Motives in Tokyo on the occasion of Shuji Saito’s 60th birthday
Date: March 26 (Mon)–March 30 (Fri) 2018
Place: Lecture Hall, Graduate school of Mathematical Sciences, The University of Tokyo
Organizers: M. Asakura, T. Geisser, N. Otsubo, K. Sato, T. Terasoma
March 26th (Monday)
10:00–11:00 K. Kato, Heights of motives and extended period domains
11:15–12:15 T. Saito, Characteristic cycle of an ℓ-adic sheaf
14:00–15:00 G. Tamme, Excision for localizing invariants
15:30–16:30 M. Kerz, Negative algebraic K-theory
16:45–17:45 B. Kahn, Albanese kernels and Griffiths groups
March 27th (Tuesday)
10:00–11:00 L. Hesselholt, K-theory and L-theory
11:15–12:15 C. Haesemeyer, On the K-theory of line bundles
14:00–15:00 K. Fujiwara, Cohomological purity from perfectoid viewpoint
15:30–16:30 O. Wittenberg, On the Lüroth problem for real varieties
16:45–17:45 M. Morrow, Motivic filtrations on K-theory and topological Hochschild homology
18:15

banquetMarch 28th (Wednesday)
10:00–11:00 S. Kelly, A more streamlined comparison of cdh and ldh cohomology
11:15–12:15 A. Krishna, Zero cycles and K-theory on schemes
14:00–15:00 J.-L. Colliot-Thélène, Disproving stable rationality (survey)
15:30–16:30 M. Levine, Quadratic Euler characteristics
16:45–17:45 J. Ayoub, Proof of the conservativity conjecture
March 29th (Thursday)
10:00–11:00 T. Yamazaki, Mixed Hodge structures with modulus
11:15–12:15 F. Binda, Rigidity for relative 0-cycles
14:00–15:00 K. Rülling, Reciprocity sheaves and conductors
15:30–16:30 J. Fresán, Hodge theory of Kloosterman connections
16:45–17:45 P. A. Østvær, Bott inverted algebraic cobordism
March 30th (Friday)
10:00–11:00 T. Abe, Ramification theory and homotopies
11:15–12:15 H. Esnault, Rigid local systems and integrality
14:00–15:00 S. Bloch, Motivic Gamma functions1

Supported by• JSPS KAKENHI Grants: 15H02048 (T. Terasoma), 15H03606 (S. Saito), 15H03607 (K. Fujiwara), 15K04769 (M. Asakura), 16K05072 (K. Sato), 16K05106 (Y. Miyaoka), 17H02836 (S.
Kobayashi), 17K18723 (S. Saito),• JSPS Core-to-Core program

