scholarly journals Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties

2020 ◽  
Vol 26 (3) ◽  
Author(s):  
Atsushi Ito ◽  
Makoto Miura ◽  
Shinnosuke Okawa ◽  
Kazushi Ueda
Keyword(s):  
Abelian Varieties ◽  
K3 Surfaces ◽  
Derived Equivalence ◽  
Grothendieck Ring

scholarly journals Ordinary K3 surfaces over a finite field

2020 ◽  
Vol 2020 (761) ◽  
pp. 141-161
Author(s):  
Lenny Taelman
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Abelian Varieties ◽  
K3 Surfaces ◽  
Small Degree ◽  
Stable Reduction

AbstractWe give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over {{\mathbf{Z}}}. This gives an analogue for K3 surfaces of Deligne’s description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Our main result is conditional on a conjecture on potential semi-stable reduction of K3 surfaces over p-adic fields. We give unconditional versions for K3 surfaces of large Picard rank and for K3 surfaces of small degree.

Isogenies Between K3 Surfaces Over

2020 ◽  
Author(s):  
Ziquan Yang
Keyword(s):  
Abelian Varieties ◽  
K3 Surfaces ◽  
Main Step ◽  
Shimura Varieties ◽  
K3 Surface ◽  
Perfect Field ◽  
Finite Height ◽  
Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ 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 $p$.

scholarly journals A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces

2008 ◽  
Vol 17 (3) ◽  
pp. 481-502 ◽  
Author(s):  
Alexei N. Skorobogatov ◽  
Yuri G. Zarhin
Keyword(s):  
Abelian Varieties ◽  
K3 Surfaces ◽  
Brauer Group ◽  
Finiteness Theorem

On the Chow ring of K3 surfaces and hyper-Kahler manifolds

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Projective Space ◽  
Abelian Varieties ◽  
K3 Surfaces ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Chow Ring ◽  
Open Set

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.

scholarly journals Finiteness theorems for K3 surfaces and abelian varieties of CM type

2018 ◽  
Vol 154 (8) ◽  
pp. 1571-1592 ◽  
Author(s):  
Martin Orr ◽  
Alexei N. Skorobogatov
Keyword(s):  
Abelian Varieties ◽  
Algebraic Closure ◽  
Complex Multiplication ◽  
Uniform Boundedness ◽  
Invariant Subgroup ◽  
Smooth Projective Variety ◽  
Number Fields ◽  
K3 Surfaces ◽  
Isomorphism Classes ◽  
Finiteness Theorems

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

scholarly journals Cremona transformations and derived equivalences of K3 surfaces

2018 ◽  
Vol 154 (7) ◽  
pp. 1508-1533 ◽  
Author(s):  
Brendan Hassett ◽  
Kuan-Wen Lai
Keyword(s):  
K3 Surfaces ◽  
Cremona Transformation ◽  
Grothendieck Ring ◽  
Cremona Transformations ◽  
Derived Equivalences ◽  
Affine Line ◽  
The Difference ◽  
Base Loci

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

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