Schedule and Abstracts

The motivic Galois group is most naturally considered as an object in spectral algebraic geometry. However, deep conjectures in the theory of motives imply that the motivic Galois group is classical, i.e., has no higher derived information. We will discuss some recent attempts to verify the classicality of the motivic Galois group.

TBA

Greenberg transform is a useful tool for studying schemes over an artinian local ring. It was introduced by S. Lang in his thesis and later studied by M.J. Greenberg. In this talk, after recalling the classical construction I will present some results obtained in collaboration with E. Previato, A. Saha, and T. Suzuki.

We prove analogs of the classical Atiyah-Bott localization and Bott residue theorem in the setting of an action of the normalizer of the torus in SL_2, where the target cohomology theory is the equivariant cohomology of the sheaf of Witt rings. With Sabrina Pauli, we apply this to give a ``quadratic’’ count of the twisted cubics on a smooth quintic threefold and other hypersurfaces and complete intersections of suitable (multi-) degree. This gives as a special case signed counts of real twisted cubics on these varieties.

I will report on a joint work with Luca in which we map a rigid additive category universally to an abelian one, and more recent developments.

Joint work with O. Brinon and N. Mazzari

Nori motives can be obtained via the free abelian category construction followed by Serre localisation. This is described in a paper with Luca Barbieri-Viale and, including the tensor structure, also with Annette Huber-Klawitter. I will discuss various aspects of this, including the model-theoretic point of view on these constructions.

All periods of algebraic varieties, say over Q, can be expressed by volumes of semi-algebraic sets. This motivates the question whether it is possible to extend algebraic de Rham cohomology and the period isomorphism from algebraic varieties to the category of semi-algebraic sets. We give a negative answer by studying the corresponding categories of Nori motives.

To study the extensions of a valuation of a field K to an algebraic or transcendental cyclic extension L=K(x), following a previous work of S. MacLane we introduce the notion of "augmented valuations" and of "admitted family" of valuations of the polynomial ring K[X]. We show how we can associate to any valuation of L=K(x) an admitted family, and how this family gives information on the extension of valued fields.

We shall prove a duality theorem for the p-adic etale cohomology of an open subvariety U of a smooth projective variety over a finite field. We shall also derive a class field theory for certain quotients of $\pi^{\ab}_1(U)$. As a consequence, we shall produce a counterexample to Nisnevich descent for Chow groups with modulus. This talk is based on a joint work with Rahul Gupta.

We will review some recent results by Iovita-Morrow-Zaharescu about p-adic uniformization of abelian varieties with good reduction. Most of it relies on the theory developed by Fontaine especially about almost Cp-representations. Then we will focus on some really basic example, like the Tate elliptic curve, in order to explain how the main results of IMZ generalise to semistable abelian varieties and 1-motives.

This is a report on joint work with H. Esnault and M. Schusterman. Recall that the etale fundamental group of a variety over an algebraically closed field of characteristic 0 is known to be a finitely presented profinite group; this is proved by first reducing to varieties over the complex numbers, and then comparing with the topological fundamental group. In positive characteristics, even if we restrict to smooth varieties, finite generation fails in general for etale fundamental groups of non-proper varieties (eg, for the affine line).For a smooth variety with a smooth, projective compactification with a SNC boundary divisor, we show that the tame fundamental group is a finitely presented profinite group. In particular, this holds for the fundamental groups of smooth projective varieties.