We extend Deligne's results on regular connections (LNM 163) to log schemes over C. More precisely, we develop a notion of regularity (at infinity) for integrable connections on idealized log smooth log schemes, and prove that the analytification functor induces an equivalence between the categories of regular algebraic connections and of integrable connections on the analytification. By the work of Ogus, the latter category can be described in terms of certain constructible sheaves on the Kato-Nakayama space. This is part of a project in progress whose goal is to obtain a Riemann-Hilbert correspondence for smooth rigid-analytic spaces over C((t)).
The variety of maximal tori in a split reductive group G can also be described as the quotient variety G/N where N is the normalizer of a maximal torus in G. Using motivic Becker-Gottlieb transfers Marc Levine has shown that invertibility of the A1-Euler characteristic of G/N considered as an element of the Grothendieck-Witt ring of quadratic forms governs some generalized splitting principles in the motivic setting. In my talk I will discuss how one may establish this invertibility property studying Betti realizations of G/N and using the classical theory of algebraic groups over the reals.
In this talk I will continue the discussion of the properties of the category of log motives over a field, introduced in a recent joint work with D. Park and P.A. Østvaer. I will explain the relationship with Voevodksy’s DM, the construction of Thom spaces in the logarithmic setting and Gysin sequences. I will also discuss some conjectures related to the étale variant of logDM, and connections to the theory of reciprocity sheaves à la Kahn-Saito-Yamazaki.
In motivic stable homotopy theory, there are two approaches to calculating the motivic stable homotopy groups of spheres: the slice spectral sequence and the motivic Adams spectral sequence. Both have been studied extensively and have produced a number of interesting calculations. In this talk, I will discuss a link between these two spectral sequences and carry out some example calculations. Time permitting, I will also discuss how to derive C_2 equivariant slice differentials from real motivic homotopy theory.
Orientation theory plays a crucial role in stable homotopy, via, for example, chromatic homotopy theory. It has been faithfully transported to motivic homotopy theory and presents the same features allowing one to understand motivic cohomology, algebraic K-theory and algebraic cobordism. Panin and Walter have constructed a vast generalization of this theory which takes into account Chow-Witt groups, hermitian K-theory and higher Witt groups. The picture is still very close to classical orientation theory after the work of many, but a few points remain open. I will present the results of a collaboration with Jean Fasel which aims to answer the following two questions: what is the analogue of Quillen's theory of associated formal group laws? What is the analogue of the Chern character?
We will survey recent developments on the construction and classification of algebraic vector bundles on smooth schemes over a field k. In the affine case, we will recall some classical problems and explain how motivic methods can sometimes solve them. We’ll then pass to more general schemes and highlight some interesting questions arising in this situation.
Projective varieties containing cylinders, more ideally affine spaces, start to receive plenty of attention from the viewpoint of a study of unipotent group actions on certain affine varieties. Such projective varieties are birational to suitable Mori Fiber Spaces (MFS), which play nowadays an important role as possible outcomes of minimal model program. Thus, in some sense, it is essential to find the affine spaces in MFS. In the talk, we will observe how to construct systematically completions of the affine spaces and exotic ones, e.g., Koras-Russell threefold into MFS by making use of linear pencils. This is based on a joint work with Adrien Dubouloz (Dijon, France) and Karol Palka (Warsaw, Poland).
It is well-known that the derived category of an algebraic variety is not a complete invariant, in the sense that there exist non-isomorphic varieties with equivalent derived categories. In this talk I will explain joint work with Max Lieblich investigating possible additional structures one might consider to conjecturally remedy this.
There are many non A^1-invariant cohomology theories like Hodge cohomology. To incorporate such theories into a framework of motives the current A^1-motivic homotopy theory is not enough. In this talk I will explain the development of a motivic homotopy theory of schemes using logarithmic geometry in which many cohomology theories have potential to be representable. This work is joint with Federico Binda and Paul Arne Østvær.
Motivated by Morel's degree in A1-homotopy theory which takes values in the Grothendieck-Witt ring of a field k, Kass and Wickelgren define the Euler number of an oriented vector bundle valued in GW(k) to be the sum of local A1-degrees of the zeros of a generic section. Using this definition they get an enriched count of lines on a smooth cubic surface in GW(k). In my talk I will compute several Euler numbers valued in GW(k). In particular, I will count lines on a general quintic threefold using a dynamic approach. In addition, I will give a geometric interpretation of the local contribution of a line on a quintic threefold to the enriched Euler number. When k = R this geometric interpretation agrees with the Segre type defined by Finashin and Kharlamov.
I will describe a version of A^1-homology for smooth schemes over a field, which is often entirely computable when the smooth scheme admits an increasing filtration by open subschemes with cohomologically trivial closed strata. I will also discuss some explicit computations and applications with emphasis on split reductive groups and generalized flag varieties. The talk is based on joint work with Fabien Morel.
Donaldson-Thomas (DT) theory is an enumerative theory which produces a count of ideal sheaves of 1-dimensional subschemes on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will give a brief introduction to motivic DT theory, in particular the role of d-critical locus structure in the definition of motivic DT invariant. I will also discuss a result on this structure on the Hilbert schemes of zero dimensional subschemes on a range of local toric Calabi-Yau threefolds. This is joint work in progress with Sheldon Katz.
I will describe a theory of genus zero curve enumeration in surfaces over arbitrary fields currently being developed in joint work with J. Kass, M. Levine and K. Wickelgren. I will discuss the problem of extending this theory to higher dimensions, and how it was solved in the related context of open Gromov-Witten theory in joint work with S. Tukachinsky.
Computation of the classical stable homotopy groups of spheres is a fundamental problem in topology. A recent breakthrough on this classical problem uses motivic homotopy theory over C in a fundamental way: By defining a t-structure on a subcategory of p-completed cellular objects of the motivic stable homotopy category over C, its heart can be identified as the abelian category of p-completed MU*MU-comodules, which is central to chromatic homotopy theory and very computable.
I will talk about a recent result that generalizes the Chow t-structure above (1) to an integral one (vs p-completed), (2) to the entire motivic stable homotopy category (vs cellular), and (3) to other base fields (vs over C). I will discuss ongoing projects on applications of computing Adams differentials and therefore stable homotopy groups of spheres, classically, over C, R and finite fields.
This talk is based on several joint works with Tom Bachmann, Robert Burklund, Bogdan Gheorghe, Dan Isaksen, Hana Jia Kong and Guozhen Wang.
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 discuss, in which sense K-theory is universal as a cohomology theory with such covariance. As one of the applications, we will obtain Hilbert scheme models for the K-theory space and for higher spaces of the very effective K-theory spectrum kgl. Based on joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro, and on the work of Tom Bachmann.