Unramified cohomology, derived categories, and rationality

February 26-28, 2016

Objective

Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other.

In the mini-course, we will explore two potential measures of rationality. The first is the universal triviality of the Chow group of 0-cycles, which is related to the Chow-theoretic decomposition of the diagonal of Bloch and Srinivas. A powerful degeneration method introduced by Voisin, and refined by Colliot-Thèlène and Pirutka, has led to a recent breakthrough in the stable rationality problem.

The second is categorical representability, which is defined by the existence of semiorthogonal decompositions of the derived category into components whose dimensions can be bounded. We will give a precise definition of this notion and present many examples. Categorical representability in codimension 2 is conjectured for rational varieties, and we will explain how known examples can be understood via a theory of non-commutative Chow motives (which we will only be able to sketch).

Furthermore, in the mini-course, we would like to explore how these Chow-theoretic and derived categorical measures of rationality can contrast and reflect each other. One of the motivating topics in this circle of ideas is the relationship, for complex surfaces, between Bloch's conjecture, the universal triviality of the Chow group of 0-cycles, and the existence of phantoms in the derived categories. Another motivating topic is the rationality problem for cubic fourfolds and its connections between derived categories, Hodge theory, as well as Voisin's recent results on the universal triviality of the Chow group of 0-cycles for certain loci of special cubic fourfolds.

List of relevant references.

Main lecturers of the workshop are Asher Auel and Marcello Bernardara.

Program

All talks are in JWB 335 (Math Department), 155 South 1400 East

Date & Time	Speaker	Title
Fri Feb 26
3:00-4:00	Mircea Mustata	Preliminaries on Chow groups
4:00-4:30	Coffee break
4:30-5:30	Ian Shipman	Preliminaries on derived categories
Sat Feb 27
9:00-9:30	Coffee
9:30-10:30	Tommaso de Fernex	Unramified cohomology and decomposition of the diagonal
11:00-12:00	Christopher Hacon	Cubic threefolds and special cubic fourfolds
12:00-2:00	Lunch break
2:00-3:00	Marcello Bernardara	Categorical representability and rationality
3:00-3:30	Coffee break
3:30-4:30	Asher Auel	Rationality and 0-cycles
Sun Feb 28
9:00-9:30	Coffee
9:30-10:30	Marcello Bernardara	Categorical representability in higher dimension
11:00-12:00	Asher Auel	0-cycles on cubics

Venue

University of Utah, February 26-28, 2016

All talks are in JWB Builing of the Math Department, Room JWB 335. (PLEASE NOTE CHANGE OF ROOM!) During the weekend one of the doors of the building will be open. If you find the door closed, please rty the door on the other side of the building.

Please refer to the interactive campus map to locate the JWB Building as well as the University Guest House.

To reach campus or the Guest House from the airport, taxis are about $30. Alternatively, there is the Trax (a city train, please refer to Rail System Map) which run from the airport to downton (green line, from Airport to Courthouse), where you change to another Trax train that brings you to campus and the Guest House (red line, from Courthouse to Stadium 1349 E 500 S and, respectively, Fort Douglas, 200 S Wasatch Dr)

Registration

If you would like to attend and request financial support and/or logding arrangemenents, please send an email to Tommaso de Fernex (cc Gail Howick) including the following information:

• Your name and affiliation. If you are a student, include name of advisor.
• Are you applying for financial support?
• Do you need accommodation? If yes, provide dates of travel. You can email them later if not known them yet, but remember that we cannot make a reservation until we have this information.

Participants will be lodged at the University Guest House. We have blocked some rooms at the Guest House. The cutoff date is January 26. After that, reservations are subject to availability. We expect to be able to fully fund all participants.

If you have any question, please contact Tommaso de Fernex.