## Periodic Reporting for period 2 - K3CRYSTAL (Moduli of Crystals and K3 Surfaces)

Reporting period: 2018-04-01 to 2019-09-30

PROJECT 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.

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.

"WORK ALREADY PERFORMED WITHIN THE PROJECT

Before one can study Torelli theorems, one often first needs theorems on ""good reduction"", which are due to Néron, Ogg, Shafarevich, Serre, and Tate in the case of Abelian varieties. For K3 surfaces, a complete and thorough treatment of such Néron-Ogg-Shafarevich theorems has now been achieved in (5), (11), and (12). Such results are important in themselves, but also crucial for understanding the image of the period map and thus, they are first steps towards Torelli theorems. A first step, namely a Torelli theorem for ordinary Enriques surfaces was established in (1), the supersingular case will be dealt with by my Ph.D. student Kai Behrens. My Ph.D. student Daniel Boada constructs moduli spaces for bielliptic surfaces.

When studying supersingular and uniruled varieties, one has to understand existence and deformations of rational curves on these varieties. In (7), we studied the deformation theory of rational curves and established rigidity criteria, which is an important step towards understanding supersingular and uniruled varieties. Moreover, in (2) and (3), we studied deformations, degenerations, and regenerations of arbitrary curves on K3 surfaces, which uses rational curves, and which has led to a proof of the longstanding conjecture that every K3 surface in characteristic zero contains infinitely many rational curves. Currently, we continue these studies via positivity properties of cotangent bundles.

To understand the boundary of the image of a period maps, one may study maximally degenerating varieties. In (4), we studied this problem for Abelian varieties, which are easier to handle than K3 surfaces, from the point of view of filtered (\phi,N)-modules. This already led to disproving a conjecture of Raskind, which shows that some ideas in the field were too naive. In (8), Picard numbers of Abelian varieties are studied in positive characteristic, which shows that our understanding of them is still at the very beginning. The study of crystals for K3 surfaces relies upon a better understanding of the deRham-Witt complex and using this language, foundational results were established in (9) and (10). We now turn to the study of moduli spaces and period maps of supersingular varieties.

EVENTS RELATED TO THE PROJECT

In April 2018, we organized a conference ""Crystals and Geometry in Characteristic p"" with 50 participants, among them 12 speakers from all over the world.

In July 2019, we organized a workshop on ""Logarithmic Algebraic Geometry"" with 11 participants and 9 mostly speakers from Germany and Italy.

PROJECT RELATED PREPRINTS AND PUBLICATIONS

One problem in pure mathematics is that the publication process takes extremely long time (one to two years from a preprint to a published article is rather normal). Therefore, there are currently only two publications mentioned. The other work done is within this project are preprints, most of which are already submitted to peer reviewed journals.

(1) Roberto Laface, Sofia Tirabassi, Ordinary Enriques surfaces in positive characteristic, preprint, 12 pages.

(2) Xi Chen, Frank Gounelas, Christian Liedtke, Curves on K3 surfaces, preprint (submitted), 38 pages.

(3) Xi Chen, Frank Gounelas, Christian Liedtke, Rational curves on lattice-polarised K3 surfaces, preprint (submitted), 37 pages.

(4) Oliver Gregory, Christian Liedtke, p-adic Tate conjectures and abeloid varieties, preprint (submitted), 45 pages.

(5) Bruno Chiarellotto, Christopher Lazda, Christian Liedtke, Good Reduction of K3 Surfaces in Equicharacteristic p, preprint (submitted), 15 pages.

(6) Frank Gounelas, John Ottem, Remarks on the positivity of the cotangent bundle of a K3 surface, preprint (submitted), 14 pages.

(7) Kazuhiro Ito, Tetsushi Ito, Kazuhiro Ito, Deformations of Rational Curves in Positive Characteristic, preprint (submitted), 40 pages.

(8) Roberto Laface, On the Picard numbers of Abelian varieties in positive characteristic, preprint (submitted),"

Before one can study Torelli theorems, one often first needs theorems on ""good reduction"", which are due to Néron, Ogg, Shafarevich, Serre, and Tate in the case of Abelian varieties. For K3 surfaces, a complete and thorough treatment of such Néron-Ogg-Shafarevich theorems has now been achieved in (5), (11), and (12). Such results are important in themselves, but also crucial for understanding the image of the period map and thus, they are first steps towards Torelli theorems. A first step, namely a Torelli theorem for ordinary Enriques surfaces was established in (1), the supersingular case will be dealt with by my Ph.D. student Kai Behrens. My Ph.D. student Daniel Boada constructs moduli spaces for bielliptic surfaces.

