Schedule for: 18w5021 - Affine Algebraic Groups, Motives and Cohomological Invariants

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The importance of the notion of anisotropy was noticed for quite a while, most notably, in the theory of quadratic forms. I will discuss the “anisotropic motivic category” which embodies this notion. This category introduced originally with the aim of studying the Picard group of Voevodsky's category appeared to have many interesting and unexpected connections.

We discuss relations between the geometry of a smooth projective variety equipped with an involution and the geometry of its fixed locus, using Chern numbers.

Let $p$ be a prime number, and let $S$ be a scheme of characteristic $p>0$. For any integer $n>1$, one can build the scheme of Witt vectors of length $n$ of $S$, denoted by $W_n(S)$. It is a natural thickening of characteristic $p^n$ of $S$. Let us say "$W_n$-bundle" (or Witt vector bundle if $n$ is understood) for "vector bundle over $W_n(S)$". In this talk, we consider the following Question. Let $V$ be a vector bundle over $S$. $Q(n,V)$: Does $V$ extend to a $W_n$-bundle? I will first give precise (hopefully elementary) definitions, and basic properties of Witt vector bundles. I will then show that the answer to question $Q$ is positive, when $V$ is the tautological vector bundle of the projective space of a vector bundle, defined over an affine base. I will discuss counterexamples in the general setting -- in a Galois-theoretic way. If time permits, I plan to discuss the main motivation for raising question $Q$: lifting Galois representations. This is joint work, with Charles De Clercq and Giancarlo Lucchini Arteche.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The 2007 determination of the essential dimension of spin groups over the complex numbers exploited knowledge of which representations of algebraic groups are generically free, showing the value of proving an analogue of that classification for algebraically closed fields of prime characteristic. Changing the characteristic adds extra complications because the representation theory is more complicated and also because generic stabilizers need not be smooth. In a series of joint papers with Bob Guralnick and Ross Lawther, we have determined the generic stabilizers for all irreducible representations of simple algebraic groups. This work has applications to essential dimension, to proving the existence of a stabilizer in general position, and to determining rings of invariants.

Let $K$ be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple $K$-group $G$ is quasi-split if and only if it quasi–splits after a finite tamely ramified extension of $K$.

Let $G$ be an absolutely almost simple algebraic group over a field $K$, which we assume to be equipped with a natural set $V$ of discrete valuations. In this talk, our focus will be on the $K$-forms of $G$ that have good reduction at all $v$ in $V$ . When $K$ is the fraction field of a Dedekind domain, a similar question was considered by G. Harder; the case where $K=\mathbb{Q}$ and $V$ is the set of all $p$-adic places was analyzed in detail by B.H. Gross and B. Conrad. I will discuss several emerging results in the higher-dimensional situation, where $K$ is the function field $k(C)$ of a smooth geometrically irreducible curve $C$ over a number field $k$, or even an arbitrary finitely generated field. These problems turn out to be closely related to finiteness properties of unramified cohomology, and I will present available results over various classes of fields. I will also highlight some connections with other questions involving the genus of $G$ (i.e., the set of isomorphism classes of $K$-forms of $G$ having the same isomorphism classes of maximal $K$-tori as $G$), Hasse principles, etc. The talk will be based in part on joint work with V. Chernousov and A. Rapinchuk.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Hasse-Minkowski theorem says that a quadratic form over a global field admits a nontrivial zero if it admits a nontrivial zero everywhere locally. Over more general fields of arithmetic and geometric interest, the failure of the local-global principle is often controlled by auxiliary structures of interest: 2-torsion points of the Jacobian and elements of Tate-Shafarevich groups for quadratic forms over function fields of curves, and the Brauer group over function fields of surfaces. I will explain recent work with V. Suresh on constructing failures of the local-global principle for quadratic forms over function fields of higher dimension varieties. These counterexamples are controlled by higher unramified cohomology groups and involves the study of certain Calabi-Yau varieties of generalized Kummer type that originally arose from number theory.

