Orateur
Description
Let X be a complex variety. In Deligne’s work on Hodge theory, he explained how to use the algebraic structure of X to endow the rational Betti cohomology H∗(X(C); Q) with a functorial increasing weight filtration. Deligne’s construction of the weight filtration is quite ingenious, and it isn’t a priori clear if the weight filtration can be refined to a filtration on the integral Betti cohomology H∗(X(C); Z). Much later, Gillet and Soulé showed that this is the case. In this talk, we’ll explain joint work with Piotr Pstrągowski that shows that for any complex-orientable cohomology theory A∗, there is a functorial weight filtration on A∗(X(C)) that recovers the Deligne and Gillet–Soulé filtrations in the cases of rational and integral cohomology. To do this, we give a new description of the the ∞-category SH(C) of motivic spectra over C as sheaves on a subcategory of compact pure motives; this generalizes Bondarko’s weight structure on MGL-modules. From this description we also refine Betti realization to a functor landing in Pstrągowski’s ∞-category of synthetic spectra, extending a construction of Pstrągowski for cellular motivic spectra to all motivic spectra.