# International Workshop on motives in Tokyo, 2023

### Organizing Committee:

Thomas Geisser (Rikkyo University), Shane Kelly (Tokyo University), Hiroyasu Miyazaki(NTT Institute for Fundamental Mathematics), Shuji Saito (Tokyo University)### Confirmed Speakers

Tomoyuki Abe (IPMU)

Toni Annala iIAS)

Aravind Asok (University of Southern California)

Federico Binda (Milano)

Christian Haesemeyer (Melbourne)

Isamu Iwanari (Tohoku)

Ryomei Iwasa (CNRS, Orsay)

Junnosuke Koizumi (Tokyo)

Achim Krause (Muenster)

Matthew Morrow (CNRS, Orsay)

Thomas Nikolaus (Muenster)

Jinhyun Park (KAIST)

Piotr Pstragowski (IAS)

Kay Ruelling (Wuppertal)

Vladimir Sosnilo (Regensburg)

Tomohide Terasoma (Hosei)

Maria Yakerson (Jussieu)

### This workshop is supported by

JSPS Grant-in-aid (B) #20H01791 representative Shuji Saito,

JSPS Grant-in-aid (C) #18K03258 representative Thomas Geisser,

JSPS Grant-in-aid (Early-Career Scientists) #19K14498 representative Shane Kelly

JSPS Grant-in-aid (Early-Career Scientists) #21K13783 representative Hiroyasu Miyazaki

### Program

### 13(Mon)/Feb:

10:00-11:00 , Ryomei Iwasa: P-homotopy invariance and algebraic cobordism.

11:30-12:30 , Isamu Iwanari: Variations of Hodge structures of stable categories

14:00-15:00 , Kay Ruelling: Hodge-Witt cohomology with modulus and duality

15:30-16:30 , Tomohide Terasoma: Depth filtration of multiple zeta value and mixed elliptic motives

### 14(Tue)/Feb:

10:00-11:00 , Aravind Asok: On P^1-stabilization in motivic homotopy theory

11:30-12:30 , Vladimir Sosnilo: Weighted A^1-invariance and the Atiyah-Segal completion theorem

14:00-15:00 , Matthew Morrow: Motivic cohomology of equicharacteristic schemes

15:30-16:30 , Thomas Nikolaus: (Generalized) Prismatic cohomology and the motivic filtration

### 15(Wed)/Feb:

10:00-11:00 , Tomoyuki Abe: Functoriality of characteristic cycles

11:30-12:30 , Junnosuke Koizumi: A motivic construction of the de Rham-Witt complex

Afternoon: Excursion

### 16(Thu)/Feb:

10:00-11:00 , Pjotr Pstragowski: Motives and chromatic homotopy theory

11:30-12:30 , Maria Yakerson: Universality of algebraic K-theory

14:00-15:00 , Federico Binda: Motivic monodromy and p-adic cohomology theories

15:30-16:30 , Achim Krause: On the K-theory of Z/p^n

### 17(Fri)/Feb:

9:30-10:30 , Toni Annala: Derived algebraic cobordism

10:45-11:45 , Jinhyun Park: A cycle model for motivic cohomology of fat points via formal / rigid geometry

12:00-13:00 , Christian Haesemeyer: Local t-structures and reconstruction

### Abstract

### Abe :

The characteristic cycle of l-adic sheaf was introduced by T. Saito after the construction of the singular support by Beilinson. We wish to show the functoriality of the characteristic cycle (up to p-torsion), which was conjectured by Saito. In the proof, six functor formalism of motives a la Ayoub plays an essential role. In the talk, I plan to focus on how motivic formalism is used.

### Annala :

In my thesis, I constructed a cohomology theory of schemes called algebraic cobordism ƒ¶^*, extending the similarly named theory of Levine-Morel from smooth varieties in characteristic 0 to all finite-Krull-dimensional Notherian schemes admitting an ample line bundle. This is a non-A^1-invariant cohomology theory that is closely related to the (higher) algebraic cobordism MGL that is constructed in an upcoming work with Marc Hoyois and Ryomei Iwasa. In particular, we expect that ƒ¶^n(X) = MGL^{2n,n}(X) whenever both sides are well defined. I will discuss results about ƒ¶^* related to this comparison, and explain why obtaining the desired isomorphism, or an interesting weakening of it, seems hopeless unless someone has a smart idea, or alternatively someone proves resolution of singularities in positive characteristic.

