# International Workshop on motives in Tokyo, 2014

### Organizing Committee:

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

Giuseppe Ancona (Duisburg-Essen)

Masanori Asakura (Hokkaido)

Thomas Geisser (Nagoya)

Lars Hesselholt (Copenhagen, Nagoya)

Marc Hoyois (MIT)

Isamu Iwanari (Tohoku)

Uwe Jannsen (Regensburg)

Max Karoubi (Paris)

Shane Kelly (Titech)

Matthew Morrow (Bonn)

Andreas Rosenschon (Muenchen)

Kanetomo Sato (Chuo Univ.)

Marco Schlichting (Warwick)

Rin Sugiyama (Duisburg-Essen)

Georg Tamme (Regensburg)

### 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) #23340004 representative Thomas Geisser,

### Program

### 15(Mon)/Dec:

11:00-12:00 Hesselholt, "Topological Hochschild homology and periodicity."

14:00-15:00 Sato, "p-adic etale Tate twists with negative log poles and with modulus."

15:30-16:30 Morrow, "Deformation of algebraic cycles, I."

16:45-17:45 Schlichting, "Homology stability for SL_n and E_n."

### 16(Tue)/Dec:

9:30-10:30 Iwanari, "Derived Tannaka dulaity and motivic Galois groups."

10:45-11:45 Tamme, "An analytic version of Lazard's isomorphism."

15:30-16:30 Ancona, "On the motive of a commutative group scheme."

16:45-17:45 Morrow , "Deformation of algebraic cycles, II."

18:30-20:30: Reception

### 17(Wed)/Dec:

9:30-10:30 Karoubi, "Hermitian K-theory invariants in Topology and Algebraic Geometry."

10-45-11:45 Schlichting, "Universal Euler class groups."

Free afternoon

### 18(Thu)/Dec:

9:30-10:30 Sugiyama , "Motivic homology of semi-abelian varieties."

10:45-11:45 Hoyois , "A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula."

14:00-15:00 Kelly , "Differential forms in positive characteristic avoiding resolution of singularities."

15:30-16:30 Rosenschon, "Torsion in the Lichtenbaum Chow group of arithmetic schemes."

### 19(Fri)/Dec:

9:30-10:30 Asakura , "Period, regulator of fibration with CM structure and the generalizaed hypergeometric function."

10:45-11:45 Geisser , "A duality between mod m and m-torsion of motivic cohomology."

13:00-14:00 Jannsen , "Duality for logarithmic de Rham-Witt sheaves and wildly ramified class field theory of varieties over finite fields."

### Abstracts

### Ancona:

We will show that the motive of G, a commutative group scheme over a noetherian and finite dimensional base, is the symmetric algebra of the 1-motive of G. A corollary is the decomposition of the Chow groups of G, generalizing the result of Beauville for abelian varieties. This is a joint work with A. Huber and S. Pepin Lehalleur.

### Asakura:

We discuss H^2 of a fibration over P^1 equipped with multiplication on generic fiber by a number field. Our first theorem is that the period is written by a certain product of Gamma values, as predicted by the period conjecture of Deligne-Gross. Our second result is that the regulator of certain elements of motivic cohomology group is a linear combination of Gamma values and unit argument of the generalized hypergeometric function 3F2. This is a joint work with Noriyuki Otsubo.

### Geisser:

We will construct a pairing between motivic cohomology mod m and the m-torsion of motivic cohomology (and variations of this for other motivic theories). We give applications to the Brauer group and Neron-Severi group.

### Hesselholt:

In this talk, I will explain my calculation of the topological Hochschild homology and the related theories TR, TF, and TC of the valuation ring in the perfectoid field of p-adic complex numbers. As a consequence of this calculation, one obtains, for any algebra over this ring, a periodicity operator similar to Connes' S-operator in cyclic homology.

### Hoyois:

Let X be a smooth projective variety over the real numbers and let f: X X be a self-map. To X one can associate a real manifold X(R) and a complex manifold X(C). l-adic cohomology gives a purely algebraic description of the Lefschetz number of f(C), but the Lefschetz number of f(R) is invisible to l-adic cohomology. I will explain how the Lefschetz number of f(R) is a motivic homotopy invariant and how a motivic version of the Lefschetz fixed-point formula for f subsumes the topological fixed-point formulas for f(C) and f(R). I will then consider the situation over an abstract field and formulate an analogous refinement of the l-adic Grothendieck-Lefschetz-Verdier trace formula.

### Iwanari:

I will talk about Tannaka duality for stable infinity-categories and Galois groups for mixed motives. I present a new notion of fine tannakian infinity-categories and a tannakian characterization in derived algebraic geometry. I will explain how it brings ideas and methods to consructions and studies of motivic Galois groups of mixed motives. The talk also covers such related topics as Kimura finiteness of Chow motives, motivic unipotent fundamental groups of curves, and tannakization.

### Jannsen:

This is a report on joint work with Shuji Saito. For smooth proper varieties over a perfect field we introduce logarithmic de Rham-Witt sheaves with modulus supported in a divisor with normal crossings. We establish a duality for these sheaves describing the coverings ramified along this divisor.