Foundation of a Global Research Cooperative Center in Mathe-matics focused on Number Theory and Geometry” (M. Kurihara),• Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, The Universityof TokyoAbstracts
T. Abe:I will report on the on-going project with D. Patel on characteristic cycles and epsilon factors. I will
first recall de Rham epsilon factor after Beilinson and Patel, and talk about the localization formula.
A goal of our project is to realize this picture in the ℓ-adic setting. I will talk about possible strategy,and formulate a conjecture which might be useful to construct a theory of epsilon factors.
J. Ayoub:
I’ll give an overview of the proof of the conservativity conjecture for the classical realisations of
mixed motives in characteristic zero.
F. Binda:
In this talk, we will present a relation between the classical Chow group of 1-cycles on a regular
scheme X , projective and flat over an excellent Henselian discrete valuation ring A with perfectresidue field k, and the so-called cohomological Chow group of zero cycles of the special fiber, introduced by Levine and Weibel in the 1980s. If k is algebraically closed and with finite coefficients(prime to the residue characteristic) these groups turn out to be isomorphic. This generalizes a previous argument due to Bloch and Esnault-Kerz-Wittenberg to the case of regular models with arbitrary
reduction. From this, one can re-prove in case of bad reduction that the etale cycle class map for relative 0-cycles with finite coefficients on X is an isomorphism, a result due to Saito-Sato in the case of(quasi) semi-stable reduction. If time permits, we will discuss how the techniques can be extended tothe case where X is only quasi-projective, obtained from a projective scheme X by removing a horizontal SNC divisor, giving what we might call a proper base change theorem ”with compact support”
for relative 0-cycles. This is a joint work with Amalendu Krishna.
S. Bloch:
The study of solutions of Picard-Fuchs differential equations in the neighborhood of singularities
with maximal unipotent monodromy (MUM) leads to Mellin transforms which resemble the classicalGamma function. This is joint work with M. Vlasenko based on ideas of V. Golyshev and D. Zagier.
2J.-L. Colliot-Thélène:
For more and more classes of rationally geometricallly connected varieties, specialisation methods
have established nonstable rationality for very general members of the class. In recent work of
Schreieder, higher unramified cohomology and Pfister quadrics of arbitrary dimension play a keyrole. This leads to striking results on Fano hypersurfaces.
H. Esnault:(joint work with Michael Groechenig) We prove that a cohomologically rigid complex local systemwith finite determinant and quasi-unipotent monodromies at infinity is integral. This answers positively a conjecture by Carlos Simpson. More generally Simpson conjectures that rigid complex local
systems are of geometric origin. We’ll show how to use Langlands’ correspondence to treat (part of)
Simspon’s conjectures.
J. Fresán:
Recently, Broadhurst studied the L-functions associated with symmetric powers of Kloosterman sums
and conjectured a functional equation after extensive numerical experiments. I will explain how theirregular Hodge filtration allows one to explain the gamma factors at infinity in a similar way to
Serre’s recipe for usual motives. Based on work in progress with Claude Sabbah and Jeng-Daw Yu.
K. Fujiwara:
For a given cohomology theory, the cohomological purity, i.e., the vanishing of local cohomology
groups in low codimension, is a basic question. In this talk I plan to discuss purity of étale and flat
cohomologies of regular schemes in the mixed characteristic case via perfectoid methods.
C. Haesemeyer:
We discuss joint work with C. Weibel analysing the K-theory of (the total space a geometric) line
bundle relative to the base, viewed as a generalisation of Bass’ Nil K-groups. In particular we willtalk about one of the core technical ingredients, a globalisation of the Kassel-Sletsjœspectral sequencefor Hochschild homology.
L. Hesselholt:
Arrows have a very particular property: They can be reversed. More precisely, the infinity-category ofinfinity-categories has one non-trivial automorphisms, which takes an infinity-category to its opposite
infinity category. In this talk, I will explain how this structure interacts with K-theory.
B. Kahn:
We describe the Griffiths group of the product of a curve C and a surface S as a quotient of theAlbanese kernel of S over the function field of C. When C is a hyperplane section of S varying
in a Lefschetz pencil, we prove the nonvanishing in Griff(C × S) of a modification of the graph ofthe embedding C ,→ S for infinitely many members of the pencil, provided the ground field k is ofcharacteristic 0, the geometric genus of S is > 0, and k is large or S is “of motivated abelian type”.
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K. Kato:
I explain ideas on heights of motives. I explain how we can formulate the motive versions of Vojta
conjectures and Manin conjectures for motives, looking at analytic theories of period domains andtoroidal partial compactifications of period domains.
S. Kelly:
Half of the proof of the main result of Ast. 391 is a comparison of the cdh and ldh cohomologies ofcertain ”nice” presheaves. There, ”nice” is an awkward list of very special properties that don’t give
much insight into why the cohomologies should agree. Worse, the proof is a poorly structured pileof lemmas, which are difficult to arrange into some kind global narrative. In this talk we present aclearer proof that the cdh and ldh cohomologies of the relevant presheaves agree, isolating universal
homeomorphism invariance of the presheaves as the key ingredient (in addition to a structure oftraces). The study of finite flat algebras over (not necessarily Nagata) henselian valuation rings plays
an important role.
M. Kerz:
In the talk I will explain a proof of Weibel’s conjecture on the vanishing of negative algebraic Kgroups (joint work with F. Strunk and G. Tamme).
A. Krishna:
The relation between 0-cycles and K-theory (especially, the Grothendieck group of vector bundles) isa well established theory for smooth varieties. In recent times, a theory of cycles with modulus has
emerged, and these cycles are expected to be directly related to the relative K-theory. In this talk, weshall present some results in this direction. In particular, we shall see that the picture for 0-cycles
with modulus is now sufficiently complete for affine schemes. An important tool in this study is thetheory of 0-cycles on singular schemes. We shall show how these are all related.
M. Levine:
We will describe a series of interpretations of the “motivic” Euler characteristic of a smooth projective
variety over a perfect field k. The motivic Euler characteristic an element of the Grothendieck-Wittring of quadratic forms over k; we will show how this refines the usual topological Euler characteristicand how one can compute it using Euler classes in various cohomology theories.
M. Morrow:
In joint work with Bhargav Bhatt and Peter Scholze we constructed a motivic filtration on the topological Hochschild and cyclic homologies of a class of characteristic p (or, more generally, p-adic)
rings; the graded pieces, i.e., the corresponding motivic cohomologies, are given by a form of derived
crystalline or syntomic cohomology. Combining this with joint work with Dustin Clausen and Akhil
Mathew, in which we identify p-adic étale K-theory with topological cyclic homology, leads to an
Atiyah-Hirzebruch-style filtration on p-adic K-theory in new settings. In particular, in the context of4

