School of Mathematics Seminars and Lectures

`Laumon 1-motives and motives with modulus'

by Dr. Federico Binda (University of Regensburg, Germany)

Wednesday, February 7, 2018 from to (Asia/Kolkata)
at TIFR, Mumbai ( AG-77 )
Description
Abstract:
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.
Material: