# Motivic Geometry Seminar Series

If you are interested in attending the seminar series, please register by following this link. You will then receive an invitation to the Zoom meetings when they are available.

Check out the seminar's YouTube channel for the recorded seminars.

## Next seminar: Oliver Röndigs

### Endomorphisms of the projective plane

April 21, 16:00 CEST

The endomorphism ring of the projective plane over a field F of characteristic not two or three is slightly more complicated in the Morel-Voevodsky motivic stable homotopy category than in Voevodsky’s derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root.

## Hana Jia Kong: The motivic Chow t-structure and computational tools for the R-motivic homotopy groups

February 10, 16:00 CET

I will first talk about the joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu on motivic Chow t-structure. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe--Wang—Xu. We identify the heart of this t-structure, and expand the results of Gheorghe—Wang--Xu to integral results on the entire motivic category over general base fields. It connects to the equivariant theory when field extensions are involved.

The Chow t-structure story leads to computational applications in determining the Adams spectral sequence differentials. Another useful computational tool is the effective spectral sequence. Work by Röndigs—Spitzweck—Østvær computes the 1-line of the motivic sphere over base fields of characteristic not two. In the on-going work with Eva Belmont and Dan Isaksen, we use the effective spectral sequence method and compute motivic homotopy groups over real numbers in a larger range.

## Andrei Druzhinin: Strict A1-invariance over the integers

February 17, 16:00 CET

The talk discusses the difference of the stable motivic homotopy categories over fields and positive dimensional schemes, such as the shift of the homotopy t-structure in the connectivity theorem. We consider ways how to separate the parts of the (stable motivic) localisation functor over base schemes that are similar and different to the base field case. In particular, for the mentioned purpose we define so-called trivial fiber topology that is trivial in the base field case, and obtain a generalisation of the strict homotopy invariance theorem to the base scheme case. This is joint work with Håkon Kolderup and Paul Arne Østvær.

## Stephen McKean: Multivariate Bezoutians and A1-degrees

February 24, 16:00 CET

The theme of this talk is “the unreasonable effectiveness of commutative algebra in motivic homotopy theory.” While Morel’s GW(k)-valued Brouwer degree is an extremely useful tool, one would like to compute the degree in a formulaic way. Cazanave gave a formula for the global degree of endomorphisms of P^1 in terms of the Bézoutian, while Kass and Wickelgren gave a commutative algebraic formula for the local degree at rational points. After discussing these results, I will describe forthcoming work (joint with Thomas Brazelton and Sabrina Pauli) on computing the local and global degrees in terms of multivariate Bézoutians. Our results generalize Cazanave’s theorem on global degrees to higher dimensions, as well as remove any residue field assumptions on local degree computations.

## Elden Elmanto: Motivic topology and purity for torsors

March 3, 16:00 CET

If X is a smooth variety over a field and G is a connected, reductive group scheme then the purity question of Colliot-Thélène and Sansuc asks if G-torsors over X are extended from codimension one subschemes. This is trivially satisfied for Serre's special groups and positive results were obtained by Panin and Panin-Chernousov, while negative ones were obtained by Antieau and Williams using topology, in the setting of complex varieties. In this talk, I will explain a reformulation of purity which works in a large class of cases in terms of motivic homotopy theory. This is a by-product of a computation of the sheaf of A^1-connected components of the classifying stacks (aka the "etale classifying space") of G which also settles Morel's pi_0-conjecture in these cases. Time permitting, we will discuss possible applications to new (and systematic) counterexamples to purity via Postnikov tower methods.

This is a report of joint work in progress with Girish Kulkarni and Matthias Wendt.

## Ran Azouri: Motivic nearby cycles and quadratic conductor formulas

March 10, 16:00 CET

Various tools may be used to investigate degenerations in a motivic setting: The nearby cycles functor of Ayoub in motivic homotopy theory; nearby cycles in the context of motivic integration; comparing the Euler characteristics of the singular and generic fibers. I will report on a quadratic conductor formula for hypersurfaces in a local setting (recent work by Levine, Pepin Lehalleur and Srinivas) with the motivic, compactly supported Euler characteristic, which takes values at the Grothendieck-Witt ring of the base field. Then I will show how reinterpreting it in terms of motivic nearby cycles and computing it along certain coverings (defined by Denef and Loeser) of pieces of the singular fiber, allows to extend the formula to a more general degeneration with a few (quasi-)homogeneous singularities.

## Lorenzo Mantovani: On the Chow-Witt ring of M_1,1

March 10, 16:45 CET

We outline a strategy for computing the Chow-Witt ring of the algebraic stacks $\overline{\mathcal{M}}_{1,1}$ and $\mathcal{M}_{1,1}$.

## Michael Temkin: Logarithmic resolution of singularities

March 17, 16:00 CET

I will talk about a recent series of works with Abramovich and Wlodarczyk, where a logarithmic analogue of the classical resolution of singularities of schemes in characteristic zero is constructed. Already for usual schemes, the logarithmic algorithm is faster and more functorial, though as a price one has to work with log smooth ambient orbifolds rather than smooth ambient manifolds. But the main achievement is that essentially the same algorithm resolves log schemes and even morphisms of log schemes, yielding a major generalization of various semistable reduction theorems.

