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Brauer groups and obstruction problems: moduli spaces and arithmetic

February 25 to March 1, 2013

at the

American Institute of Mathematics, Palo Alto, California

organized by

Asher Auel, Brendan Hassett, Anthony V'arilly-Alvarado, and Bianca Viray

This workshop, sponsored by AIM and the NSF, will be devoted to studying elements of the Brauer group from both an arithmetic perspective and a Hodge-theoretic and derived categorical perspective.

Elements of the Brauer group arise in the theory of obstructions to the existence of rational points on varieties and to the existence of universal objects of moduli spaces of stable sheaves. The goal of this workshop is to bring together researchers in number theory, complex algebraic geometry, derived categories, and Hodge theory to study connections between these perspectives on Brauer group elements.

The main topics for the workshop are

  1. Brauer classes arising in twisted derived equivalence as obstructions to rational points.
  2. Transcendental Brauer classes arising in Brauer--Manin obstructions as providing twisted derived equivalences.
  3. Explicitly representing Brauer classes as symbols and via non-fine moduli spaces of stable objects.
There has been a flurry of activity in recent years on semi-orthogonal decompositions of the derived category, twisted derived equivalences, the theory of stability conditions, and the birational geometry of cubic hypersurfaces. Questions central to the workshop concern the investigation of well-known cases of twisted derived equivalences and semiorthogonal decomposition (Kummer surfaces, K3 surfaces, elliptic threefolds, cubic threefolds, conic bundles over rational surfaces, abelian varieties, quadric fibrations, and quadric intersection fibrations) for instances providing Brauer--Manin obstructions. Do Brauer--Manin obstructions exist when purely Hodge-theoretic criteria provide nontrivial transcendental Brauer classes? Can specific arithmetic techniques be brought to bear in constructing examples of new twisted derived equivalences?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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