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Motivic equivalence for algebraic groups and critical varieties

(by Charles De Clercq & Anne Quéguiner-Mathieu)

The Chow motif of a projective homogeneous variety under some algebraic group contains some information on the splitting properties of the underlying algebraic objects. Considering all projective homogeneous varieties under the action of a given group leads to the notion of motivic equivalence for algebraic groups. The aim of this course is to state and prove a criterion of motivic equivalence in terms of Tits’ indices of algebraic groups, and to prove the existence of critical varieties, which are test-varieties for motivic equivalence. 

The course will focus on algebraic groups of classical type. The first two lectures will cover required material such as classification of classical algebraic groups, twisted flag varieties, Tits’s indices, Chow motives, Rost’s nilpotence principle, and upper motives.

p-group actions and Chern numbers of varieties (by Olivier Haution)

The course will concern the study of actions of finite p-groups on algebraic varieties, and more precisely the use of certain numerical invariants of varieties to detect fixed points. We will thus discuss various fixed point theorems, present methods to prove them, and illustrate them by applications and examples. Among those numerical invariants are the Chern numbers, whose consideration will lead us to introduce the cobordism ring. We will review an elementary approach to cobordism due to Merkurjev, and illustrate how the cobordism ring can be used to interpret the fixed point theorems, and more generally to understand better how the geometry of the fixed locus is related to that of the ambient variety (time permitting).

As prerequisites we will assume familiarity with basic algebraic geometry, the Chow group and K-theory (only K_0)

Motives without A^1-invariance (by Vova Sosnilo)

A motivic oo-category contains all the information about certain cohomology theories on schemes and forgets all the non-additive information the category of schemes has. More precisely, such an additive oo-category should admit a functor from the category of schemes and every cohomology theory on schemes in an appropriate sense should factor through this oo-category.

One common requirement for being a cohomology theory in this context is the A^1-invariance property. The oo-category of A^1-invariant motivic spectra has been studied extensively over the past 20 years, which gave us fertile soil for studying K-theory, hermitian K-theory, algebraic cobordism and many other cohomology theories of regular schemes. However, over non-regular schemes K-theory is not A^1-invariant and some new methods are being called for. 

The goal of this short series of lectures is to construct new motivic oo-categories of non-A1-invariant motivic spectra, based on the work of Annala and Iwasa, and to show how these can be used to prove new results about K-theory of non-regular schemes.

Isotropic motives (by Alexander Vishik)

The homological properties of algebraic varieties are encoded in their motives. These can be considered as linearizations of varieties. The category of motives, although much handier than the category of varieties themselves, is still pretty large and complicated.  One may try to read motivic information by applying some realization functor with values in a small and well-understood category. One possibility is to consider the topological realization, thus replacing algebraic varieties by topological spaces of their complex points (if the ground field is embedded into complex numbers). The motivic version of this functor takes values in the category of "topological motives'', which is the derived category of abelian groups. This category is small and simple, but the functor looses a lot of information. One would want to supplement it with other similar realization functors, so that the resulting family would be reasonably conservative. Isotropic realizations provide a large supply of such functors. These are parametrized by prime numbers and (equivalence classes of) finitely generated field extensions of the ground field.

I will start by introducing Chow motives and motivic cohomology and recalling basic facts about them. Then I will move to  anisotropic varieties and flexible fields and will introduce isotropic realizations. A particular attention will be paid to isotropic Chow motives, where Hom's are described by isotropic Chow groups. The latter groups should coincide with Chow groups (with finite coefficents) modulo numerical equivalence, and so, are much simpler than the usual Chow groups. I will discuss various corollaries of this conjecture and will prove it for divisors. One of the consequences is that isotropic realizations should provide points for the Balmer's tensor triangulated spectrum of the Voevodsky category. Finally, I will introduce Čech simplicial schemes and will discuss the calculation  of the isotropic motivic cohomology of a point, for p=2.


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