## Alberto Merici: Connectivity and purity of logarithmic motives

March 24, 16:00 CET

I will present a generalisation of Morel’s Connectivity Theorem to the category of logarithmic motives over a field of Binda–Park–Østvær, leading to the construction of a homotopy t-structure. The heart of this t-structure is the category of strictly cube-invariant logarithmic sheaves, which generalises Voevodsky’s category of homotopy invariant sheaves. If time permits, I will explain the relation with the category of reciprocity sheaves of Kahn–Saito–Yamazaki–Rülling. The talk is based on a joint work with Federico Binda.

## Luca Barbieri Viale: Theoretical motives revisited and developed

March 31, 16:00 CEST

Abstract: The approach to mixed motives through universal cohomology theories is revisited and developed. These universal theories are obtained via representing 2-functors yielding universal abelian categories of T-motives. Eilenberg-Steenrod axioms are included: there are ordinary homologies with values in abelian categories for any homological structure on a category and the ordinary theories are representable. The universal ordinary theory on CW-complexes yields hieratic R- modules and considering the theory generated by singular homology we get back R-modules. For geometric categories like schemes or manifolds, adding vanishing and duality axioms, shall be sufficient to get mixed motives as T-motives.

## Giuseppe Ancona: The standard conjecture of Hodge type for abelian fourfolds

April 7, 16:00 CEST

Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V. The intersection product defines a non degenerate quadratic form on V. The Hodge index theorem says that it is of signature (1,d-1).

In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.

## Baptiste Calmes: Hermitian K-theory of quadratic functors and Karoubi periodicity

April 14, 16:00 CEST

Joint work with E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus and W. Steimle

In order to classify quadratic forms, Karoubi and Villamayor introduced Hermitian K-theory in the 70's. The subject has been in constant evolution since then, and has received recent attention from the motivic community when Morel showed it naturally occurred as endomorphisms in the stable homotopy category of schemes.

A new definition of Hermitian K-theory using quadratic functors and stable infinity-categories enables us to simplify and generalize its classical properties, study the relationship between different objects of quadratic nature such as symmetric bilinear forms or quadratic ones, and completely remove or clarify the invertibility of 2 assumptions that plagued the theory until now.

I'll give an overview of the construction and explain how it applies to prove periodicity conjectures of Karoubi and Giffen.

Slides are located here.

## Oliver Röndigs: Endomorphisms of the projective plane

April 21, 16:00 CEST

The endomorphism ring of the projective plane over a field F of characteristic not two or three is slightly more complicated in the Morel-Voevodsky motivic stable homotopy category than in Voevodsky’s derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root.

## Denis-Charles Cisinski: TBA

April 28, 16:00 CEST

May 5, 16:00 CEST

## Marc Levine: TBA

May 12, 16:00 CEST

## Wieslawa Niziol: TBA

May 19, 16:00 CEST

## Joseph Ayoub: TBA

June 2, 16:00 CEST

## Marc Hoyois: Introduction to framed correspondences

August 27, 28, 31 at 2 pm CET

I will give an overview of the theory of framed correspondences in motivic homotopy theory. Motivic spaces with framed transfers are the analogue in motivic homotopy theory of E_∞-spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite P^1-loop spaces. I will discuss the foundations of the theory (following Voevodsky, Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the computations of the infinite loop spaces of the motivic sphere and of algebraic cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open problems.

YouTube recordings: Part 1, Part 2, Part 3.

## Grigory Garkusha: Semilocal Milnor K-theory

September 3 at 2 pm CET

In this talk I will tell about semilocal Milnor K-theory of fields. A strongly convergent spectral sequence relating semilocal Milnor K-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups HpZar(k,ℤ(2)), p≤1, are computed as semilocal Milnor K-theory groups KˆM2,3−p(k). The following applications are given: (i) several criteria for the Beilinson-Soulé Vanishing Conjecture; (ii) computation of K4 of a field; (iii) the Beilinson conjecture for rational K-theory of fields of prime characteristic is shown to be equivalent to vanishing of rational semilocal Milnor K-theory.

## Kay Rulling: The cohomology of reciprocity sheaves

September 22, 23, 24 at 2 pm CET

We start by introducing the theory of reciprocity sheaves and the necessary background of modulus sheaves with transfers, as developed by B. Kahn, H. Miyazaki, S. Saito, and T. Yamazaki.

Then we will explain some basic examples of reciprocity sheaves with a special emphasis on - and an introduction to the de Rham-Witt complex. After an overview of some fundamental results, we will proceed to recent joint work of F. Binda, S. Saito, and myself on the cohomology of reciprocity sheaves. In particular, we prove a projective bundle formula, a blow-up formula as well as a Gysin sequence, generalizing work of Voevodsky on homotopy invariant sheaves with transfers. From this we construct pushforwards under a projective morphism and an action of projective Chow correspondences on the cohomology of reciprocity sheaves. This generalizes several such constructions which originally relied on Grothendieck duality for coherent sheaves and gives a motivic view towards these results.

