A Stronger Derived Torelli Theorem for K3 Surfaces

Lieblich, Max; Olsson, Martin

doi:10.1007/978-3-319-49763-1_5

Cited by 7 publications

(6 citation statements)

References 27 publications

(23 reference statements)

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“…The moduli theory of twisted sheaves for complex K3's was initiated by S. Mukai and its generalization to positive characteristic has been studied by Lieblich, Maulik, Olsson and Snowden (c.f. [20], [21], [22]). Huybrechts has shown that in fact, over C, every isogeny can be realized by a sequence of such operations [13, Theorem 0.1].…”

Section: S (Cspin(l D ) ω)mentioning

confidence: 99%

“…We remark that in fact any two supersingular K3 surfaces X, X ′ over Fp for p > 2 are isogenous according to our definition, as Pic (X) Q ∼ = Pic (X ′ ) Q as Q-quadratic lattices. By works of Artin and Radukov-Shafarevich, for any supersingular X over Fp , Pic (X) ∼ = N p,σ for some 1 ≤ σ ≤ 10, where N p,σ is the unique even, non-degenerate Z-lattice with signature (1,21) and discriminant group (Z/pZ) 2σ . One may check that N p,σ ⊗Q ∼ = N p,σ ′ ⊗Q for any two different σ, σ ′ by the Hasse principle: N p,σ ⊗R ∼ = N p,σ ′ ⊗R as a real quadratic form is determined by its signature.…”

Section: Constructing Liftingsmentioning

confidence: 99%

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Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

Yang¹

2018

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We generalize Mukai and Shafarevich's definition of isogenies between K3 surfaces over C to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over Fp by prescribing linear algebraic data when p is large. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on p.Surjectivity of the period map gives us a complex analytic K3 X ′ which corresponds to the induced Hodge structure on Λ. Then one can use [3, IV Theorem 6.2] to see that X ′ has to be algebraic.We give an analogue of Theorem 1.2 for quasi-polarized K3 surfaces over Fp , using K3 crystals and Z p -lattices to replace Z-lattices. K3 crystals were originally defined by Ogus to characterize the F-crystals that arise from K3 surfaces (c.f. [34, Definition 3.1]): Definition 1.3. Let k be a perfect field of characteristic p > 0 and let σ be the lift of Frobenius on W (k). A K3 crystal over k is a finitely generated free W (k)-module D equipped with a σ-linear injection ϕ : D → D and a perfect symmetric bilinear pairing −, − such that ϕ(x), ϕ(y) = p 2 σ( x, y ), p 2 D ⊂ ϕ(D), and rank ϕ⊗k = 1.Recall that a quasi-polarized K3 surface is a pair (X, ξ) where X is a K3 surface and ξ is a primitive big and nef line bundle on X. We call the self-intersection number of ξ the degree of (X, ξ). Customarily, we write P 2 * (X) := ch * (ξ) ⊥ ⊂ H 2 * (X) for the primitive cohomology of (X, ξ) ( * = dR, cris, ét, etc). Our main theorem states:

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Section: S (Cspin(l D ) ω)mentioning

confidence: 99%

Section: Constructing Liftingsmentioning

confidence: 99%

Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

Yang¹

2018

Preprint

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“…We also point out here that the proofs of these results go via lifting to characteristic zero and thus use the Hodge theoretic description given by Mukai and Orlov. Furthermore, Lieblich-Olsson [46] also proved the derived version of the Torelli theorem using the Crystalline Torelli theorem for supersingular K3 surfaces.…”

Section: Introductionmentioning

confidence: 96%

On Derived Equivalences of K3 Surfaces in Positive Characteristic

Srivastava¹

2018

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For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide an upper bound on the kernel of this natural cohomological descent map.Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product. ContentsAcknowledgement 1 1. Introduction 2 1.1. Conventions and Notations 3 2. Preliminaries on K3 Surfaces and Derived Equivalences 3 3. Derived Autoequivalences of K3 Surfaces in Positive Characteristic 14 3.1. Obstruction to Lifting Derived Autoequivalences 14 3.2. The Cone Inversion Map 20 4. Counting Fourier-Mukai Partners in Positive Characteristic 23 4.1. The Class Number Formula 27 5. Appendix: F-crystal on Crystalline Cohomology 27 References 31 AcknowledgementThe results contained in this article are a part of PhD thesis written under the supervision of Prof. Dr. Hélène Esnault. I owe her special gratitude for guidance, support, continuous encouragement and inspiration she always provided me. I thank Berlin Mathematical School for the PhD fellowship. I am greatly thankful to

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“…This explains why the conclusions of Theorem 1.2 and [BM,Theorem 5.1] are not completely symmetric. Finally, we point out that filtered equivalences have been introduced in [LibO1], together with other stronger versions in [LibO2], to establish a derived version of Torelli theorem for K3 surfaces in characteristic p = 2. Now we go back to the proof of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

2019

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We prove that any Fourier-Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the Fourier-Mukai set of canonical covers of hyperelliptic and Enriques surfaces over an algebraically closed field of characteristic greater than three is trivial. These results extend to positive characteristic earlier results of Bridgeland-Maciocia and Sosna.

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A Stronger Derived Torelli Theorem for K3 Surfaces

Cited by 7 publications

References 27 publications

Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

On Derived Equivalences of K3 Surfaces in Positive Characteristic

Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

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