Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

“…Other examples include Calabi-Yau threefolds from Grassmannians of type G 2 [IMOU16a,Kuz16]. Kuznetsov and Shinder [KS16] have formulated general conjectures relating derived equivalence to zero-divisors in the Grothendieck ring; our example is an instance of [KS16, Conj. 1.6].…”

Section: Introductionmentioning

confidence: 99%

Cremona transformations and derived equivalences of K3 surfaces

Hassett

Lai

2018

Compositio Math.

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“…Specifically, for Calabi–Yau threefolds

X

Y

in the so‐called Pfaffian–Grassmannian correspondence, the classes satisfy

[X] \neq [Y]

and

{double-struckL}^{n} \cdot false([X] - [Y] false) = 0,

where one can take any

n ⩾ 6

[4, 24]. Following [22], we say that smooth projective connected varieties

X

and

Y

are L‐equivalent if equation (1.1) holds for some

n ⩾ 1

, and we say that

X

and

Y

are non‐trivially L‐equivalent if in addition

[X] \neq [Y]

. If

X

and

Y

are not covered by rational curves and

X

and

Y

are not birational, then an L‐equivalence between them is automatically non‐trivial (see, for example, [22, Proposition 2.2]).…”

Section: Introductionmentioning

confidence: 99%

L‐equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

Shinder

Zhang

2020

Bull. London Math. Soc.

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We construct non-trivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence.The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2, 5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg-Shafarevich theory of twisting for elliptic surfaces.Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron-Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.

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“…one could study their classes [S] and [S ′ ] in the Grothendieck ring of varieties K 0 (Var(k)) or their associated motives h(S) and h(S ′ ) in the category of Chow motives Mot(k). In the recent articles [7,14,16], examples in degree 8 and 12 have been studied for whichThe question whether the Chow motives h(S) and h(S ′ ) in Mot(k) are isomorphic was first addressed and answered affirmatively in special cases in [23]. Assuming finite-dimensionality of the motives the question was settled in [3].…”

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confidence: 99%

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confidence: 99%

Motives of derived equivalent K3 surfaces

Huybrechts

2017

Abh. Math. Semin. Univ. Hambg.

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Abstract. We observe that derived equivalent K3 surfaces have isomorphic Chow motives.There are many examples of non-isomorphic K3 surfaces S and S ′ (over a field k) with equivalent derived categories D b (S) ≃ D b (S ′ ) of coherent sheaves. It is known that in this case S ′ is isomorphic to a moduli space of slope-stable bundles on S and vice versa, cf. [10,20,22]. However, the precise relation between S and S ′ still eludes us and many basic questions are hard to answer. For example, one could ask whether S ′ contains infinitely many rational curves, expected for all K3 surfaces, if this is known already for S. Or, if k is a number field, does potential density for S imply potential density for S ′ ?As a direct geometric approach to these questions seems difficult, one may wonder whether S and S ′ can be compared at least motivically. This could mean various things, e.g. one could study their classes [S] and [S ′ ] in the Grothendieck ring of varieties K 0 (Var(k)) or their associated motives h(S) and h(S ′ ) in the category of Chow motives Mot(k). In the recent articles [7,14,16], examples in degree 8 and 12 have been studied for whichThe question whether the Chow motives h(S) and h(S ′ ) in Mot(k) are isomorphic was first addressed and answered affirmatively in special cases in [23]. Assuming finite-dimensionality of the motives the question was settled in [3].In this short note we point out that the available techniques in the theory of motives are enough to show that two derived equivalent K3 surfaces have indeed isomorphic Chow motives.Theorem 0.1. Let S and S ′ be K3 surfaces over an algebraically closed field k. Assume that there exists an exact k-linear equivalence D b (S) ≃ D b (S ′ ) between their bounded derived categories of coherent sheaves. Then there is an isomorphismin the category of Chow motives Mot(k).The assumption on the field k can be weakened, it suffices to assume that ρ(S) = ρ(Sk). We had originally expected that the invariance of the Beauville-Voisin ring as proved in [11,12] would be central to the argument However, it turns to out to have no bearing on the problem, but it implies that a distinguished decomposition of the motives in their algebraic and transcendental parts is preserved under derived equivalence.Acknowledgements: I am very grateful to Charles Vial for answering my questions and to him and Andrey Soldatenkov for comments on a first version and helpful suggestions.

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Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

Cited by 28 publications

References 24 publications

Cremona transformations and derived equivalences of K3 surfaces

Cremona transformations and derived equivalences of K3 surfaces

L‐equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

Motives of derived equivalent K3 surfaces

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