When studying supersingular and uniruled varieties, one has to understand existence and deformations of rational curves on these varieties. In (7), we studied the deformation theory of rational curves and established rigidity criteria, which is an important step towards understanding supersingular and uniruled varieties. Moreover, in (2) and (3), we studied deformations, degenerations, and regenerations of arbitrary curves on K3 surfaces, which uses rational curves, and which has led to a proof of the longstanding conjecture that every K3 surface in characteristic zero contains infinitely many rational curves. Currently, we continue these studies via positivity properties of cotangent bundles.

To understand the boundary of the image of a period maps, one may study maximally degenerating varieties. In (4), we studied this problem for Abelian varieties, which are easier to handle than K3 surfaces, from the point of view of filtered (\phi,N)-modules. This already led to disproving a conjecture of Raskind, which shows that some ideas in the field were too naive. In (8), Picard numbers of Abelian varieties are studied in positive characteristic, which shows that our understanding of them is still at the very beginning. The study of crystals for K3 surfaces relies upon a better understanding of the deRham-Witt complex and using this language, foundational results were established in (9) and (10). We now turn to the study of moduli spaces and period maps of supersingular varieties.

EVENTS RELATED TO THE PROJECT

In April 2018, we organized a conference ""Crystals and Geometry in Characteristic p"" with 50 participants, among them 12 speakers from all over the world.

In July 2019, we organized a workshop on ""Logarithmic Algebraic Geometry"" with 11 participants and 9 mostly speakers from Germany and Italy.

PROJECT RELATED PREPRINTS AND PUBLICATIONS

One problem in pure mathematics is that the publication process takes extremely long time (one to two years from a preprint to a published article is rather normal). Therefore, there are currently only two publications mentioned. The other work done is within this project are preprints, most of which are already submitted to peer reviewed journals.

(1) Roberto Laface, Sofia Tirabassi, Ordinary Enriques surfaces in positive characteristic, preprint, 12 pages.

(2) Xi Chen, Frank Gounelas, Christian Liedtke, Curves on K3 surfaces, preprint (submitted), 38 pages.

(3) Xi Chen, Frank Gounelas, Christian Liedtke, Rational curves on lattice-polarised K3 surfaces, preprint (submitted), 37 pages.

(4) Oliver Gregory, Christian Liedtke, p-adic Tate conjectures and abeloid varieties, preprint (submitted), 45 pages.

(5) Bruno Chiarellotto, Christopher Lazda, Christian Liedtke, Good Reduction of K3 Surfaces in Equicharacteristic p, preprint (submitted), 15 pages.

(6) Frank Gounelas, John Ottem, Remarks on the positivity of the cotangent bundle of a K3 surface, preprint (submitted), 14 pages.

(7) Kazuhiro Ito, Tetsushi Ito, Kazuhiro Ito, Deformations of Rational Curves in Positive Characteristic, preprint (submitted), 40 pages.

(8) Roberto Laface, On the Picard numbers of Abelian varieties in positive characteristic, preprint (submitted),"

CRITERIA FOR GOOD REDUCTION AND TOWARDS A LOCAL TORELLI THEOREM

In (5), (11), and (12), a complete and satisfactory Néron-Ogg-Shafarevich criterion for K3 surfaces was found, which establishes an important tool in itself and will be important for the image of period maps and Torelli theorems. In (4), we explored period maps under maximal degeneration and disproved a conjecture of Raskind, which is an important result beyond the state of the art. Moreover, a Torelli theorem for ordinary Enriques surfaces in characteristic p>0 was established in (1), which is an important step towards Torelli theorems for such surfaces in the arithmetic situation.

RATIONAL CURVES AND SUPERSINGULAR VARIETIES

A quantitative approach to unirationality and supersingularity via deformations and rigidity of rational curves was established in (7). Moreover, in (2) and (3), we explored a new deformation and a new degeneration technique for curves on K3 surfaces. As an application, which established a long-standing conjecture concerning infinity of rational curves on K3 surfaces, at least in characteristic zero. In (8), the situation of Picard ranks for Abelian varieties in characteristic p was clarified, which complements the previous results.

In (5), (11), and (12), a complete and satisfactory Néron-Ogg-Shafarevich criterion for K3 surfaces was found, which establishes an important tool in itself and will be important for the image of period maps and Torelli theorems. In (4), we explored period maps under maximal degeneration and disproved a conjecture of Raskind, which is an important result beyond the state of the art. Moreover, a Torelli theorem for ordinary Enriques surfaces in characteristic p>0 was established in (1), which is an important step towards Torelli theorems for such surfaces in the arithmetic situation.

RATIONAL CURVES AND SUPERSINGULAR VARIETIES

A quantitative approach to unirationality and supersingularity via deformations and rigidity of rational curves was established in (7). Moreover, in (2) and (3), we explored a new deformation and a new degeneration technique for curves on K3 surfaces. As an application, which established a long-standing conjecture concerning infinity of rational curves on K3 surfaces, at least in characteristic zero. In (8), the situation of Picard ranks for Abelian varieties in characteristic p was clarified, which complements the previous results.