# International Workshop on motives in Tokyo, 2013

### Organizing Committee:

Thomas Geisser (Nagoya University), Tomohida Terasoma (Tokyo University), Shuji Saito (TIT)### Speakers

Aravind Asok (USC)

Daniel Grayson (Illinois)

Jean Fasel (Muenchen)

Florian Ivorra (Rennes)

Rob de Jeu (Amsterdam)

Shane Kelly (TITech)

Matthew Morrow (Hausdorff Center Bonn)

Jinhyun Park (KAIST)

Gereon Quick (Muenster)

Oliver Roendigs (Osnabrueck)

Andreas Rosenschon (Muenchen)

Joe Ross (USC)

Shuji Saito (TITech)

Marco Schlichting (Warwick)

Markus Spitzweck (Osnabrueck)

Sinan Unver (Instanbul)

Jo"rg Wildeshaus (Universite' Paris 13)

Tako Yamazaki (Tohoku University)

Takehiko Yasuda (Ohsaka University)

Inna Zakharevich (IAS)

### This workshop is supported by

Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology,

Graduate School of Mathemtaical Sciences, the University of Tokyo, Leading Graduate Course for Frontiers of Mathematical Sciences and Physics

JSPS Grant-in-aid (B) #23340001 representative Tomohide Terasoma,

JSPS Grant-in-aid (B) #30571963 representative Thomas Geisser,

JSPS Grant-in-aid (B) #23340016 representative Lars Hesselholt.

JSPS Grant-in-aid (S) #22224001 representative Shigeyuki Kondo.

### Program

### 25(Mon)/Nov:

9:15-10:15 Rosenschon, "Etale motivic cohomology and algebraic cycles."

10:30-11:30 Schlichting, "The Grayson Spectral Sequence for Grothendieck-Witt groups."

13:30-14:25 Fasel, "A^1-homotopic classification of vector bundles I."

14:35-15:30 Spitzweck, "Triangulated Categories of Motives I."

### 26(Tue)/Nov:

9:30-10:30 Yasuda, "Quotient singularities and local Galois representations."

10:45-11:45 Ross, "Intersection theory on singular varieties."

14:00-15:00 Ivorra, "Motives of rigid analytic tubes and nearby motivic sheaves."

15:30-16:30 Spitzweck, "Triangulated Categories of Motives II."

16:45-17:45 Fasel, "A^1-homotopic classification of vector bundles II."

18:30-20:30: Reception

### 27(Wed)/Nov:

9:00-10:00 Yamazaki, "Reciprocity sheaves: towards non-homotopy invariant motive theory."

10:15-11:15 Saito, "Relative motivic complexes with moduli."

11:30-12:30 Quick, "Some applications of homotopy theory in algebraic geometry."

Free afternoon

### 28(Thu)/Nov:

9:30-10:30 Park, "Algebraic cycles and crystalline cohomology."

10:45-11:45 Zakharevich, "On the Grothendieck spectrum of varieties."

14:00-15:00 Morrow, "Infinitesimal deformation of cycles in finite characteristic."

15:30-16:30 de Jeu, "The syntomic regulator for K_4 of curves ."

16:45-17:45 Unver, "Cyclotomic p-adic multi-zeta values."

### 29(Fri)/Nov:

9:30-10:30 Asok, "Unstable A^1-homotopy sheaves of spheres."

10:45-11:45 Kelly, "Proper descent for homotopy invariant K-theory."

14:00-15:00 Wildeshaus, "Weights and conservativity."

15:30-16:30 Roendigs, "Some comments on the Grothendieck ring of varieties."

16:45-17:45 Grayson, "An elementary approach to long exact sequences in algebraic K-theory."

### Abstracts

### Asok:

I will describe some known and conjectural computations of unstable A^1-homotopy sheaves of punctured affine spaces. Time permitting, I will sketch an application to the question of when topological vector bundles on certain smooth complex affine varieties are algebraizable.

### de Jeu:

Let C be a curve defined over a discrete valuation field of characteristic zero, where the residue field has positive characteristic p. Assuming that C has good reduction over the residue field, we compute a certain p-adic (syntomic) regulator on a certain part of K_4^(3) of the function field of the C. The result can be expressed in terms of p-adic polylogarithms and Coleman integration, or by using a trilinear map (triple index) on certain functions. This is joint work with Amnon Besser.

### Fasel:

In these talks, we will survey recent results on the classification of vector bundles over smooth affine schemes using the A^1-homotopy theory of schemes. Time permitting, we will also indicate how to compute some motivic homotopy groups of algebraic spheres.

### Grayson:

I will show how to use elementary techniques to produce the long exact sequence of algebraic K-theory (associated to an exact functor between exact categories), thereby placing it intermediate in complexity between the long exact sequence of homotopy theory and the long exact sequence of homological algebra. Every group in the sequence is the Grothendieck group of a cube of exact categories, which are constructed using chain complexes and "binary" chain complexes in ways recognizable from homological algebra.

### Ivorra:

Let k be a field of characteristic zero, R=k[[t]] the ring of formal power series and K its fraction field. Let X be a finite type R-scheme with smooth generic fiber and Z a locally closed subset of the special fiber of X. We establish a relation between the rigid motive of the tube ]Z[ of Z and the restriction to Z of the nearby motivic sheaf associated with X. Our main result can be seen as an analog of a theorem of V. Berkovich.

### Kelly:

This will be a survey talk. We will discuss Cisinski's proof of the representability of homotopy invariant K-theory in the Morel-Voevodsky stable homotopy category, and how this representability is used with Ayoub's proper base change theorem to show that homotopy invariant K-theory satisfies cdh descent. If there is time we will show how this can be combined with Gabber's theorem on alterations to give an unconditional proof of some cases of Weibel's vanishing conjecture. Some of these cases were not known, even assuming resolution of singularities.

