Calabi–Yau and fractional Calabi–Yau categories

Kuznetsov, Alexander

doi:10.1515/crelle-2017-0004

Cited by 95 publications

(105 citation statements)

References 28 publications

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“…Therefore, for any

F \in R

we have

τ^{- 1} false(R_{a} ({double-struckL}_{fraktura} false(τ (F) false)) false) ≅ τ^{- 1} false(τ (F) false) ≅ F .

On the other hand,

{double-struckL}_{fraktura} false(τ (τ^{- 1} false(R_{a} (F) false)) false) ≅ {double-struckL}_{fraktura} false(R_{a} (F) false) ≅ F

for any

F \in {fraktura}^{⊥}

, hence

τ^{- 1} \circ {double-struckR}_{fraktura}

is indeed the inverse of

τ_{R}

. Finally, let us check that

τ_{R}

is a polarization of

scriptR

of index

m

(this is a combination of [, Lemmas 2.6 and 3.13]). Indeed, for any

i ⩽ m

we have

\begin{matrix} true \\ right τ_{scriptR}^{i} & left ≅ L_{a} \circ τ \circ L_{a} \circ τ \circ L_{a} \circ τ \circ \dots \circ L_{a} \circ τ \\ left ≅ L_{a} \circ (τ \circ {double-struckL}_{fraktura} \circ τ^{- 1}) \circ \end{matrix}

…”

Section: Residual Categoriesmentioning

confidence: 97%

“…Therefore, the Serre functor of the phantom category

{scriptC}_{k, n} \subset {scriptR}_{k, n}

(see Remark ) is also trivial, that is,

C_{k, n}

is a Calabi–Yau category. But a Calabi–Yau category with zero Hochschild homology is itself zero by [, Corollary 5.3]. Thus,

{scriptC}_{k, n} = 0

□

…”

Section: Lefschetz Decompositions For Grassmanniansmentioning

confidence: 99%

“…So, we only need to check that

τ_{R}

has all the necessary properties. Most of the arguments can be found in and , where the functors

{double-struckL}_{fraktura} \circ τ

come under the name rotation functors . We give a complete proof for reader's convenience.…”

Section: Residual Categoriesmentioning

confidence: 99%

“…Remark In , many fractional Calabi–Yau categories were constructed. Note that by definition, each of these categories is a residual category for an appropriate Lefschetz decomposition.…”

Section: Introductionmentioning

confidence: 99%

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On residual categories for Grassmannians

Kuznetsov

Smirnov

2019

Proc. London Math. Soc.

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We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian G(k,n), we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for k=p, a prime number, modulo completeness of Fonarev's collection (and for p=3 we check this completeness).

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“…Therefore, for any

F \in R

we have

τ^{- 1} false(R_{a} ({double-struckL}_{fraktura} false(τ (F) false)) false) ≅ τ^{- 1} false(τ (F) false) ≅ F .

On the other hand,

{double-struckL}_{fraktura} false(τ (τ^{- 1} false(R_{a} (F) false)) false) ≅ {double-struckL}_{fraktura} false(R_{a} (F) false) ≅ F

for any

F \in {fraktura}^{⊥}

, hence

τ^{- 1} \circ {double-struckR}_{fraktura}

is indeed the inverse of

τ_{R}

. Finally, let us check that

τ_{R}

is a polarization of

scriptR

of index

m

(this is a combination of [, Lemmas 2.6 and 3.13]). Indeed, for any

i ⩽ m

we have

\begin{matrix} true \\ right τ_{scriptR}^{i} & left ≅ L_{a} \circ τ \circ L_{a} \circ τ \circ L_{a} \circ τ \circ \dots \circ L_{a} \circ τ \\ left ≅ L_{a} \circ (τ \circ {double-struckL}_{fraktura} \circ τ^{- 1}) \circ \end{matrix}

…”

Section: Residual Categoriesmentioning

confidence: 97%

“…Therefore, the Serre functor of the phantom category

{scriptC}_{k, n} \subset {scriptR}_{k, n}

(see Remark ) is also trivial, that is,

C_{k, n}

is a Calabi–Yau category. But a Calabi–Yau category with zero Hochschild homology is itself zero by [, Corollary 5.3]. Thus,

{scriptC}_{k, n} = 0

□

…”

Section: Lefschetz Decompositions For Grassmanniansmentioning

confidence: 99%

“…So, we only need to check that

τ_{R}

has all the necessary properties. Most of the arguments can be found in and , where the functors

{double-struckL}_{fraktura} \circ τ

come under the name rotation functors . We give a complete proof for reader's convenience.…”

Section: Residual Categoriesmentioning

confidence: 99%

“…Remark In , many fractional Calabi–Yau categories were constructed. Note that by definition, each of these categories is a residual category for an appropriate Lefschetz decomposition.…”

Section: Introductionmentioning

confidence: 99%

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On residual categories for Grassmannians

Kuznetsov

Smirnov

2019

Proc. London Math. Soc.

Self Cite

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“…Remark 1.9. We expect that, when Twist X is an autoequivalence, it can be expressed as a twist of a spherical functor as in [A2,A1,AL,K3]. However, we do not address this question in this paper, as it would not simplify our proofs.…”

Section: Introductionmentioning

confidence: 98%

Noncommutative enhancements of contractions

Donovan

Wemyss

2019

Advances in Mathematics

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Given a contraction of a variety X to a base Y , we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.

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

Pertusi

Robinett

2023

Mathematische Nachrichten

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We show that the stability conditions on the Kuznetsov component of a Gushel–Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of GL2+(prefixdouble-struckR)$\text{GL}^+_2(\operatorname{\mathbb {R}})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel–Mukai fourfolds.

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Calabi–Yau and fractional Calabi–Yau categories

Cited by 95 publications

References 28 publications

On residual categories for Grassmannians

On residual categories for Grassmannians

Noncommutative enhancements of contractions

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

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