SCIENTIFIC PROGRAMS AND ACTIVITIES

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

SL-oriented cohomology theories

The basic and most fundamental computation for an oriented cohomology theory is thE projective bundle theorem claiming A(P^n_k) to be a truncated polynomial ring over A(k) with an explicit basis given by the powers of a Chern class. Having this result at hand one can introduce characteristic classes and carry out a variety of <<geometric>> computations. We establish analogous results for a representable SL-oriented cohomology theory A_\eta with the stable Hopf map inverted. A typical example of a cohomology theory with the prescribed properties is given by the derived Witt groups with the special linear orientation defined via Koszul complexes. It turns out that in this setting one should look at the varieties SL_{n+2}/(SL_2x SL_n) instead of the projective spaces P^n.

Orthogonal group schemes with simple degeneration

Over a base scheme, I will discuss a class of quadratic forms that have the simplest type of nontrivial degeneration along a divisor. Such forms naturally arise in number theory and algebraic geometry; I will give examples related to Gauss composition and to cubic fourfolds containing a plane. Quadratic forms with such simple degeneration turn out to be torsors for orthogonal group schemes that are smooth, yet not reductive, over the base. I will describe the local structure of these orthogonal group schemes, which are interesting objects in their own right.

Semiorthogonal decomposition for twisted Grassmannians

A basic way to study a derived category of coherent sheaves is to decompose it into simpler subcategories and this can be implemented by using the notion of semi orthogonal decomposition. Orlov gave the semiorthogonal decompositions for projective, grassmann, and flag bundles, which generalize the full exceptional collections on the corresponding varieties by Beilinson and Kapranov. In the case of projective bundles, Bernardara extended the semiorthogonal decomposition to the twisted forms. In this talk, we present, in a similar way, semiorthogonal decompositions for twisted forms of grassmannians.

Torsors over general bases

In these lectures, I will give concrete descriptions of categories of torsors under various classical reductive groups. The emphasis will be on working over a general base S rather than over a field. I will also explain how these torsors are mapped to each other using well-known exact sequences of algebraic groups between simply connected forms, adjoint forms, etc. The framework of Giraud's "Cohomologie non abélienne" will be used, but I will try to keep everything elementary, so that someone who is not familiar with stacks, gerbes, etc. should get a first idea of these concepts, without being lost in their generality.

Toric Varieties and Severi-Brauer Varieties

A Severi-Brauer variety is a twisted form of projective space. I consider twisted forms of toric varieties as a natural generalization of Severi-Brauer varieties and discuss how many wellknown structural results have extensions to this more general setting. The main tool is a description of the automorphisms of the Cox ring of a toric variety (a notion closely related to universal torsors).

Let k be a field, let G be a linear algebraic group over k and let V be a generically free linear representation of G. Noether's problem consists in determining whether the (birational) quotient V/G is stably rational over $k$. The answer does not depend on the choice of V, thanks to the well-known 'no-name lemma'. In this talk, we consider the following question. Assume Noether's problem for G has a positive answer. Given a generically free representation V of G, is the birational quotient V/G -rational- over k? To my knowledge, there is no known counterexample to this question. We shall focus on particular cases where G is a rational variety and a special group- in which case, of course, Noether's problem has a positive answer. This is joint work with Michel Van Garrel.

Cohomological invariants of exceptional groups

We survey what is known about the cohomological invariants of exceptional groups. For each of the groups, we discuss: Do we know all the cohomological invariants? What are the fibers of the known invariants? Do the values of the cohomological invariants determine the Tits index of the corresponding twisted group?

Introduction to Chow groups and Chow motives

Chow groups (pull-back, push-forward, homotopy invariance, localization). Characteristic classes. Basics on the Intersection theory. Chow motives. Motives of flag varieties (cellular decomposition, Bruhat-Tits decomposition). Rost nilpotence and the Krull-Schmidt Theorem.

