9 September 2024 – 14.30 Aula Dottorato
Swann Tubach (ENS Lyon)

Higher enhancements of mixed Hodge modules

Let X be a smooth complex algebraic variety. Deligne proved that the isomorphism between singular and de Rham cohomology of X give rise to a mixed Hodge structure, which led to a very fruitful area of research. M. Saito constructed a theory of Mixed Hodge modules, which is a sheafy version of mixed Hodge structures, and in which the complex of mixed Hodge structures computing the cohomology of X is naturally a derived pushforward of an object living on X, as in ell-adic cohomology or singular cohomology. We prove that the constructions of Saito (that is the derived category of mixed Hodge modules together with the 6 sheaf operations) can be lifted to the world of infinity-categories in a very coherent way. This is done using a « change of variables »: adapting a proof of Nori, we can prove that the derived category of (perverse) mixed Hodge modules is also the derived category of constructible Hodge modules. This enhancement has useful applications, such as a Hodge realisation of motives commuting with the 6 operations and an extension to stacks of the formalism of mixed Hodge modules. This also applies to perverse Nori motives.