International Workshop on motives in Tokyo, 2012

Program

10(Mon)/Dec:

9:30-10:30 Merkurjev, "Generalized Rost motives I."

10:45-11:45 Zhong, "On the torsion of Chow group of complete spin flags."

14:00-15:00 Ramachandran, "The Lichtenbaum-Milne conjecture on special values I."

15:30-16:30 Wildeshaus, "Compactifications and purity."

16:45-17:45 Ruelling, "K-groups of reciprocity functors I."

11(Tue)/Dec:

9:30-10:30 Ramachandran, "The Lichtenbaum-Milne conjecture on special values II."

10:45-11:45 Merkurjev, "Generalized Rost motives II."

14:00-15:00 Tabuada, "Noncommutative motives."

15:30-16:30 Ruelling, "K-groups of reciprocity functors II."

16:45-17:45 Yasuda, "p-cyclic quotient singularities and motivic integration."

18:30-20:30: Reception

12(Wed)/Dec:

9:30-10:30 Sato, "Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance."

10:45-11:45 Gillet, "Rational Points on Varieties I."

Free afternoon

13(Thu)/Dec:

9:30-10:30 Gillet, "Rational Points on Varieties II."

10:45-11:45 Kerz, "Skeleton sheaves over finite fields."

14:00-15:00 Kelly, "ldh cohomological descent for motives."

15:30-16:30 Sugiyama, "Remarks on Lefschetz classes on simple CM abelian varieties."

16:45-17:45 Wittenberg, "Degrees of zero-cycles and Euler characteristics over Henselian fields I."

14(Fri)/Dec:

9:30-10:30 Asakura, "Real regulator on K_1 of elliptic surfaces."

10:45-11:45 Hesselholt, "Algebraic K-theory, the failure of excision, and what to do about it."

14:00-15:00 Wittenberg, "Degrees of zero-cycles and Euler characteristics over Henselian fields II."

15:30-16:30 Brown, "The motivic Lie algebra of the category of mixed Tate motives over Z."

Abstract

Asakura:

In this talk I will explain a certain method for computation of real regulator of K_1 of elliptic surface by using the Picard-Fuchs operator.

Brown:

The category of mixed Tate motives over Z is a Tannakian category whose Lie algebra is freely generated by one element in every odd degree >1. An important open problem in Grothendieck-Teichmuller theory is to construct these generators explicitly. A very closely related problem, due to Drinfeld, is to give an explicit construction of all rational associators. This would have applications in knot theory, deformation quantization, the Kashiwara-Vergne problem, and the theory of multiple zeta values. In this talk I will describe some elementary recipes for solving both these problems, modulo some standard conjectures. If time permits, I will explain a connection with modular forms which answers a question posed by Y. Ihara and M. Matsumoto in their study of the stable derivation algebra.

Gillet:

In the first lecture I will give an over view of some of the questions and results in the area, including recent work using motivic techniques. A central example is: Conjecture (Bombieri-Lang): Let V be a variety of general type defined over a number field K then the set V(K) of rational points in V is not Zariski dense in V. In the second lecture I will discuss criteria for descent, and some recent work on a special case of the analog of the Bombieri-Lang conjecture for varieties over function fields of characteristic p.

Kelly:

We discuss how a theorem of Gabber on alterations can be used to replace the hypothesis of resolution of singularities in Voevodsky's work on motives, if we are willing to work with Z[1/p] coefficients (where p is the characteristic of the perfect base field).

Kerz:

Recently Deligne and Drinfeld studied compatible systems of lisse l-adic sheaves on the subcurves of a variety over a finite field. It is expected that any such system should be induced from a lisse l-adic sheaf on the variety if a certain ramification condition is verified. In this talk I present joint work with Shuji Saito on the rank one case of this problem.

Merkurjev:

We will discuss properties of varieties possessing a generalized Rost motive and applications.

Ramachandran:

This talk will be about the conjecture of Lichtenbaum-Milne on special values of zeta functions of varieties over finite fields. We will begin with a gentle introduction and then move to the recent results about the p-part.

Ruelling:

Given semi abelian varieties over a field G_1,..., G_r, Somekawa introduced the K-group K(G_1,..., G_r), which is a functor on field extensions of k, having traces for finite field extensions. Taking G_i to be G_m one recovers Milnor K-theory, taking G_i to be the Jacobians of smooth projective curves C_i one can describe the kernel of the Albanese map for the product of the C_i's via K(G_1,..., G_r), by work of Spiess and Raskind. Recently Kahn and Yamazaki generalized this definition to the case where the G_i's are allowed to be homotopy invariant Nisnevich sheaves with transfer (HINST) and assuming the ground field to be perfect relate this K-group to the tensor product of the G_i's in the category of HINST. In my talk I want to introduce reciprocity functors following Serre's treatment of commutative algebraic groups. These functors are defined on finitely generated field extensions over a given ground field, they have traces and they have symbols indexed by closed points in regular projective curves over some field, which satisfy a reciprocity law. Examples of reciprocity functors are smooth commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, Rost cycle modules and Ka"hler differentials. Then I introduce a Somekawa-type K-group of a finite family of reciprocity functors, which is itself a reciprocity functor and satifies a certain universal property. We compute these K-groups in various cases. In particular we can describe Milnor K-theory, the tensor product of HINST and - in characteristic zero- the Ka"hler differentials in terms of these K-groups. If time permits we discuss some open questions and problems at the end.

Sato:

I will formulate axioms of cohomology theory on a big site which include neither homotopy invariance nor purity. I will also talk about a Riemann-Roch theorem without denominators for Chern class maps to such cohomology theory.

Sugiyama:

In this talk, I will talk about a necessary and sufficient condition for all Hodge (resp. Tate) classes to be are generated by Lefschetz classes on a simple CM abelian variety over a number field (resp. finite field). Here Lefschetz classes are Hodge (resp. Tate) classes of degree one. For certain simple CM abelian variety $A$ over a number field, I explain the relationship between the condition for $A$ and for a reduction of $A$.

Tabuada:

I will describe the recent developments in the theory of noncommutative motives.

Wittenberg:

The index of a variety is the smallest positive degree of a zero-cycle lying on it. I will explain a relation between the index of a variety defined over a strictly Henselian local field, Euler characteristics of coherent sheaves, and the cobordism class of the variety. As an application, rationally connected varieties over the maximal unramified extension of a p-adic field have p-power index. A conjecture of Kato and Kuzumaki about Fano hypersurfaces over p-adic fields will also be discussed. (Joint work with H. Esnault and M. Levine.)

Yasuda:

This talk is concerned with quotient singularities associated with modular representations of a cyclic group of prime order. I will present a way to compute stringy invariants of such singularities by using the motivic integration. The computed invariants provide information on resolution of singularities if any.

Zhong:

In this talk I will talk about an upper bound of the annihilator of the torsion part of the Chow group CH^d(X) where $X$ is the variety of complete flags of a linear algebraic group $G$ of type B_n or D_n. Such upper bound does not depend on the rank of the group $G$ but only on the codimension d. It is obtained by studying the characteristic maps of Chow groups and K_0 of flag varieties. This is joint work with S. Baek and K. Zainoulline.