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

Original Announcement

This workshop 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?

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:
Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces
On a local-global principle for H3 of function fields of surfaces over a finite field
ACM bundles on cubic fourfolds containing a plane
Universal spaces for unramified Galois cohomology
Derived equivalences and rational points of twisted K3 surfaces
Twisted derived equivalences for affine schemes
The Brauer group is not a derived invariant
Computing Néron-Severi groups and cycle class groups
Brauer groups on K3 surfaces and arithmetic applications
Moriwaki divisors and the augmented base loci of divisors on the moduli space of curves
Unramified Brauer classes on cyclic covers of the projective plane
Universal unramified cohomology of cubic fourfolds containing a plane
Descente galoisienne sur le second groupe de Chow : mise au point
Hypersurfaces quartiques de dimension 3 : non rationalité stable
Unirational threefolds with no universal codimension 2 cycle