# 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.