# New Trends in Arithmetic and Geometry of Algebraic Surfaces (17w5146)

Arriving in Banff, Alberta Sunday, March 12 and departing Friday March 17, 2017

## Organizers

(Harvard University)

(Leibniz University Hannover)

Keiji Oguiso (The University of Tokyo)

## Objectives

The interplay of arithmetic and geometry has been a driving force in the study of algebraic curves, culminating in Faltings' finiteness theorem for rational points on curves of general type. For algebraic surfaces, such deep structures are mostly still conjectural, but great progress has been made in recent years following this leitmotif. The most spectacular achievement of the last few years might have been the proof of the Tate conjecture for K3 surfaces due to Madapusi, Maulik and Charles. However, this is only the brightest star among a plentitude of amazing results manifesting the intertwining of arithmetic and geometry (and also initiating new directions, for instance in dynamics). In the following, we shall highlight a few streams of research which we consider of utmost relevance for our workshop.

K3 surfaces are the most prominent player in our story, notably because of their versatility and also because of their relevance to neighboring areas such as differential geometry and physics. Beyond the breakthrough on the Tate conjecture, there have several further important developments on K3 surfaces in the last few years:

(1) Good reduction and Honda-Tate (Matsumoto, Liedtke, Taelman);

(2) unirationality (Liedtke, Lieblich);

(3) moduli of K3 surfaces, in particular relating to double sextics, elliptic K3 surfaces and degenerations of K3 surfaces and their relation with arithmetic (Alexeev, Brunyate, Elkies, Hacking, Kumar, Laza, Thompson);

(4) dynamics (Blanc, Cantat, Esnault, McMullen, Oguiso, Schütt).

We would also like to emphasize the experimental approach towards arithmetic and geometry of K3 surfaces (closely related to the developments sketched above). As an illustration, consider the important problem how the Picard number of a K3 surface $X$ defined over some number field behaves under reduction. By work of Li and Liedtke, this has important implications for rational curves on $X$, since an infinitude of places with the Picard number increasing upon reduction can be used to produce an infinitude of rational curves on the original surface. For odd Picard number, this crucial reduction property follows from the Tate conjecture, but for even Picard number, it seems only to be known in special cases such as Kummer surfaces of product type (Charles). This is where experiments using zeta functions (Elsenhans, Jahnel) and $p$-adic cohomology (Costa, Harvey) enter. We expect that this area of ideas might see important progress until 2017.

Enriques surfaces are closely related to K3 surfaces, yet they come with intriguing subtleties which have been a driving force for the investigation of the deep structures of algebraic surfaces. Historically, they have been among the first surfaces (together with Godeaux surfaces) which were shown to be non-rational despite sharing the $\mathbf Q$-cohomology with the projective plane blown up in a finite number of points (which thus led Castelnuovo to formulate his rationality criterion in terms of the second plurigenus). From today's perspective, this special role of Enriques surfaces manifests itself prototypically in the study of finite group actions. For instance, there are complex Enriques surfaces with finite groups of automorphisms acting trivially on cohomology (with $\mathbf Q$ or even $\mathbf Z$ coefficients). In a similar direction, Enriques surfaces with finite automorphism group are very special; both properties are completely contrary to what happens for K3 surfaces. We would like to highlight the following recent projects (partly ongoing):

(1) Semi-symplectic finite group actions on Enriques surfaces and their relation to the Mathieu group $M_{12}$ (Mukai, Ohashi)

(2) Moduli of polarized complex Enriques surfaces (Gritsenko, Hulek)

(3) Enriques surfaces in characteristic $2$ (Katsura, Kondo, Liedtke, Shepherd-Barron)

Surfaces of general type form the most mysterious class of algebraic surfaces. There are still many open problems about them, such as the classification of surfaces of general type and their moduli spaces, and often rather surprising results! We anticipate that the workshop will feature lectures on algebraic surfaces of general type, but from today's perspective it is not so clear in which direction research on surfaces of general type will head in the next years. Therefore, next to above classical topic, we only emphasize the central role that derived categories have lately played for surfaces of general type, in particular for Godeaux surfaces, Burniat surfaces and Barlow surfaces. Notably, there have been deep results on exceptional collections and phantom categories (Alexeev, Böhning, Katzarkov, Orlov, Sosna).

For the precise conference program, we will also take into account the latest developments that might succeed this proposal, thus giving an up-to-date account of the arithmetic and geometry of algebraic surfaces.

The conference is meant to foster the interactions between experts working on algebraic surfaces. The inspiring atmosphere of the BIRS will allow them to share their ideas and latest results, and hopefully it will initiate new collaborations and programs. At the same time, we hope to attract many junior participants from Canada and the United States and abroad and give them the opportunity to gain insight into the latest developments in algebraic and arithmetic geometry and learn about new methods directly from the inventors.

In particular we hope to attract many female participants. Among some 75 potential participants as listed below, we are targeting 19 female researchers at all career stages. Thus we are optimistic to have the community of women working on the topic of the conference represented very well at BIRS.