and fractional Calabi–Yau categories

Kuznetsov

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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101

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Abstract. We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories.

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Derived Categories of Cubic Fourfolds

Abstract: Abstract. We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.

Cited by 145 publications

References 7 publications

Noncommutative motives of Azumaya algebras

Noncommutative motives of Azumaya algebras

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Calabi–Yau and fractional Calabi–Yau categories

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