Courses

Courses:

• • Jean-Louis Colliot-Thélène

  • Stable rationality over arbitrary fields

  • In this course the following topics will be discussed:

    • • R-equivalence on rational points and Chow groups of zero-cycles;

      • Unramified cohomology, basic properties;

      • Specialisation theorem for R-equivalence, Fulton's specialisation theorem for the Chow group of zero-cycles;

      • How to disprove stable rationality over nonclosed fields: specialisation method with arbitrary residue field, cubic hypersurfaces over nonclosed fields;

      • Rationality versus stable rationality for some classes of geometrically rational surfaces;

      • Computing the Brauer group, over nonclosed fields, and over an algebraically closed field;

      • Computing higher unramified invariants;

      • Application to families of Pfister quadrics over complex projective space.

• • • Notes on personal web page: Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder and Rationalité stable sur les corps quelconques.

• • Daniel Huybrechts

  • Hodge theory of cubic fourfolds, their Fano varieties and associated K3 categories

  • We will discuss the period domain for smooth cubic hypersurfaces and study various types of special divisors. The origins of the theory go back to Hassett’s thesis in which he exhibits special divisors related to moduli spaces of polarized K3 surfaces (of certain degrees), others are related to moduli spaces of twisted K3 surfaces. There are various ways to think about the involved Hodge theory (classical, using Mukai lattices which is more appropriate for derived categories, etc.) which shall all be explained.

  • Lecture notes: link.

  • Emanuele Macrì

  • Categorical aspects of the birational geometry of hypersurfaces

  • In these lectures, we will introduce Kuznetsov components and present their basic properties, with particular emphasis on cubic fourfolds. Topics will include: Serre functors and the Calabi-Yau property, Examples and Conjectures, Bridgeland stability conditions, and Moduli Spaces.

• • • Notes: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces (together with Paolo Stellari).

• • Claire Voisin

  • Decomposition of the diagonal and stable rationality

  • We will discuss various notions of decomposition of the diagonal (cohomological, Chow) and the stable birational invariants they control. This notion originates in the work of Bloch and Srinivas where it is used (with rational coefficients) to prove a generalized version of Mumford's theorem and various consequences. The similar formalism with integral coefficients leads to interesting stable birational invariants, which are obstructions to the existence of a decomposition of the diagonal. Not all are known to be possibly nonzero for some rationally connected varieties and this leaves open very interesting geometric questions. We will also describe the specialization (or degeneration) method and some of its main consequences on rationality questions.