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Moduli of Crystals and K3 Surfaces

Total Cost €


EC-Contrib. €






Project "K3CRYSTAL" data sheet

The following table provides information about the project.


Organization address
address: Arcisstrasse 21
postcode: 80333

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Germany [DE]
 Project website
 Total cost 1˙328˙710 €
 EC max contribution 1˙328˙710 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-CoG
 Funding Scheme ERC-COG
 Starting year 2016
 Duration (year-month-day) from 2016-10-01   to  2021-09-30


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

Algebraic geometry deals with algebraic varieties, that is, systems of polynomial equations and their geometric interpretation. Its ultimate goal is the classification of all algebraic varieties. For a detailed understanding, one has to construct their moduli spaces, and eventually study them over the integers, that is, in the arithmetic situation. So far, the best results are available for curves and Abelian varieties.

To go beyond the aforementioned classes, I want to study arithmetic moduli spaces of the only other classes that are currently within reach, namely, K3 surfaces, Enriques surfaces, and Hyperkähler varieties. I expect this study to lead to finer invariants, to new stratifications of moduli spaces, and to open new research areas in arithmetic algebraic geometry.

Next, I propose a systematic study of supersingular varieties, which are the most mysterious class of varieties in positive characteristic. Again, a good theory is available only for Abelian varieties, but recently, I established a general framework via deformations controlled by formal group laws. I expect to extend this also to constructions in complex geometry, such as twistor space, which would link so far completely unrelated fields of research.

I want to accompany these projects by developing a general theory of period maps and period domains for F-crystals, with an emphasis on the supersingular ones to start with. This will be the framework for Torelli theorems that translate the geometry and moduli of K3 surfaces, Enriques surfaces, and Hyperkähler varieties into explicit linear algebra problems, thereby establishing new tools in algebraic geometry.


year authors and title journal last update
List of publications.
2018 Christian Liedtke, Yuya Matsumoto
Good reduction of K3 surfaces
published pages: 1-35, ISSN: 0010-437X, DOI: 10.1112/s0010437x17007400
Compositio Mathematica 154/01 2019-06-18
2019 Bruno Chiarellotto, Christopher Lazda, Christian Liedtke
A Néron–Ogg–Shafarevich criterion for K3 surfaces
published pages: 469-514, ISSN: 0024-6115, DOI: 10.1112/plms.12237
Proceedings of the London Mathematical Society 119/2 2019-06-06

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The information about "K3CRYSTAL" are provided by the European Opendata Portal: CORDIS opendata.

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