Base change for semiorthogonal decompositions

Kuznetsov, Alexander

doi:10.1112/s0010437x10005166

Cited by 108 publications

(121 citation statements)

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“…On the other hand, Kuznetsov has shown that, for any admissible subcategory

{sans-serifA}_{X} \subset {sans-serifD}^{normalb} false(X false)

, the projection functor (that is, the right adjoint to the embedding functor) is a Fourier–Mukai functor. Hence, the choice of such a functor endows

A_{X}

with a dg‐structure, which is in principle not unique, but depends on the choice of projection.…”

Section: Semiorthogonal Decompositions and Categorical Representabilitymentioning

confidence: 99%

“…Given a smooth projective variety

X

over a field

k

, and fixing a separable closure

k^{italics}

k

, we are interested in comparing

k

‐linear semiorthogonal decompositions of

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

and

k^{italics}

‐linear semiorthogonal decompositions of

{sans-serifD}^{normalb} false(X_{k^{s}} false)

. The general question of how derived categories and semiorthogonal decompositions behave under base field extension has started to be addressed by several authors [, § 2; ]. Galois descent does not generally hold for objects in a triangulated category, due to the fact that cones are only defined up to quasi‐isomorphism.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

“…These questions also intertwine with concurrent developments in equivariant and descent aspects of triangulated and dg‐categories in the context of noncommutative geometry, see .…”

Section: Introductionmentioning

confidence: 96%

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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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“…On the other hand, Kuznetsov has shown that, for any admissible subcategory

{sans-serifA}_{X} \subset {sans-serifD}^{normalb} false(X false)

, the projection functor (that is, the right adjoint to the embedding functor) is a Fourier–Mukai functor. Hence, the choice of such a functor endows

A_{X}

with a dg‐structure, which is in principle not unique, but depends on the choice of projection.…”

Section: Semiorthogonal Decompositions and Categorical Representabilitymentioning

confidence: 99%

“…Given a smooth projective variety

X

over a field

k

, and fixing a separable closure

k^{italics}

k

, we are interested in comparing

k

‐linear semiorthogonal decompositions of

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

and

k^{italics}

‐linear semiorthogonal decompositions of

{sans-serifD}^{normalb} false(X_{k^{s}} false)

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

See 1 more Smart Citation

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

Auel

Bernardara

2018

Proc. London Math. Soc.

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“…It was shown by Kuznetsov that the projection functors, α i are represented by unique integral transforms [Ku1].…”

Section: Be a Semi-orthogonal Decomposition For Any T ∈ T The Diagmentioning

confidence: 99%

On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

Favero

Iliev

Katzarkov

2014

Pure and Applied Mathematics Quarterly

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A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. This serves as partial confirmation of seminal work of Candelas, Derrick, and Parkes describing a cubic 7-fold as a mirror to a rigid CY 3-fold.Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold X 3 , the fivefold quartic double solid X 4 , and the fivefold intersection of a quadric and a cubic X 2.3 . We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a general complete intersection FCY manifolds, X 4 and X 2.3 , is also infinitely generated.Received December 11, 2012. David Favero, Atanas Iliev and Ludmil KatzarkovIn the case of X 4 , we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for X 2.3 there is a noncommutative CY 3-fold, B, such that the Griffiths group of X 2.3 surjects on to the Griffiths group of B. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back toq honest algebraic varieties such as products of elliptic curves and K3-surfaces.

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“…Let n > r > 0 with 2|r and now let Y be the variety of skew symmetric n × nmatrices of rank ≤ r. If n is odd then in [19] we constructed an NCCR Λ for k [Y ] (the existence of the resulting strongly crepant categorical resolution of Y was conjectured in [10,Conj. 4.9]).…”

Section: Introductionmentioning

confidence: 99%