Titles and Abstracts

Derived categories and functors among them are important objects of study in modern algebraic geometry. It turns out that almost every geometrically meaningful functor between the derived categories of two algebraic varieties or schemes is a Fourier-Mukai functor, namely it is induced in a natural way by an object in the derived category of their product.

After introducing the theory of Fourier-Mukai functors in the classic purely triangulated setting, the main focus will be on the improved insight provided by the use of higher, and specifically dg (differential graded), categorical methods. In fact, it is well known that some intrinsic pathologies of triangulated categories can be overcome by enhancing them to (pretriangulated) dg categories. An advantage of dg categories is that the so-called quasi-functors (which correspond to morphisms in the homotopy category of dg categories) admit a nice description, and, in particular, every quasi-functor between dg enhancements of the derived categories of two (good enough) schemes can be said to be of Fourier-Mukai type. On the other hand, the precise relation between the triangulated and the dg level is not yet completely clear, despite the progress made in recent years through the contribution of several authors.

The main results in this direction (like the first examples of non Fourier-Mukai functors) will be presented, together with some challenging open problems.

Grothendieck discovered the six functors formalism as a way of extending Serre’s duality. Fully realized in the setting of étale sheaves, it then provides the ground for incredibly many advances in algebraic geometry : Lefschetz formulas, vanishing cycles, Bloch-Ogus theory, Deligne’s theory of weights, perverse sheaves, Laumon’s Fourier transform… The list goes on even today! Belinson refundation of Grothendieck’s motivic dream was naturally based right away on the six functors formalism. It was fully realized in Voevodsky’s motivic homotopy theory, through Voevodsky’s cross functors formalism in Ayoub’s Ph. D. Nowadays, the theory has even evolved in a complete and deep formalism encompassing several directions : continuity, cohomological descent, orientations, purity and duality, and even t-structures.

The course proposes to apply the motivic six functors formalism to get an extension of classical geometrical questions such as Zariski-Mumford’s study of the topology of (normal) singularities, Freudenthal space of ends, tubular neighborhoods, and intersection cohomology. We will in particular explain a construction, found in collaboration with Adrien Dubouloz and Paul Arne Ostvaer which surprinsingly allow to synthetize all these theories in algebraic geometry, and open the way to new directions in arithmetic and various quadratic refinments. This will be the occasion to apply the six functors formalism concretely, and to relate it to classical results in geometry and topology.

The plan of the course will be as follows :
- we will first explain the relation between Grothendieck’s motivic point of view and the six functors formalism, and through examples gives interpretation of phenomena like fundamental classes, purity, duality and Künneth formulas.
- Second, we will explain our main synthetic definition, say quickly the algebraic analog of a small ball around a singular point, and exemplify this notion through examples, concrete realizations, and properties such as birational/formal invariance. We will also explain its deep link with vanishing cycles, homotopy at infinity and boundary motives.
- Lastly, we will give the main computation, for «regular crossing» singularities, and illustrate this both with motivic and A1-homotopical computations, introducing in particular the «quadratic Mumford matrix».

The lectures will be concerned with the following three questions regarding six-functor formalisms:

Why care about them? In the first lecture we are going to look in a leisurely fashion at many examples (from topology and algebraic geometry) of cohomological phenomena and exhibit these as shadows of phenomena occuring at a higher level, namely at the level of six-functor formalisms. In other words, we are going to see how six-functor formalisms enhance cohomology theories.

What are they? In the second lecture we are going to describe in some detail a convenient framework for speaking about six-functor formalisms. It is both general enough to encompass most six-functor formalisms considered and specific enough to recover many of the phenomena encountered in the first lecture. A second goal is to get an impression of the framework’s rich structure.

How to construct them? In the third lecture we will highlight some of the difficulties in constructing six-functor formalisms as well as solutions that have been employed in the literature. By way of both example and outlook, we will end with some recent successes in establishing six-functor formalisms in rigid-analytic geometry.