Brauer groups and obstruction problems: moduli spaces and arithmetic
February 25 to March 1, 2013
American Institute of Mathematics,
Palo Alto, California
and Bianca Viray
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
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?
- Brauer classes arising in twisted derived equivalence as
obstructions to rational points.
- Transcendental Brauer classes arising in Brauer--Manin obstructions
as providing twisted derived equivalences.
- Explicitly representing Brauer classes as symbols and via non-fine
moduli spaces of stable objects.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.