### Morrow:

Given a proper, smooth family of varieties over a perfect field of finite characteristic, and a cycle on a fibre within the family, I will explain a result which gives a necessary and sufficient condition, in terms of crystalline Chern characters, for the cycle to infinitesimal extend inside the family. This variational Hodge type result will be obtained via new results in topological cyclic homology which are joint work with B. Dundas.

### Park:

Berthelot's crystalline cohomology theory is a Weil cohomology theory for smooth projective varieties over a field of characteristic p>0. In the late 70s, Bloch gave a description of it in terms of relative higher algebraic K-groups of Quillen, and llusie gave a description in terms of the de Rham-Witt complexes. Subsequently, de Rham-Witt complexes were generalized to the big de Rham-Witt complexes by Hesselholt and Madsen. In this talk, we give a description of Zariski sheaf of big de Rham-Witt complexes on smooth varieties in terms of additive higher Chow groups. From this, we deduce a new description of crystalline cohomology in terms of algebraic cycles.

### Quick:

We present recent applications of homotopy theory to detect interesting examples of algebraic cycles. In particular, we present Totaro's topological method to find cycles in the Griffiths group of varieties and discuss how this idea can be extended to different classes of examples (this is joint work with Michael Hopkins). In the end we will give an outlook on how these methods yield interesting new invariants in Arakelov geometry (this is joint work in progress with Jakob Scholbach).

### Roendigs:

This talk will discuss some properties of the Grothendieck ring of varietes and related constructions from the perspective of motivic stable homotopy theory.

### Rosenschon:

We consider etale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give a geometric interpretation of Lichtenbaum cohomology, and use it to show that the usual integral cycle maps extend to cycle maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. This is a report on joint work with V. Srinivas.

### Ross:

We introduce some ideas from motivic (co)homology into the study of singular varieties. Our approach is modeled on the intersection homology of Goresky-MacPherson; we aim to intersect cycles on a stratified singular variety provided the cycles do not meet the strata too badly. We define "perverse" analogues of Chow groups and motivic (co)homology. Properties include homotopy invariance, a localization theorem, and a splitting theorem. As a consequence we obtain pairings between certain "perverse" cycle groups on a singular variety. This is joint work with Eric Friedlander.

### Saito:

This is a joint work with M. Kerz. We introduce relative motivic complex as a complex of Zariski sheaves on X for a pair (X,D) of a smooth variety and an effective (non-reduced) Cartier divisor. Its cohomology groups called relative motivic cohomology are related to various non-homotopy invariants such as relative Picard groups, relative Chow groups with moduli, and additive higher Chow groups by Bloch-Esnault-Park. The main results are computation of the motivic complex in weight one, and computation of relative motivic cohomology using relative Milnor K-groups, and the construction of regulator maps to a relative version of Deligne cohomology, which provides Abel-Jacobi maps with additive parts.

### Schlichiting:

In the 90s, Grayson constructed a spectral sequence for K-theory using commuting automorphisms. Later, Suslin showed that the E_2-term of Grayson's spectral sequence is given by motivic cohomology. In my talk I will explain a generalisation of Grayson's spectral sequence to symmetric monoidal categories. This yields a "motivic" spectral sequence for Grothendieck-Witt theory which is different from the Roendigs-Ostvaer spectral sequence one obtains using the slice filtration. Applied to K-theory, our proof is different, and we believe simpler, than Grayson's original proof. This is joint work with Simon Markett.

### Spitzweck :

In the first talk I will review constructions of triangulated categories of motives due to Voevodsky and others over fields and other base schemes representing in some cases motivic cohomology (or motivic cohomology with rational coeffcients). In the second talk I will define motivic categories for general base schemes with integral coefficients, building on previous work in motivic homotopy theory (e.g. the proof of the Bloch-Kato conjecture).

### Unver:

We will define the generalization of p-adic multi-zeta values to roots of unity and compute these values for depth less than or equal to 2. The method is to solve the fundamental differential equation satisfied by the crystalline frobenius using rigid analytic methods and to study the coefficients of its power series expansion.

### Yamazaki:

In his fundamental work on triangulated category of mixed motives, Voevodsky developed a theory of homotopy invariant presheaves with transfers. We explain our attempt to generalize it to non-homotopy invariant theory. The basic idea is to replace homotopy invariance by Weil reciprocity. (Joint work with B. Kahn and S. Saito.)

### Yasuda:

I will talk about a conjecture that stringy motivic invariants of quotient singularities are equal to weighted (motivic) counts of local Galois representations. It would relate Bhargava's mass for etale extensions of a local field and the Hilbert scheme of points on an affine plane. Weights used here of local Galois representations are closely related to the Artin conductor. This is a joint work with Melanie Wood.

### Zakharevich:

Algebraic $K$-theory provides a generalization of the Grothendieck group to an entire spectrum of invariants. The classical examples in which it is used involve algebraic examples, such as modules over a given ring. In this talk we present an approach which allows us to construct algebraic $K$-theory spectra for more geometric problems, such as the Grothendeick ring of varieties, definable sets, and scissors congruence groups. Although this approach does not yield any new computations as yet it has several advantages. Firstly, it presents a spectrum, rather than just a group, invariant of the problem. One advantage of this approach is that it allows us to construct filtrations by filtering the set of generators of the groups, rather than the group itself. This last observation allows us to construct a filtration on the Grothendieck spectrum of varieties that does not (necessarily) exist on the ring.