## Periodic Reporting for period 1 - K3CRYSTAL (Moduli of Crystals and K3 Surfaces)

**Reporting period:**2016-10-01

**to**2018-03-31

## Summary of the context and overall objectives of the project

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 performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"WORK DONE WITHIN THE PROJECT

Work by the PI:

- before one can study Torelli theorems, one oftentimes first needs so-called theorems on ""good reduction"" (due to Neron, Ogg, Shafarevich, Serre, and Tate in the case of Abelian varieties). More precisely, these are important to show that the period map, which lies at the heart of Torelli theorems, is surjective. For K3 surfaces, this was largely achieved by the PI and led to the article (joint with Yuya Matsumoto) ""Good reduction of K3 surfaces"" (see Publications section and below). It turned out that the situation in the K3-case is more complicated than in the case of the Abelian varieties. The occuring subtleties have been overcome by the PI in the preprint (joint with Bruno Chiarellotto and Christopher Lazda) ""A Néron-Ogg-Shafarevich criterion for K3 surfaces"" (submitted, currently under review, see below), which solves this problem in the framework of F-crystals, as well as l-adic Galois representations. From there, I am currently working with my PostDoc Oliver Gregory on local period domains, which is the next step towards arithmetic Torelli theorems for K3 surfaces

- when developping a general theory of supersingular varieties, one has to understand deformations of rational curves on these varieties. (Conjecturally, these varieties are covered by rational curves.) This leads to the question, when or whether such rational curves are rigid, i.e., do not deform. It turns out that such rational curves have to be rather singular (with respect to the characteristic of the ground field). This has been made precise in a preprint of the PI (joint with Tetsushi Ito and Kazuhiro Ito) ""Deformation of rational curves in positive characteristic"". This is in important step towards a general theory of supersingular varieties, as envisioned in the project.

Work by the two PostDocs, who are employed within the project:

- Roberto Laface: an important aspect of supersingular varieties in positive characteristic is that their Picard numbers should be maximal. Before understanding the problem of Picard numbers of supersingular varieties in detail, it turned out that even elementary cases, namely Picard numbers of Abelian varieties in positive characteristic are poorly understood. This has been addressed and solved by him. From there, he will start working on stable curves of degree 6 in the projective plane, which is the key to the stability of supersingular K3 surfaces, which is the key to (the missing cases of) Ogus' crystalline Torelli theorem in characteristic 3.

- Oliver Gregory: the study of crystals for K3 surfaces relies upon a better understanding of the deRham-Witt complex. For example, to understand limits and boundary of moduli spaces, the geometric part is good reduction theorems (see above), but on the level of period spaces, Torelli theorems, and period maps, one has to understand the semi-stable case of F-crystals, which he has achieved in two preprints, which are submitted to journals, and which are currently under review (see below). In the preprint ""Overconvergent de Rham-Witt cohomology for semistable varieties"" (see below), the framework for these degenerations are laid. When establishing period maps, one might have to work within category of displays, which generalizes the category of crystals. Some foundations of this were laid in the preprint ""Higher displays arising from filtered de Rham-Witt complexes"". Both preprints will most likely be important steps and frameworks for arithmetic Torelli theorems, the ultimate goal of the project.

Work by the two Ph.D students, who are employed within the project:

- Daniel Boada: he studies arithmetic moduli spaces for (quasi-)elliptic surfaces. These are related to Abelian varieties of dimension two, and are close relatives of K3 surfaces. However, this class of surfaces is much easier to study and to understand. Therefore, we expect to solve some of the difficult questions raised in the ERC project in a particular-ea"

Work by the PI:

- before one can study Torelli theorems, one oftentimes first needs so-called theorems on ""good reduction"" (due to Neron, Ogg, Shafarevich, Serre, and Tate in the case of Abelian varieties). More precisely, these are important to show that the period map, which lies at the heart of Torelli theorems, is surjective. For K3 surfaces, this was largely achieved by the PI and led to the article (joint with Yuya Matsumoto) ""Good reduction of K3 surfaces"" (see Publications section and below). It turned out that the situation in the K3-case is more complicated than in the case of the Abelian varieties. The occuring subtleties have been overcome by the PI in the preprint (joint with Bruno Chiarellotto and Christopher Lazda) ""A Néron-Ogg-Shafarevich criterion for K3 surfaces"" (submitted, currently under review, see below), which solves this problem in the framework of F-crystals, as well as l-adic Galois representations. From there, I am currently working with my PostDoc Oliver Gregory on local period domains, which is the next step towards arithmetic Torelli theorems for K3 surfaces

- when developping a general theory of supersingular varieties, one has to understand deformations of rational curves on these varieties. (Conjecturally, these varieties are covered by rational curves.) This leads to the question, when or whether such rational curves are rigid, i.e., do not deform. It turns out that such rational curves have to be rather singular (with respect to the characteristic of the ground field). This has been made precise in a preprint of the PI (joint with Tetsushi Ito and Kazuhiro Ito) ""Deformation of rational curves in positive characteristic"". This is in important step towards a general theory of supersingular varieties, as envisioned in the project.

Work by the two PostDocs, who are employed within the project:

- Roberto Laface: an important aspect of supersingular varieties in positive characteristic is that their Picard numbers should be maximal. Before understanding the problem of Picard numbers of supersingular varieties in detail, it turned out that even elementary cases, namely Picard numbers of Abelian varieties in positive characteristic are poorly understood. This has been addressed and solved by him. From there, he will start working on stable curves of degree 6 in the projective plane, which is the key to the stability of supersingular K3 surfaces, which is the key to (the missing cases of) Ogus' crystalline Torelli theorem in characteristic 3.

