In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules.
In this thesis defense, we would like to explain a simpler proof in the case of semisimple local systems, using a more geometric approach adapting de Cataldo- Migliorini. On the one hand, we complement Simpson’s theory of weights for local systems by proving a global invariant cycle theorem in the setting of local systems. On the other hand, we define a notion of polarization via Hermitian forms on pure twistor structures. This is partially based on joint work with Chuanhao Wei.
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