QZP asks if two quadrics of the same dimension over a field that are stably birationally equivalent are in fact birationally equivalent. This is a notoriously difficult problem in the algebraic theory of quadratic forms. Only partial results are known (quadrics of small dimension or quadrics associated to certain subforms of multiples of Pfister forms). In this talk we present progress towards the solution of QZP for certain types of ground fields including global fields.

Let $k$ be a number field or a $p$-adic field and $\ell$ a prime. The class field theory asserts that every element in $H^2(k, \mu_\ell)$ is a symbol. Let $F$ be the function field of a curve over $k$. Suppose that $K$ contains a primitive $\ell^{\mathrm{th}}$ root of unity. If $\ell = 2$, then assume that $K$ is a totally imaginary number field. We show that every element in $H^3(F, \mu_\ell^{\otimes 3})$ is a symbol. This leads to the finite generation of the Chow group of zero-cycles on a quadric fibration of a curve over a totally imaginary number field.

An "arithmetic toric variety" is a normal variety with a faithful action of an algebraic torus having a dense open orbit. When the base field is algebraically closed, there is only one torus in every dimension and one can identify the torus with its orbit. Over a general field, there may be many non-isomorphic tori of the same dimension. Moreover, it is no longer possible to identify the torus with its orbit since there may not exist any rational points. Exceptional collections are one way of describing the bounded derived categories of coherent sheaves on a variety. The existence of exceptional collection is a very strong condition but, nevertheless, Kawamata showed that all smooth projective toric varieties possess exceptional collections when the ground field is algebraically closed. For a general field, this immediately fails even in dimension 1. However, if one allows an arithmetic generalization of the "usual" notion of exceptional object, then the theory is again interesting.

For an element of a group $G$ representable as a product of commutators, one can define its commutator length as the smallest number of commutators needed for such a representation, by definition the other elements are of infinite length. The commutator width of $G$ is defined as supremum of the lengths of its elements. Recently it was proven that all finite simple groups have commutator width one. On the other hand, there are examples of infinite simple groups of arbitrary finite width and of infinite width. In a similar manner, one can define the bracket width of a Lie algebra. It is known that all finite-dimensional simple Lie algebras over an algebraically closed field have bracket width one. Our goal is to present first examples of simple Lie algebras of bracket width greater than one. This talk is based on a work in progress, joint with A. Regeta.

We introduce and study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the celebrated projection of the root system of type E8 onto the subring of icosians of the quaternion algebra which gives the root system of type H4. Using moment graph techniques we show that such a twisted folding induces a surjective map at the equivariant cohomology level. This is a joint project with Martina Lanini.

Although the Cremona groups are infinite-dimensional, there are many striking analogies between them and affine algebraic groups. The talk is aimed to discuss their similarities, differences, and problems related to this phenomenon.

In our previous joint work with L. Boelaert and T. De Medts, we have constructed a categorical equivalence between simple structurable division algebras over a field $k$ up to isotopy, and simple algebraic groups of $k$-rank 1 up to isogeny. In the present talk we explain how this equivalence extends to all simple structurable algebras and all isotropic simple algebraic groups over a field.