### Karoubi:

Hermitian K-theory (aka Witt groups) is the analog of Algebraic K-theory, defined through the orthogonal or symplectic group in lieu of the general linear group. Although for C* algebras we get essentially the same invariants as for usual K-groups, the theory is quite different for general algebras with involution in Algebra and Topology. A related invariant is Michael Atiyah's KR theory associated to a space with involution. It has connections with real elliptic operators on one hand and real Algebraic Geometry on the other hand. In this lecture, following Atiyah's ideas, we define a map from the Witt group associated to a real algebraic variety to the Witt analog of KR-theory. We show that this map is very close to be an isomorphism between two invariants of seemingly different natures. The proofs use different ideas and results from Algebra and Topology: solution of the Quillen-Lichtenbaum conjecture in an Hermitian framework, Bott periodicity, etc.. This is joint work with Charles Weibel.

### Kelly:

Recently, Huber-J?rder studied the h-sheafification of the sheaves of K?hler differentials in characteristic zero. They showed that these sheaves have a number of good properties, and they unify and simplify various ad hoc definitions of differential forms on singular varieties appearing in the literature. We will discuss to what extent these techniques can and can't be extended to positive characteristic. This is joint work with Huber and Kebekus.

### Morrow:

I will survey some recent developments in the deformation theory of algebraic cycles in families, in the style of variational Hodge and Tate conjectures, as well as explaining the classical theory. In particular, I will discuss the formulation of a crystalline variational Tate conjecture in characteristic p, its proof for line bundles, and an infinitesimal variant for higher codimension cycles. Similar ideas in characteristic 0 lead earlier to the formulation and proof of an infinitesimal Hodge conjecture, which was independently considered by S. Bloch, H. Esnault, and M. Kerz after their work on the analogous p-adic result.

### Rosenschon:

We give an example of a smooth arithmetic scheme X over the spectrum of a Dedekind domain and primes p with the property that the p-primary torsion subgroup of the Lichtenbaum Chow group CH^2_L(X){p} has positive corank. This is a report onjoint work with V. Srinivas.

### Sato:

For a regular generalized semistable family X over a p-adic integer ring A, we have a complex T_n(r) of etale Z/p^n-sheaves which plays analogous roles with the r-fold tensor product of the sheaf of p^n-th roots of unity in characteristic zero. In this talk, I will explain the construction of a version of the complex T_n(r) with modulus and partial results on some fundamental properties of this new object.

### Schlichting (I) :

We prove the SL_n and E_n (=group generated by elementary matrices) analog of a result of Nesterenko and Suslin on homology stability for rings with many units. For such rings, this verifies a conjecture of Bass from 1972 on stability of unstable Quillen K-theory. For commutative local rings with infinite residue field, the result is that H_i(SL_n,SL_{n-1}) = 0 for i < n and is Milnor-Witt K-theory for i=n. There are two innovations which make this possible: the correct SL_n-analog of Suslin's homology computation of affine groups and a new presentation of Milnor-Witt K-theory.

### Schlichting (II) :

An Euler class theory for rank r vector bundles is a cohomology theory E^r (for the Zariski topology) together with an Euler class map, that is, a map of presheaves of spaces e:BSL_r \to E^r whose restriction to BSL_{r-1} is (homotopically) trivial. Examples are weight r motivic cohomology, algebraic K-theory, weight r A1-homology groups and higher Grothendieck-Witt theory GW^r. By definition, an Euler class theory comes with an Euler class e(V) in E^r_0(X) for every oriented rank r vector bundle V on X. In this talk I shall construct the universal Euler class theory E^r. Using our restults on homology stability for SL_n, I will show that for a noetherian scheme X of dimension r whose ring of regular functions has many units that the resulting universal Euler class group in degree zero E^r_0(X) is Zariski cohomology of X with coefficients in the Milnor-Witt K-sheaf K^{MW}_r. I conjecture that if X = Spec R is affine with many units, dim X = rank V =r then e(V) = 0 if and only if V has a nowhere vanishing section. We show that the conjecture holds in the following situations: 1) R regular noetherian containing a field, 2) R smooth over a Dedekind domain with perfect residue fields, 3) R of finite type over an algebraically closed field, 4) dim R \leq 3 and 2\in R^*.

### Sugiyama:

In this talk, I generalize classical results of Beauville and Bloch on vanishing of some eigenspaces of the Chow group of abelian varieties to motivic (co)homology of semi-abelian varieties, by using a result of Ancona-Enright-Ward-Huber.

### Tamme:

In his seminal paper on p-adic analytic groups, Lazard proves two important theorems on the cohomology of such groups. Firstly, he gives a comparison of continuous group cohomology with Lie algebra cohomology and, secondly, a comparison of continuous with locally analytic group cohomology. In the talk, I will give a new, simple proof of the comparison between locally analytic group cohomology and Lie algebra cohomology. It generalizes Lazard's result to analytic groups over arbitrary non-archimedean ground fields and arbitrary locally analytic representations as coefficients.