A monodromy criterion for the good reduction of $K3$ surfaces

2020 ◽  
Author(s):  
Genaro Hernandez-Mada
Keyword(s):  
K3 Surfaces ◽  
Good Reduction

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 Good reduction criterion for K3 surfaces

2014 ◽  
Vol 279 (1-2) ◽  
pp. 241-266 ◽  
Author(s):  
Yuya Matsumoto
Keyword(s):  
K3 Surfaces ◽  
Good Reduction

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

scholarly journals Good Reduction of K3 Surfaces in Equicharacteristic p

2021 ◽  
pp. 15
Author(s):  
Bruno Chiarellotto ◽  
Christopher Lazda ◽  
Christian Liedtke
Keyword(s):  
K3 Surfaces ◽  
Good Reduction

scholarly journals Motivic integral of K3 surfaces over a non-archimedean field

2011 ◽  
Vol 228 (5) ◽  
pp. 2688-2730 ◽  
Author(s):  
Allen J. Stewart ◽  
Vadim Vologodsky
Keyword(s):  
K3 Surfaces

scholarly journals Verlinde formulae on complex surfaces: K-theoretic invariants

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Göttsche ◽  
M. Kool ◽  
R. A. Williams
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Complex Surfaces ◽  
Type Formula ◽  
Verlinde Formula ◽  
Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

scholarly journals Curves on K3 surfaces in divisibility 2

2021 ◽  
Vol 9 ◽  
Author(s):  
Younghan Bae ◽  
Tim-Henrik Buelles
Keyword(s):  
Modular Forms ◽  
K3 Surfaces ◽  
Initial Condition ◽  
Smooth Curves ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Degeneration Formula ◽  
The Relationship ◽  
Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

scholarly journals Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

2021 ◽  
Author(s):  
Alice Garbagnati
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Minimal Resolution ◽  
Elliptic Fibrations ◽  
Smooth Model ◽  
Birational Model ◽  
Hodge Numbers ◽  
Rational Elliptic Surface

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

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