Seminar at UiO (Oslo University)

On March 13th I will give a talk in the geometry seminar at the University of Oslo (UiO). Here’s the abstract: for more details see the preprint  arXiv:1802.00165.

Title: Rigidity for relative 0-cycles.
In this talk, we will present a relation between the classical Chow group of relative 0-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring $A$ with perfect residue field $k$, and the so-called cohomological Chow group of zero cycles of the special fiber. 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 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 étale cycle class map for relative $0$-cycles with finite coefficients on $\mathcal{X}$ is an isomorphism, a result due to Saito and Sato in the case of semi-stable reduction. This is a joint work with Amalendu Krishna.

Seminar at T.I.F.R.

On Wednesday, 07 February, 2018, I will give a lecture in the Geometry Seminar of the Tata Institute of Fundamental Research, Mumbai. Here’s the abstract.

Laumon 1-motives and motives with modulus

In 1974, Deligne introduced the category $\mathcal{M}_{1}$ of 1-motives (built out of semi-abelian varieties and lattices) as algebraic analogue of the category of mixed Hodge structures of level $\leq 1$. Today, thanks to the works of Ayoub, Barbieri-Viale, Kahn, Orgogozo and Voevodsky, we know that the derived category $D^b(\mathcal{M}_{1, \mathbb{Q}})$ can be embedded as a full subcategory of $\mathbf{DM}^{eff}_{gm}(k)\otimes \mathbb{Q}$, and that this embedding admits a left adjoint, the so-called “motivic Albanese functor”. Deligne’s original definition was later generalised by Laumon, introducing what are now known as “Laumon 1-motives”, to include in the picture all commutative connected group schemes (rather then only semi-abelian varieties). Due to the presence of unipotent groups (such as $\mathbb{G}_a$), the derived category of this bigger category cannot be realised as a full subcategory of Voevodsky’s motives. In this talk, we will explain how at least a piece of this category (the “\’etale part”) can be embedded in the bigger motivic category $\mathbf{MDM}^{eff}(k)$ of “motives with modulus”, recently introduced by Kahn-Saito-Yamazaki, and that this embedding also admits a left adjoint (a generalized motivic Albanese functor).

This is a joint work with Shuji Saito.

SPP 1786 Jahrestagung

I’ll be giving a talk at the Jahrestagung for the SPP project 1786 “Homotopy Theory and Algebraic Geometry” at the University of Wuppertal. The Conference takes place from March 21 till  March 24.

Here‘s a link to the conference website.

Upcoming seminars

I’ll be giving a talk at Tohoku University (Algebra Seminar) next week.

Title: Towards a motivic homotopy theory without A¹-invariance.

Abstract: Motivic homotopy theory as conceived by Morel and Voevodsky is based on the crucial observation that the affine line A¹ plays in algebraic geometry the role of the unit interval in algebraic topology. Inspired by the work of Kahn-Saito-Yamazaki, we constructed an unstable motivic homotopy category “with modulus”, where the affine line is no longer contractible. In the talk, we will sketch this construction and we will explain why this category can be seen as a candidate environment for studying representability problems for non A¹-invariant generalised cohomology theories.



Hi all!

I’ve just moved from the Essen Seminar for Algebraic Geometry and Arithmetic (ESAGA) to the new research group SFB 1085 “Higher Invariants” at the University of Regensburg.

Check here for events, seminars, etc.