Rational points, zero cycles of degree one, and A^1 homotopy theory.

We report on joint work with Aravind Asok investigating how having a rational point or zero cycle of degree one is reflected in the A^1-homotopy type of a complete variety. It turns out that from this point of view, the difference between having a rational point and admitting a zero cycle of degree one is a process of stabilization; this insight relies on a computation of the zeroeth stable homotopy sheaf of a variety in terms of Chow-Witt groups.

Duality theorems over a p-adic function field.

Let K be the function field of a curve over a p-adic field. We prove Poitou-Tate-like duality theorems for K-tori and finite Galois modules over K, and give applications to the arithmetic of torsors under K-tori (joint work with Tamas Szamuely).

Local-global principles in the theory of linear algebraic groups

In this course, we consider local-global principles for torsors when the base field is an algebraic function field over a complete discretely valued field. We compute the obstructions to these principles with respect to certain other families of overfields. The results then give insight about the original local-global map with respect to discrete valuations. The proofs use patching methods.

Singularities of codimension two and algebraic cycles

Using Lipman's work on resolution of two-dimensional singularities, I will provide a form of resolution of singularities of codimension two for excellent schemes. I will then discuss applications to the study of algebraic cycles : integrality of the Chern character, Steenrod squares, operational Chow groups.

Witt kernels in characteristic 2 for algebraic extensions.

A natural question in the algebraic theory of quadratic forms is the determination of Witt kernels, i.e. the kernel of the restriction map when passing from the Witt ring or Witt group of a field to that of a field extension. In general, this is a difficult problem. For odd degree field extensions, the Witt kernels are zero due to a theorem of Springer. For degree 2 extensions, Witt kernels have been known for quite some time (in any characteristic). For degree 4 extensions, these kernels have been determined completely by Sivatski in characteristic not 2. We determine Witt kernels for degree 4 extensions in characteristic 2, extending the partial results that have been known so far. In characteristic 2, there is an added difficulty because of possible inseparability of the extensions
leading.

Simplicial sheaves, cocycles and torsors

This talk gives a rapid introduction to simplicial sheaves, their
homotopy types, and homotopy theoretic classifications of torsors. The
method for comparing torsors to homotopy types uses a generalized
theory of cocycles.

The twisted gamma-filtration and algebras with orthogonal involution

For the Grothendieck group of a split simple linear algebraic group, the twisted gamma filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this project, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.

Mini-lecture on Patching and a local-global principle

In these lectures, I'll describe how one applies the techniques of field patching to obtain local-global principles for Galois cohomology.

Oriented cohomology of algebraic groups and motives of flag varieties

In this talk we will discuss the application of the technique developed by Calmes-Petrov Zainoulline to oriented cohomology of algebraic groups and motives of twisted flag varieties. In particular we will show how one can compare oriented cohomology of algebraic group to its chow ring. As an example, we will be able to compute algebraic cobordism of some groups of small ranks. Also we will discuss the relation between the Chow motive of a twisted flag variety and its h-motive for an oriented cohomology h.

On the Tate conjecture for integral classes on cubic fourfolds.

Let X be a smooth projective variety defined over a finite field. The Tate conjecture predicts that the cycle class map from the Chow groups of X with rational coefficients to the l-adic étale cohomology groups is surjective. The integral version, which is known not to be true in general, investigates the similar question for integral coefficients. In this talk we will explain how to prove this integral version for codimension two cycles on a cubic fourfold. The strategy is very much inspired by the approach of Claire Voisin used in the context of the integral Hodge conjecture. This is a joint work with F. Charles.

