Harmonic bundle and pure twistor D-module. Lecture 4

The classical theorem of Corlette says that there is a correspondence between semisimple flat bundles and harmonic bundles on a smooth projective variety. Rather recently, it has been generalized to the correspondence between polarizable pure twistor D-modules and semisimple holonomic D-modules. It enables us to use techniques in global analysis for the study on D-modules. As a remarkable application, as conjectured by Kashiwara, we obtain that a projective push-forward preserves semisimplicity of holonomic D-modules, and that a decomposition theorem holds for semisimple holonomic D-modules.
The plan of my lecture is as follows:

• 1. Introduction of harmonic bundle
• 2. Asymptotic behaviour around singularity
• 3. Kobayashi-Hitchin correspondence
• 4. Good formal structure and Stokes structure of meromorphic flat bundles
• 5. Twistor structure and Simpson's meta-theorem
• 6. Introduction to polarizable pure twistor D-module