### Asok :

I will discuss recent progress on the analysis of P^1-stabilization in motivic homotopy theory over a field based on joint work with Tom Bachmann and Mike Hopkins. After discussion of an analog of the Freudenthal suspension theorem for P^1-stabilization, I will discuss some new results about ``metastable" homotopy for punctured affine spaces.

### Binda :

In this talk, we will discuss some recent advances in the theory of motives in the context of log geometry and rigid analytic geometry. Building on work of Ayoub, Bondarko, we offer a new definition of the Hyodo-Kato cohomology, purely defined on the generic fiber, without making any reference to log schemes or the log-de Rham Witt complex. As a consequence, we can construct Clemens-Schmidt-style complexes in the mixed characteristic setting, confirming an expectation of Flach and Morin. This is a joint work in progress with Alberto Vezzani and Martin Gallauer.

### Haesemeyer :

The notion of a local t-structure (or sheaf of t-structures) on derived categories of varieties was introduced by Abramovich and Polishchuk. In this talk, I will discuss joint work with David Gepner on the general notion of a sheaf of t-categories, and how one can use their classification (recently obtained by G. Sahoo and U. Dubey generalising Alonso Tarrio et al) in the case of the Zariski site of a scheme to give a new proof of Bondal and Orlov's theorem regarding Fourier - Mukai transforms between varieties with (anti-)ample canonical bundle; the proof is similar in spirit to that in recent work of H. Matsui.

### Iwanari :

A family of algebraic varieties gives rise to a variation of Hodge structure. The subject of my talk is about its categorical generalization: we consider a family of stable infinity-categories (or the likes such as pretriangulated dg categories up to Morita equivalences). I would like to introduce two methods of constructions of Hodge theoretic objects realized as the periodic cyclic complex with a D-module structure and a filtration. Two approaches are interrelated to each another and have their own advantages. They are not analogous to procedures in the commutative case. They involve factorization homology (topological chiral homology), mapping stacks, Hochschild pairs and their moduli-theoretic interpretation, Koszul duality theorems, and the relation between deformation theory and dg Lie algebras, etc. I will discuss the circle of ideas/motivations related to results.

### Iwasa :

This is joint work with Toni Annala and Marc Hoyois. P-homotopy invariance in a category C of algebro-geometric origin refers to the following phenomenon: given any global sections of an algebraic vector bundle E on a scheme/stack X, the induced maps X\to P_X(E\oplus O) are homotopic to each other in C. I will explain that P-homotopy invariance follows from tensor-invertibility of the pointed projective line P^1 and elementary blowup excision. This allows us to do homotopy theory in algebraic geometry while keeping the affine line A^1 non-contractible, and we motivically prove, for example, Bass fundamental theorem, weighted A^1-homotopy invariance, and equivalence Grass_n=BGL_n. Then Ifll explain applications to algebraic cobordism. One of our main results is a Conner-Floyd isomorphism for algebraic K-theory of any qcqs derived scheme.

### Koizumi :

We introduce the notion of Q-modulus pair over a noetherian scheme, which generalizes the notion of modulus pair over a field introduced by Kahn-Miyazaki-Saito-Yamazaki. We prove that the de Rham-Witt complex of Hesselholt-Madsen can be written as the p-typical part of the Suslin homology of a certain Q-modulus pair, whenever there is a suitable transfers on the de Rham-Witt complex. This is a joint work with Hiroyasu Miyazaki.

### Krause :

In recent work with Antieau and Nikolaus, we develop methods to compute algebraic K-theory of rings such as Z/p^n, based on trace methods and prismatic cohomology. Our methods lead to a practical algorithm, which we use to study these K-groups. The most striking pattern we discover is that these K-groups vanish in sufficiently large even degrees, which we are able to prove. In this talk, I want to explain the ingredients behind these results.

### Morrow :

Joint work with Elden Elmanto. I will present an extension of motivic cohomology from smooth varieties to arbitrary varieties, even to all qcqs equicharacteristic schemes. It is necessarily non-A^1-invariant, as it is equipped with an Atiyah-Hirzebruch spectral sequence converging to the K-theory of the scheme, not the KH-theory. It continues to satisfy various properties akin to classical motivic cohomology, such as the projective bundle formula, some Beilinson-Lichtenbaum type formulae, and the degree 2d, weight d motivic cohomology is related to zero cycles on singular varieties. But there are also new phenomena; for example, it has a vanishing range which refines Weibelfs vanishing conjecture and it satisfies pro-cdh descent.

