Description

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.

Members

Project related preprints and publications

1. (with Gebhard Martin, Yuya Matsumoto) Torsors over the rational double points in characteristic p,
   76 pages, October 8, 2021.
2. (with Gebhard Martin, Yuya Matsumoto) Linearly reductive quotient singularities,
   preprint, 51 pages, February 2, 2021.
3. Xi Chen, Frank Gounelas, Curves of maximal moduli on K3 surfaces,
   preprint, 22 pages, July 6, 2020.
4. Kai Behrens, On the number of Enriques quotients for supersingular K3 surfaces,
   preprint, 27 pages, March 3, 2020.
5. Kai Behrens, A moduli space for supersingular Enriques surfaces, 
   preprint, 27 pages, January 24, 2020.
6. Roberto Laface, Sofia Tirabassi, On ordinary Enriques surfaces in positive characteristic,
   to appear in Nagoya Math. J.
7. Xi Chen, Frank Gounelas, Christian Liedtke, Curves on K3 surfaces,
   to appear in Duke Math. J.
8. Xi Chen, Frank Gounelas, Christian Liedtke, Rational curves on lattice-polarised K3 surfaces,
   to appear in Algebraic Geometry.
9. Olof Bergvall, Frank Gounelas, Cohomology of moduli spaces of Del Pezzo surfaces,
   preprint, 26 pages, June 7, 2019. 
10. Oliver Gregory, Christian Liedtke, p-adic Tate conjectures and abeloid varieties,
    Doc. Math. 24 (2019), 1879-1934.
11. Bruno Chiarellotto, Christopher Lazda, Christian Liedtke, Good Reduction of K3 Surfaces in Equicharacteristic p,
    to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.
12. Frank Gounelas, John Ottem, Remarks on the positivity of the cotangent bundle of a K3 surface,
    Épijournal Géom. Algébrique 4 (2020), Art. 8, 16 pp.
13. Roberto Laface, On the Picard numbers of Abelian varieties in positive characteristic,
    preprint, 17 pages, March 23, 2018.
14. Kazuhiro Ito, Tetsushi Ito, Christian Liedtke, Deformations of Rational Curves in Positive Characteristic,
    J. Reine Angew. Math. 769 (2020), 55-86.
15. Oliver Gregory, Andreas Langer, Overconvergent de Rham-Witt cohomology for semistable varieties,
    Münster J. Math. 13 (2020), no. 2, 541-571.
16. Oliver Gregory, Andreas Langer, Higher displays arising from filtered de Rham-Witt complexes,
    preprint (submitted), 13 pages, October 13, 2017.
17. Bruno Chiarellotto, Christopher Lazda, Christian Liedtke, A Néron-Ogg-Shafarevich criterion for K3 surfaces,
    Proc. Lond. Math. Soc. 119 (2019), 469-514.
18. Christian Liedtke, Yuya Matsumoto, Good Reduction of K3 Surfaces,
    Compos. Math. 154 (2018), 1–35.

Press Releases

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