Twisted projective homogeneous varieties for algebraic groups are projective varieties which are isomorphic to a given projective homogeneous variety after extension to the separable closure of the field. Important examples include Severi Brauer varieties, which are twisted forms of projective space; generalised Severi Brauer varieties, which are twisted forms of Grassmannians. We investigate conditions under which generalised Severi Brauer varieties have rational subvarieties which are forms of given Schubert subvarieties of the associated Grassmannian. For regular Grassmannians, and more generally for any projective homogeneous variety for an algebraic group, the classes of the Schubert subvarieties of a given codimension give a basis for the Chow groups of that codimension. The Chow groups, even in codimension 2, for twisted projective homogeneous varieties, are less well understood. We use our results as well as an analysis of the Chow motive and the topological filtration of the Grothendieck group of generalised Severi Brauer varieties to show that the codimension 2 Chow groups of certain generalised Severi Brauer varieties are torsion free. This is joint work with Caroline Junkins and Danny Krashen. In joint work with Caroline Junkins and Jasmin Omanovic, we investigate the analogous problem for Lagrangian involution varieties, twisted forms of Lagrangian Grassmannians. We determine conditions under which Lagrangian involution varieties have rational subvarieties which are forms of given Schubert subvarieties of the associated Lagrangian Grassmannian. We also investigate the torsion in the codimension 2 Chow groups of certain Lagrangian involution varieties.

Let $k$ be a field of characteristic not $2$, and let $k_s$ be a separable closure of $k$. Let $S$ be a cubic surface over $k$. It is well-known that $S$ has $27$ lines over $k_s$. Let $E$ be the corresponding rank $27$ etale $k$-algebra, and let $q_E$ be its trace form. The aim of this talk is to describe the quadratic form $q_E$; the proofs use Serre's Witt invariants of the group $W(E_6)$. The invariants of $q_E$ (determinant, signature...) provide some information concerning the lines of $S$. For instance, when $k$ is the field of real numbers, we recover the classical result that a cubic surface over $R$ has $27$, $15$, $7$ or $3$ lines over $R$.

Appropriately chosen combinations of the lambda-operations defined on the Gothendieck-Witt ring of a field allow to construct cohomological invariants of Witt classes. To transpose such methods to algebras with involutions, we are led to define a "mixed" Grothendieck-Witt ring, which contains both bilinear spaces over the base field and hermitian modules over the algebra with involution. One can then show that this ring has natural lambda-operations, and a natural filtration by powers of an ideal which is the equivalent of the fundamental ideal for quadratic forms. In this setting we construct invariants with values in the graded ring associated to the filtration, which in particular contains the mod 2 cohomology ring. This way we obtain various relative and absolute cohomological invariants of arbitrarily high degree for algebras with involutions, with especially nice behaviour in index 2.

Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. The Bruhat decomposition of G defines subvarieties of G/B called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. The Chow ring of G/B is generated as an abelian group by the classes of all Schubert varieties, and is "almost" generated as a ring by the classes of Schubert divisors. More precisely, an integer multiple of each element of G/B can be written as a polynomial in Schubert divisors with integer coefficients. In particular, each product of Schubert divisors is a linear combination of Schubert varieties. I am going to talk about the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination equals 1. Also, I am going to talk about possible applications of my result to the computation of so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.

For a prime number $p$ and a non-negative integer $n$ we consider a Morava $K$-theory $K(n)$ with the coefficient ring $\mathbb{Z}_{(p)}$. This is a universal oriented cohomology theory in the sense of Levine-Morel with a $p^n$-typical formal group law which has height $n$ modulo $p$. It turns out that $K(n)$ is strongly related to cohomological invariants of algebraic groups in the sense of Serre. This is our starting point to compute the Chow groups of quadrics from the powers $I^{m+2}$ of the fundamental ideal of the Witt ring up to codimension $2^m$. Moreover, the Morava $K$-theory gives a conceptual explanation of the nature of the obtained answer. This is a joint work with Pavel Sechin.

We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric $Q$ over a number field $F$. Besides, we show that such summands are twists of Rost motives. In the case when $F$ has at most one real embedding we describe a complete motivic decomposition of $Q$. This is a joint work with M. Borovoi and N. Semenov.