- Oliver Gregory: the study of crystals for K3 surfaces relies upon a better understanding of the deRham-Witt complex. For example, to understand limits and boundary of moduli spaces, the geometric part is good reduction theorems (see above), but on the level of period spaces, Torelli theorems, and period maps, one has to understand the semi-stable case of F-crystals, which he has achieved in two preprints, which are submitted to journals, and which are currently under review (see below). In the preprint ""Overconvergent de Rham-Witt cohomology for semistable varieties"" (see below), the framework for these degenerations are laid. When establishing period maps, one might have to work within category of displays, which generalizes the category of crystals. Some foundations of this were laid in the preprint ""Higher displays arising from filtered de Rham-Witt complexes"". Both preprints will most likely be important steps and frameworks for arithmetic Torelli theorems, the ultimate goal of the project.

Work by the two Ph.D students, who are employed within the project:

- Daniel Boada: he studies arithmetic moduli spaces for (quasi-)elliptic surfaces. These are related to Abelian varieties of dimension two, and are close relatives of K3 surfaces. However, this class of surfaces is much easier to study and to understand. Therefore, we expect to solve some of the difficult questions raised in the ERC project in a particular-ea"

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

"So far, it is a little bit early to make definite statements.

GOOD REDUCTION AND TOWARDS A LOCAL ARITHMETIC TORELLI

The naive analog of a good reduction criterion à la Néron-Ogg-Shafarevich for K3 surfaces is false - this was observed in the article ""Good Reduction of K3 Surfaces"" (see (5) in the list above) and a solution to this problem was found in the preprint ""A Néron-Ogg-Shafarevich criterion for K3 surfaces"" (see (4) in the list above). This detour was a little bit unforeseen and definitely beyond the state of the art, but the now that a complete and satisfactory solution is available, it follows that one has to add one more piece of data to obtain a criterion of good reduction. This slightly changes the way arithmetic Torelli theorems might work for K3 surfaces - in joint work with my PostDoc Oliver Gregory, we are currently incorporating these changes into our general framework and the strategy. We hope to obtain a local and p-adic version of an arithmetic Torelli theorem in the next year or so, and from there, we plan to tackle the global situation, which will be hopefully finished by the end of the project.

To have a framework for these results, the new results (2) and (3) from the list of publications above on higher displays (which generalize crystals to higher dimensional bases) and deRham-Witt cohomology (which is closely related to crystalline cohomology) in the semi-stable case, will most likely be needed.

SUPERSINGULAR VARIETIES

With my PostDoc's Roberto Laface's work in progress, the situation of Picard ranks and Shioda-supersingularity for Abelian varieties in characteristic p will be clarified - again, this was a little bit of a detour, since everyone (that is, many experts) expected no interesting results and no surprises here, which did arise. This is still astonishing and puzzling. Therefore, his results on Picard ranks go beyond the state of the art. From here, he will now turn to K3 surfaces in small characteristics. Within the next year, we hope to see a refined version of Ogus' supersingular Torelli theorem.

A first and novel geometric approach to unirationality and supersingularity, which goes beyond what was known, was taken in the PI's preprint (see (1) in the above list), where the role of rational curves on these varieties was studied and clarified. From here, one has to see whether rational curves can be deformed, e.g., in the twistor spaces as in my proposal. This is still highly speculative and I dare not make any predictions at this point.

"

GOOD REDUCTION AND TOWARDS A LOCAL ARITHMETIC TORELLI

The naive analog of a good reduction criterion à la Néron-Ogg-Shafarevich for K3 surfaces is false - this was observed in the article ""Good Reduction of K3 Surfaces"" (see (5) in the list above) and a solution to this problem was found in the preprint ""A Néron-Ogg-Shafarevich criterion for K3 surfaces"" (see (4) in the list above). This detour was a little bit unforeseen and definitely beyond the state of the art, but the now that a complete and satisfactory solution is available, it follows that one has to add one more piece of data to obtain a criterion of good reduction. This slightly changes the way arithmetic Torelli theorems might work for K3 surfaces - in joint work with my PostDoc Oliver Gregory, we are currently incorporating these changes into our general framework and the strategy. We hope to obtain a local and p-adic version of an arithmetic Torelli theorem in the next year or so, and from there, we plan to tackle the global situation, which will be hopefully finished by the end of the project.

To have a framework for these results, the new results (2) and (3) from the list of publications above on higher displays (which generalize crystals to higher dimensional bases) and deRham-Witt cohomology (which is closely related to crystalline cohomology) in the semi-stable case, will most likely be needed.

SUPERSINGULAR VARIETIES

With my PostDoc's Roberto Laface's work in progress, the situation of Picard ranks and Shioda-supersingularity for Abelian varieties in characteristic p will be clarified - again, this was a little bit of a detour, since everyone (that is, many experts) expected no interesting results and no surprises here, which did arise. This is still astonishing and puzzling. Therefore, his results on Picard ranks go beyond the state of the art. From here, he will now turn to K3 surfaces in small characteristics. Within the next year, we hope to see a refined version of Ogus' supersingular Torelli theorem.

A first and novel geometric approach to unirationality and supersingularity, which goes beyond what was known, was taken in the PI's preprint (see (1) in the above list), where the role of rational curves on these varieties was studied and clarified. From here, one has to see whether rational curves can be deformed, e.g., in the twistor spaces as in my proposal. This is still highly speculative and I dare not make any predictions at this point.

"