### Nikolaus :

We will explain how the motivic filtration on topoloical periodic homology (due to Bhatt-Morrow-Scholze and recenetly drastically generalized by Hahn-Raksit-Wilson) is constructed. The associated graded is given by prismatic cohomology. The main goal of the talk is to explain prismatic cohomology without assuming any preknowledge. Then we discuss a recent generalization, which is joint with Antieau and Krause. This generalization is one of the keys for the computation of K(Z/p^n) that will be explained in Achim Krause's talk.

### Park :

I sketch a potential new cycle model for the motivic cohomology of fat points using formal geometry. To suggest that this direction has a good potential, I present an example of an Artin local scheme for which one has a concrete calculation. This example suggests a few new insights that some methods from convergent formal power series as in formal / rigid geometry may be useful in understanding the motivic cohomology of schemes with singularities.

### Pstragowski :

Chromatic homotopy theory is the study of the intricate relationship between algebraic topology and the arithmetic of formal groups. In the past couple of years, the field made major advances following the discovery of the surprising connection to the theory of motives and the deformation-theoretic picture suggested by algebraic geometry. I will talk about these recent developments and how they connect to the motivic filtrations of Bhatt-Morrow-Scholze.

### Sosnilo:

The A^1-invariance property does not hold in many cases when one wishes to use the methods of motivic homotopy theory. For instance, it generally fails for algebraic K-theory and topological cyclic homology. Weighted A^1-invariance is a weakening of A^1-invariance which holds in a broader setting and, in particular, in the two cases mentioned. We will discuss some consequences of this property and use it to prove a version of the Atiyah-Segal completion theorem for topological cyclic homology.

### Terasoma :

The depth of multiple zeta values define a filtration on the space Z of multiple zeta values which is called the depth filtration. Broadhurst and Kreimer proposed a conjecture on the dimension of this filtration. In this talk we will discuss the relation between mixed elliptic motives and depth filtration on Z.

### Yakerson :

Among various features of algebraic K-theory, there is known to be covariance with respect to finite flat morphisms of schemes. In this talk we will see, in which sense K-theory is universal as a cohomology theory with such covariance. Time permitting, we will discuss an analogous universality property for hermitian K-theory. Based on joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro, and on the work of Tom Bachmann.

### Way to the university

You will need some cash, because credit cards are not as commonly used. But you can exchange money at the airport.

"From Narita:"

Don't even think of taking a taxi, the airport is about 70km from the guest house. There are convenient buses and trains, and they cost 3000Yen per person. When you leave the customs at Narita, you are in a hall where you can buy bus or train tickets, and the staff will tell you where to board the bus or train. Everything will be explained, punctual and well-organized, so relax.

Bus: The fastest way is to take a direct bus to SHIBUYA (it has two stops, take the second and last one: Shibuya Excel hotel). Then you just have to walk down two floors to the local train you need to take, see below, or take a taxi for about 1000Yen. There are buses at

13:05, 14:05, 15:05, 15:30, 16:05, 16:30, 17:05, 17:30,18:05, 20:05, 21:05.

It should work for most of you.

Train: There is a direct train to Shibuya, Narita express. There are trains at

16:15, 16:45, 17:14, 18:14, 18:46, and 19:13.

A cheaper and faster alternative is Keisei Skyliner, which also has more trains and different hours. The disadvantage is that you have to change trains one more time: Take Keisei Skyliner to Nippori, change to JR Yamenotesen, and then go to Shibuya.

After arriving in Shibuya: You have to find

INOKASHIRASEN (i.e. inokashira-line).

The good news is that it's the terminal station, so you cannot go the wrong direction. The bad news is that there at least 9 different lines meeting in Shibuya. If you took the bus, just take the stairs down, if you took the train, follow the signs or ask someone.

After you find the entrance, buy the cheapest ticket for 120 Yen at a ticket vending machine (which have an English menue). Enter through the gate, and board a train. DO NOT board an express train. The trains run every 10 minutes, so no need to hurry. On the local train, take the second stop:

KOMABATODAIMAE

Take the east exit (where you have to walk up stairs), exit the ticket gate turn left and walk down the stairs. You should be in front of the main gate of Tokyo University. Turn left at the gate, walk about 150m until you reach a french restaurant. The entrance to the guest house is on the right back of the building. Once you are on campus, you could also ask the officers at the main gate for help.

``From Haneda:"

There is a regular bus from Haneda to Shibuya station

take it to Shibuya Excel hotel and proceed as above.

Faster and cheaper is to take the train:

Take Keikyu line to Shinagawa (the train runs every 10 minutes), and transfer to JR Yamanote line (which runs every 4 minutes) to Shibuya. Then proceed as above.