Abstract

Varieties equipped with a regular function admit interesting “exponential” cohomology theories: rapid decay cohomology, twisted de Rham cohomology in characteristic 0, twisted l-adic cohomology in positive characteristic. They exhibit motivic-like properties – weights, a kind of Hodge filtration, a period isomorphism – but do not fit into the classical theory of motives. Building on ideas of Kontsevich-Soibelman and Fresán-Jossen, we construct triangulated categories of exponential Voevodsky motives equipped with functors realising exponential cohomology theories. More generally, we associate to any “six operation formalism” an exponential version. Unlike classical motivic sheaf theories, these exponential sheaf theories come with a built-in Fourier-Deligne transform, which plays a key role in the construction of exponential realisations. This is joint work in progress with Javier Fresán and Martin Gallauer.