a smooth variety X with modulus D, this produces a motivic filtration on the wild part of the relative
K-theory. I will give an overview of these results and of the techniques used.
P. A. Østvær:In joint work with Elden Elmanto, Marc Levine, and Markus Spitzweck we compare the stable motivic homotopy category SH with its etale analogue SHet . By inverting a Bott element in algebraic
cobordism we obtain an equivalence with the corresponding etale theory. Our method of proof amplifies the effects of the Beilinson-Lichtenbaum conjecture via the slice spectral sequence.
K. Rülling:(joint work with Shuji Saito) For a general reciprocity Nisnevich sheaf as defined by Kahn-SaitoYamazaki we define a motivic conductor. We give a list of properties which uniquely determine it anduse this to show that in many examples this conductor coincides with classical abelian conductors,
such as the Artin conductor of a rank 1 representation, the conductor on an algebraic group defined by
the Rosenlicht-Serre local symbol or the irregularity of a rank one connection. We also give motivic
versions of some of the more involved properties of the classical conductors as studied by Kato andothers.
T. Saito:
The characteristic cycle of an ℓ-adic sheaf on a smooth variety over a perfect field is a Z-linear
combination of irreducible components of the singular support, defined by Beilinson as a closed
conical subset of the cotangent bundle. We discuss its functorial property with respect to proper
direct image including the index formula.
G. Tamme:
A fundamental property of algebraic K-theory is that it is a localizing invariant, i.e., it sends a Verdier
quotient sequence A → B → B/A of small dg categories or small stable infinity-categories to afibre sequence of spectra. In the talk I will explain that any localizing invariant also sends certain
pullback diagrams of categories to pullback squares of spectra. If one applies this to categories ofperfect complexes over rings, one obtains a general descent result which includes Nisnevich descentand Suslin’s excision for algebraic K-theory as special cases.
O. Wittenberg:(joint work with Olivier Benoist.) The Lüroth problem asks whether unirational varieties are rational.
It has a positive answer for complex curves and surfaces; negative answers for complex threefolds
have been understood since the 70’s. I will discuss the Lüroth problem for real algebraic varieties thatare geometrically rational and explain a counterexample not accounted for by the topology of the real
locus or by a nontrivial unramified cohomology group over the reals.

5T. Yamazaki:
We define a notion of mixed Hodge structure with modulus that generalizes the classical notion ofmixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts
introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension we attach mixed Hodge structures with modulus. We combine thisconstruction with an equivalence between the category of level one mixed Hodge structures withmodulus and the category of Laumon 1-motives to generalize Kato-Russell’s Albanese varieties withmodulus to 1-motives.

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