Arbeitsgemeinschaft: Twistor D-Modules and the Decomposition Theorem

Abstract

The purpose of this Arbeitsgemeinschaft is to introduce the notion of twistor $\mathcal{D}$-modules and their main properties. The guiding principle leading this discussion is Simpson's "meta-theorem", which gives a heuristic for generalizing (mixed) Hodge-theoretic results into (mixed) twistor-theoretic results. The strength of the twistor approach is that it enables to enlarge the scope of Hodge theory not only to arbitrary semi-simple perverse sheaves, equivalently semi-simple regular holonomic $\mathcal{D}$-modules via the Riemann-Hilbert correspondence, but also to possibly semi-simple irregular holonomic $\mathcal{D}$-modules. An overarching goal for this session is Mochizuki's proof of the decomposition theorem for semi-simple holonomic $\mathcal{D}$-modules on a smooth complex projective variety, first conjectured by Kashiwara in 1996.