Swann Tubach: Higher enhancements of mixed Hodge modules

Abstract:

Let X be a smooth complex algebraic variety. Deligne proved that the isomorphism between singular and de Rham cohomology of X gives 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 enhancement has useful applications, such as a Hodge realisation of Voevodsky motives commuting with the 6 operations and an extension to stacks and simplicial schemes of the formalism of mixed Hodge modules.