The Erwin Schroedinger Institute for Mathematical Physics

Thematic Program: Advances in Birational Geometry (Vienna, April 3-May 26 , 2017)

Organizers: F. Bogomolov, J-L. Colliot-Thélène, L. Katzarkov, A. Kuznetsov, A. Pirutka, Y. Tschinkel

Birational geometry is a major branch of algebraic geometry, focused on the study of function fields of algebraic varieties. One of its main classical problems, going back to Italian algebraic geometry, is to determine whether or not an algebraic variety \(X\) of dimension \(d\) over a field \(k\) is rational, i.e., birational to projective space \(\mathbb P^d\), or equivalently, whether or not its function field \(F=k(X)\) is a purely transcendental extension over \(k\). Several related notions play an important role:

  • stable rationality : \(X\times \mathbb P^n\) is rational, for some \(n\),
  • unirationality : there exists a rational dominant map \(\mathbb P^d\rightarrow X\).

The three notions coincide for surfaces over algebraically closed fields, but differ over nonclosed fields and in higher dimensions. In addition to being of intrinsic interest, rationality and unirationality properties have various applications. For example:

  • (Inverse Galois problem) Let \(V\) be a faithful representation of a finite group \(G\) over an infinite field \(k\). Assume that the field of invariants \(F=k(V)^G\) is rational. Then \(G\) is a Galois group of \(k\). E. Noether asked whether or not all such fields are rational; first counterexamples have been constructed by Saltman and Bogomolov.
  • Unirationality of moduli spaces allows one to write down explicit equations for varieties in question, over the ground field, which is particularly useful in arithmetic geometry.

However, what makes the rationality problem especially appealing is that it has stimulated, and continues to stimulate, major advances in algebraic geometry and number theory. The most notable examples of such are the breakthrough results from 1974, concerning nonrationality of certain 3-folds:

  • Quartics in \(\mathbb P^4\), via the group of birational automorphisms (Iskovskikh--Manin). This led to the development of the birational rigidity method (Corti, Pukhlikov, Cheltsov, De Fernex) and to extensive studies of birational models of 3-folds, which culminated in the classification results of Iskovskikh, Mori--Mukai, and finally in the Minimal Model Program.
  • Cubics in \(\mathbb P^4\), via intermediate Jacobians and Hodge theory (Clemens-Griffiths). This approach was further expanded in the work of Beauville and Tyurin, and culminated in the work of Voisin.
  • Conic bundles over rational surfaces, via torsion in \(H^3(X,\mathbb Z)\), i.e., the unramified Brauer group (Artin-Mumford). This triggered systematic studies of the unramified Brauer group and higher unramified cohomology of more general varieties by Colliot-Thélène--Ojanguren, Bogomolov, Kahn, Merkurjev, Parimala, Peyre, Pirutka, Totaro, Voisin, and others. which produced, among other results, counterexamples to Noether's problem, and led to Bogomolov's anabelian geometry. The Merkurjev-Suslin theorem, connecting the Brauer group with K-theory, and via the Bloch--Ogus sequence to Chow groups, proved particularly fruitful in the work of Colliot-Thélène--Sansuc, Colliot-Thélène--Raskind, Parimala-Suresh, Kahn, Pirutka, Saito, and Voisin.

A new geometric method, via degenerations, was introduced by Kollár in 1995: he proved that very general hypersurfaces \(X_d\subset \mathbb P^{n_1}\) of degree \(d\ge 2(n+3)/3\) are not rational.

Seminal ideas on stable rationality, based on the method of specialization and decomposition of the diagonal, were recently introduced by Voisin and Colliot-Thélène--Pirutka. This is the most important achievement in this field during the last 20 to 30 years. It was followed by applications in the works of Beauville, Totaro, and Hassett-Kresch-Tschinkel.

At the same time, we do not yet have answers to some very basic questions. For example: Are cubic hypersurfaces of dimension \(\ge 3\) stably rational? Are nonrational cubic fourfolds dense in moduli?

Recent developments in Homological Mirror Symmetry and derived categories lead to the possibility of using derived categories in order to study the question of rationality. We would like to mention here the results obtained by Bondal, Orlov, Kuznetsov, Thomas, Huybrechts, Addington, Macri, Boehning, Auel, Bernardara, and many others.

This two month long program, which is focused on these recent developments, will help consolidate results that have been obtained and will help find new directions of collaboration.

The Program will contain three workshops:

  • April 24 - 28, 2017
    Recent developments in rationality questions
  • May 2 - 5, 2017
    Categorical approach to rationality
  • May 15 - 19, 2017
    Closing workshop - future directions
In addition there will be lectures by C. Voisin, J.L. Colliot-Thélène, P. Griffiths, A. Pirutka, Y. Tshcinkel, F. Bogomolov, C. Boehning, A. Kuznetsov. D. Orlov.