We use this to prove the birational invariance of the cohomology of certain classes of reciprocity sheaves in a systematic way, many of which were not considered before.

If time permits we will also explain how to apply the pushforward to define higher local symbols for reciprocity sheaves.

## Ryomei Iwasa: Chern classes with modulus

October 1 at 2 pm CET

I start with a review of the classical construction of Chern classes for a multiplicative cohomology theory E on a category of schemes. It turns out that, to construct Chern classes, we only need to assume that E has 1) proper transfers and 2) projective bundle formula.

There is additional difficulty to construct Chern classes for a relative cohomology theory, aka cohomology theory with modulus. In a joint work with Wataru Kai, we worked out the construction for the relative motivic cohomology defined by Binda-Saito. I’ll explain the construction and an application to relative motivic cohomology of henselian dvr.

## Tom Bachmann: Eta-periodic motivic stable homotopy theory

October 8 at 2 pm CET

Over any scheme on which 2 is invertible, we establish a 2-term resolution of the eta-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. As applications we determine the eta-periodized motivic stable stems and the eta-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic dedekind schemes with 2 invertible. This is joint work with Mike Hopkins.

## Ieke Moerdijk: Profinite completion of Quillen model categories

October 29 at 2 pm CET

We describe a general method for obtaining fibrantly (sic) generated model category structures on Pro-categories. As special cases, we obtain known structures such as the ones of Morel and Quick, as well as new ones such as a model category of profinite infinity-categories, resp. profinite complete Segal spaces, profinite infinity-operads and dendroidal profinite complete Segal spaces.

The talk is based on joint work with Thomas Blom.

## David Rydh: Milnor excision of stacks, pushouts and étale neighborhoods

November 5 at 2 pm CET

## Matthew Morrow: Some recent developments in p-adic motivic cohomology

November 17, 18, 20 at 2 pm CET

These talks will present an overview of recent work in p-adic étale motivic cohomology, in finite and mixed characteristic, due to various collaborations of Antieau, Bhatt, Clausen, Kelly, Lüders, Mathew, Nikolaus, Scholze, the speaker… We will introduce the motivic filtration on topological cyclic homology, whose graded pieces provide a general theory of p-adic étale motivic cohomology, and discuss its finer properties, comparisons to existing invariants, and prismatic interpretation. Along the way we will see concrete applications, such as to calculations of K-groups.

## Mattia Talpo: Kummer-étale additive invariants of log schemes

November 26 at 2 pm CET

Hagihara and Nizioł proved decomposition results for the Kummer-étale (and -flat) algebraic K-theory of some class of nice log schemes, in terms of the algebraic K-theory of the strata of the “boundary”. I will talk about some work with Sarah Scherotzke and Nicolò Sibilla about a categorification (at the level of non-commutative motives) of this decomposition. I will start by recalling some basics on log schemes, then move on to root stacks and semi-orthogonal decompositions of their derived categories, which are the main tools of our work, and to describing our results.

## Niels Feld: Milnor-Witt homotopy sheaves and Morel generalized transfers

December 3 at 2 pm CET

We present a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with generalized transfers and Milnor-Witt homotopy sheaves. As applications, we describe the essential image of the canonical functor that forgets Milnor-Witt transfers and use these results to discuss the conservativity conjecture in A^1-homotopy due to Bachmann and Yakerson.

## Alexander Schmidt: Tame cohomology of schemes and adic spaces

December 10 at 2 pm CET

Étale cohomology with non-invertible coefficients has some unpleasant properties, e.g., it is not A^1-homotopy invariant and for constructible coefficients the expected finiteness properties do not hold. To remedy this, we introduce the `tame site'. Tame cohomology coincides with étale cohomology for invertible coefficients but is better behaved in the general case. The associated higher tame homotopy groups hopefully have a better behaviour than the higher étale homotopy groups, which vanish for affine schemes in positive characteristic by a result of Achinger.

There are two approaches to the construction of a tame site. The first one considers the discretely ringed adic space Spa(X,S) associated with a scheme X over a base scheme S. In the category of adic spaces, it is natural to define tame morphisms by a tameness condition on residue field extensions. The resulting "adic tame site" Spa(X,S)_t has good local properties and it seems promising to develop technical machinery such as base change theorems and cohomological purity in this setting.

The second approach works with étale morphisms of schemes and imposes a tameness condition on coverings. This "algebraic tame site" fits nicely into the framework of motivic cohomology theory as it sits in between the étale and the Nisnevich site.

In this talk we will explain the basic properties of tame cohomology and a comparison theorem between the tame cohomology of an S-scheme X and the tame cohomology of the associated adic space Spa(X,S) in the case of pure characteristic.

## Fabio Tanania: The isotropic stable motivic homotopy category

March 24, 16:45 CET

In this talk, I will present the isotropic stable motivic homotopy category, which is a stable homotopic version of Vishik’s category of isotropic motives. Roughly speaking, it is a localization of the stable motivic homotopy category obtained by killing all anisotropic varieties. Moreover, I will discuss results about the isotropic stable motivic homotopy groups of the sphere spectrum and the structure of the category of isotropic cellular spectra.