Motivic Geometry Seminar Series

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

A. Beilinson and T. Saito have recently added more microlocal methods into the toolkit of étale cohomology, by defining the singular support of a constructible sheaf on a smooth scheme over a field, and constructing the associated characteristic cycle. This talk will be a report on a work in progress aiming at promoting these constructions to étale motivic sheaves, along with new ones, with the goal of proving Saito's conjectural index formula for characteristic cycles. In a joint work with Enlin Yang, we construct the singular support of locally constructible étale motivic sheaf. There is also a way to construct a Fourier transform à la Laumon (worked out by Adeel Khan), and to construct a monodromic (or conical) version of Ayoub's nearby cycles functor associated to Verdier's deformation to the normal cone, which altogether give a very promising candidate for the microlocalization of motivic sheaves à la Kashiwara-Schapira. This provides a new way to construct characteristic cycles which only makes sense for motivic sheaves (as opposed to classical étale sheaves, with l-adic or torsion coefficients).

Homology and homotopy groups at infinity are classical invariants of open topological manifolds. I will present a realization of a program foreseen by Asok around 2008 consisting of importing these invariants in the framework of A1-homotopy theory. The definition of the stable A1-homotopy type at infinity of a smooth scheme uses the six functors formalism, and then, the real work starts with the problem of developing techniques which allow to compute this invariant. I will explain in particular how to get alternative presentations of stable A1-homotopy types at infinity via the consideration of completions with normal crossing boundary divisors. I will illustrate these methods by examples of effective computations for certain surfaces, related on the one hand to Mumford's characterization of smooth points on complex algebraic surfaces in terms of their local topological fundamental groups and on the other hand to Ramanujam's characterization of the complex affine plane among smooth complex affine surfaces by the triviality of its topological fundamental group at infinity.

Classical Aityah-Bott localization applies to a space X with a torus action and gives an isomorphism of the equivariant cohomology of X, suitably localized, with the localized equivariant cohomology of the fixed point locus, and gives a formula for the class in the fixed-point cohomology that maps to a given equivariant Chern class of an equivariant bundle. In looking for a ``quadratic’’ analog, we consider equivariant cohomolology of a scheme with coefficients in the sheaf of Witt groups. Since the Witt-sheaf cohomology of BG_m is trivial, there is no suitable localization theorem for a torus action. Instead, we consider actions of SL_2^n or N^n, where N is the normalizer of the torus in SL_2. Under an admittedly restrictive hypothesis on the structure of the non-trivial orbits, we prove an analog of the Atiyah-Bott localization theorem and the localization formula in these two settings. We will also discuss some elementary computations as applications. Part of this relies on a construction due to Alessandro D’Angelo of equivariant Borel-Moore Witt cohomology, and some of the examples have been worked out by Sabrina Pauli.

Lecture notes can be found here.

p-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this talk, I will review p-adic Hodge Theory of algebraic varieties, present current developments in p-adic Hodge Theory of analytic varieties, and discuss some of its applications to problems in Number Theory.

Lecture notes

In recent work, Land-Tamme showed that K-theory satisfies pro-Milnor excision for rings. By looking at the category of modules over the rings, we reinterpret this as a categorical excision statement for squares of ∞-categories with one generator and prove a ‘many-generator’ version of this. As a corollary, we prove pro-Milnor excision for equivariant K-theory for the action of a linearly reductive group and use this to prove pro-cdh descent for K-theory of stacks. Adapting the ideas of Kerz-Strunk-Tamme, this yields a generalization to stacks of Weibel’s conjecture on the vanishing of negative K-groups. This is a report on joint work with Tom Bachmann, Adeel Khan and Vladimir Sosnilo.

I will report on recent work concerning the action of the motivic Galois group on Anabelian objects such as fundamental groups of algebraic varieties conveniently completed. I'll sketch the proof of a motivic analog of a theorem of Pop (aka., the Ihara-Matsumoto-Oda conjecture) yielding several interpretations of the motivic Galois group as the automorphism group of some large diagrams of anabelian objects.

Link to lecture notes

We globalize a recent duality result of Eisenbud and Ulrich, defining a pushforward in oriented Chow under a (non-regular!) closed immersion associated to certain residual intersections. This gives rise to an excess intersection formula. We will compute some examples related to enumerative geometry enriched in bilinear forms, giving arithmetic refinements of results in enumerative geometry classically valid only over algebraically closed fields. This is joint work with Tom Bachmann.

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.

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.

https://arxiv.org/abs/2007.09044

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.

Slides for the talk are available here.

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.

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.

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.

Lecture notes from the talk.

Slides for the talk.

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.

Complete notes

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.

Slides for the talk.

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.

Slides for the talk

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.