It is known that the group of type G2 can be constructed as the automorphism group of the octave algebra, while its symmetric space of type G turns out to be the variety of the quaternion subalgebras in the octave algebra. Brown has constructed a 56-dimensional non-associative algebra with involution whose automorphism group is of type E6. We consider the variety of quaternion subalgebras in this algebra and show that, on the one hand, it is the 32-dimensional symmetric space of type EIII, and, on the other hand, is an open subvariety in the complexified Cayley plane (the latter reflects the fact that EIII is a "Rosenfeld plane" over complexified octaves). The structure of the projective plane and Witt extension theorem for Hermitian forms allow to give a description of the orbits of the action of a twisted form of EIII. In terms of Galois cohomology, the orbits are the fibers of the map $H^1(F,{}^2D_5)\to H^1(F,{}^2E_6)$ (the latter map is surjective up to cubic extensions of the base field).

A vector space $V$ with a symmetric trilinear map $V\times V \times V \to V$ will be called a triple system. Freudenthal introduced and studied a certain type of triple system on a 56-dimensional vector space whose automorphism group is a simply connected group of type E$_7$. If $S$ is a subsystem generated by $n$ generic elements in a triple system $V$, then (in some situations) we can use the slice method to deduce a surjection from $\mathrm{Aut}(V,S)$-torsors to $\mathrm{Aut}(V)$-torsors. In this talk I will describe some examples of this procedure for triple systems of varying dimensions; in one example we will deduce the upper bound on essential dimension $\mathrm{ed}(\mathrm{HSpin}_{12}) \leq 6$ for characteristic not 2 (beating the previously best known bound of 26 from Garibaldi and Guralnick).

(Joint work with A. Merkurjev and V. Chernousov.) Norm principles examine the behaviour of the images of group morphisms over field extensions from a linear algebraic group into a commutative one with respect to the norm map. The classical norm principles attributed to Scharlau and Knebusch can be reformulated as a special case of norm principles for certain group homomorphisms. Norm principles have been previously studied by Merkurjev-Gille, especially in conjunction with the rationality or the R-triviality of the algebraic group in question. However a result of Merkurjev and Barquero shows that norm principle holds in general for all reductive groups of classical type without D$_n$ components, the question still remaining open for groups with type D$_n$ components. In this talk, we investigate the D$_n$ case over an arbitrary complete discretely valued field $K$ with residue field $k$ (of characteristic not 2) restricting ourselves to type D groups arising from quadratic forms. We show that if the norm principle holds for such groups defined over all finite extensions of the residue field $k$, then it holds for such groups defined over $K$. This yields examples of complete discretely valued fields with residue fields of virtual cohomological dimension 2 over which the norm principle holds for the groups under consideration. As a further application, we also relate the possible failure of the norm principle to the non-triviality of certain Tate-Shaferevich sets.

A generalized version of Noether's problem asks whether the classifying space BG of an algebraic group is stably rational or retract rational and it is still open for a connected group G over an algebraically closed field. It is well known that the nonrationality of BG can be detected by means of unramified cohomology groups and the cohomology groups of degree at most 2 are trivial. Recently, Merkurjev showed that the degree 3 unramified cohomology group of BG is trivial if G is a semisimple group of type A. Using a complete description of cohomological invariants we show that the same result holds if G is a semisimple group of type C.

Let $R$ be a discrete valuation ring with $2$ a unit in $R$, and $A$ be an $R$-Azumaya algebra with involution $\tau$ (of any kind). Denote by $k$ the residue field, by $K$ the fraction field of $R$, set $A_{S}:=S\otimes_{R}A$ for an $R$-algebra $S$. We show that there is an exact sequence of $\epsilon$-hermitian Witt groups ($\epsilon\in\pm 1$) $$ 0\longrightarrow W_{\epsilon}(A,\tau)\,\longrightarrow\, W_{\epsilon}(A_{K},\tau_{K})\,\xrightarrow{\;\partial\;}\, W_{\epsilon}(A_{k},\tau_{k})\longrightarrow 0\, , $$ where $\tau_{K}$ and $\tau_{k}$ denote the by $\tau$ induced involutions. This implies purity of hermitian Witt groups of Azumaya algebras over regular local rings of dimension $\leq 2$.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.