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 Holomorphic curves in Shimura varieties

2018 ◽  
Vol 111 (4) ◽  
pp. 379-388 ◽  
Author(s):  
Michele Giacomini
Keyword(s):  
Abelian Varieties ◽  
Holomorphic Curves ◽  
Zariski Closure ◽  
Shimura Varieties

Abstract We prove a hyperbolic analogue of the Bloch–Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.

scholarly journals Integral canonical models for Spin Shimura varieties

2015 ◽  
Vol 152 (4) ◽  
pp. 769-824 ◽  
Author(s):  
Keerthi Madapusi Pera
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
K3 Surfaces ◽  
Shimura Varieties ◽  
Good Reduction ◽  
Intersection Numbers ◽  
Orthogonal Groups ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Precise Sense

We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes$p>2$where the level is not divisible by$p$. We exhibit these models as schemes of ‘relative PEL type’ over integral canonical models of larger Spin Shimura varieties with good reduction at$p$. Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

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.

scholarly journals Autoequivalences of twisted K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 912-937 ◽  
Author(s):  
Emanuel Reinecke
Keyword(s):  
K3 Surfaces ◽  
K3 Surface ◽  
Lattice Structures ◽  
Hodge Structures ◽  
Derived Equivalences

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$or $2$in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

scholarly journals On Some Generalized Rapoport–Zink Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1111-1187
Author(s):  
Xu Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Connected Components ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Mixed Characteristic ◽  
Special Fibers ◽  
Type Spaces ◽  
Local Invariants

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

scholarly journals Ordinary reduction of K3 surfaces

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Zarhin
Keyword(s):  
Number Field ◽  
Field Extension ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Zero Set ◽  
Algebraic Field ◽  
Density Zero ◽  
Archimedean Place ◽  
Ordinary Reduction

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

The Euler number of certain primitive Calabi–Yau threefolds

2000 ◽  
Vol 128 (1) ◽  
pp. 79-86
Author(s):  
MEI-CHU CHANG ◽  
HOIL KIM
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Three Dimensional ◽  
Building Blocks ◽  
K3 Surfaces ◽  
Physical Models ◽  
Ample Divisor ◽  
K3 Surface ◽  
Superstring Theory ◽  
Two Parameters

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.

K3 Surfaces

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Moduli Spaces ◽  
Special Class ◽  
K3 Surfaces ◽  
Derived Categories ◽  
K3 Surface ◽  
Hodge Structures ◽  
Surface Theory ◽  
Torelli Theorem ◽  
Stable Sheaves ◽  
Moduli Theory

After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

T-dual solutions of the Hull–Strominger system on non-Kähler threefolds

2020 ◽  
Vol 2020 (766) ◽  
pp. 137-150
Author(s):  
Mario Garcia-Fernandez
Keyword(s):  
Moduli Spaces ◽  
Tangent Bundle ◽  
Analytical Methods ◽  
Existence Of Solutions ◽  
K3 Surfaces ◽  
Dual Solutions ◽  
K3 Surface ◽  
Yang Mills ◽  
Torus Bundles ◽  
Stable Sheaves

AbstractWe construct new examples of solutions of the Hull–Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection {\nabla} on the tangent bundle is Hermite–Yang–Mills. With this ansatz for the connection {\nabla}, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull–Strominger system on compact non-Kähler manifolds with different topology.

scholarly journals Computing cup products in integral cohomology of Hilbert schemes of points on K3 surfaces

2016 ◽  
Vol 19 (1) ◽  
pp. 78-97
Author(s):  
Simon Kapfer
Keyword(s):  
Computer Program ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Hilbert Schemes ◽  
Integral Cohomology ◽  
Cup Products ◽  
By Products

We study cup products in the integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.Supplementary materials are available with this article.

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