Melanie Raczek
l'Université de Louvain

Lecture Notes

Okubo algebras in characteristic 3 and valuations

Okubo algebras are forms of pseudo-octonion algebras, i.e. octonion algebras with a twisted product. An Okubo algebra in characteristic different from 3 and without nonzero idempotents is described as a subspace of a degree 3 central division algebra endowed with the Okubo product. Given an Okubo algebra S in characteristic 0 contained in a division algebra D which is endowed with a valuation with residue characteristic 3, I prove that the residue of S is an Okubo algebra (in characteristic 3) if and only if the residue division algebra has dimension 9 over the ground field and the height of D is maximal. Moreover Okubo algebras in characteristic 3 are always the residue of some Okubo algebra in characteristic 0.

On division algebras having the same maximal subfields.

The talk will address the following question: Let D and T be central division algebras over a field K. When does the fact that D and T have the same maximal subfields imply that D and T are actually isomorphic over K? I will discuss various motivations for this question and some recent results. Time permitting, I will also indicate some variations of this question and its generalizations to algebraic groups. This is a joint work with V. Chernousov and I. Rapinchuk.

An introduction to the theory of essential dimension

The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. The course will survey some of this research. It will be started with the definition of essential
dimension and conclude with a discussion of open problems.

The course will survey some of this research. It will be started with the definition of essential dimension and conclude with a discussion of open problems.

Degree 3 Cohomological Invariants of Split Semisimple Groups

As is well-known, the Rost invariant describes the degree 3 cohomological invariants of a semisimple simply connected group. In a recent preprint, A. Merkurjev has extended this result by producing an exact sequence which includes as one of the terms the group of degree 3 normalized cohomological invariants of a semisimple algebraic group G. He then uses this sequence to describe the invariants for all adjoint groups of inner type. We use his sequence to study the degree 3 invariants of split groups of types A and D that are neither simply connected nor adjoint. As a consequence, we recover relations to many of the known results and find new invariants that have not been previously described in the literature. (Joint work with H. Bermudez)

J.-L. Colliot-Thélène
CNRS/Paris-Sud

Lecture Notes

Espaces homogènes sur les corps de fonctions de courbes sur un corps local

Over such a function field F, D. Harbater, J. Hartmann and D. Krashen have proved a localglobal principle for the existence of rational points on principal homogeneous spaces under a connected linear algebraic group G over F when the underlying variety of G is F-rational, i.e. birational to affine space over the field F. In recent work with Parimala and Suresh, we show that this local-global principle may fail when the group G is not F-rational. The obstruction we use comes from the Bloch-Ogus complex for étale cohomology over an arithmetic surface extending the curve. One may then ask when this new obstruction is the only obstruction to the existence of rational points.

The discriminant of symplectic involutions

A cohomological invariant of degree 3 for symplectic involutions on central simple algebras has been defined in a joint work with Garibaldi and Parimala. A new approach to its definition for algebras of degree 8 has been developed jointly with Becher and Grenier-Boley. This talk will describe this new approach and the relation with an invariant of Barry, which gives a criterion for the quadratic descent of biquaternion algebras.

Kirill Zainoulline
University of Ottawa

Lecture Notes

Motives and algebraic cycles on twisted flag varieties

Motives of twisted flag varieties is an important tool in the study of splitting properties of torsors, in the geometric theory of quadratic forms and central simple algebras with involutions. For instance, motives of Pfister quadrics played a key role in the proofs of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture. Another applications include cohomological invariants, the theory of canonical and essential dimensions of linear algebraic groups.

The course will survey some of this research. It will be started with an introduction to the theory motives of twisted flag varieties and conclude with a discussion of open problems. We will introduce and study the discrete motivic invariant of a torsor (the J-invariant), explain relations to canonical dimensions, K-theory, Chow groups, algebraic cobordism of twisted flag variaties and linear algebraic groups.

On the gamma filtration of oriented cohomology of flag varieties

In this talk I will first briefly recall the formal group algebra and the characteristic map defined by Calmes-Petrov-Zainoulline, then I will talk about the deformation map between formal group algebras of two distinct formal group laws. Such map can be applied to study the gamma filtration of oriented cohomology of flag varieties.