Cohomology of semisimple local systems and the Decomposition theorem

Abstract

In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we establish a generalization of Hodge-Riemann bilinear relations. For a semisimple local system on a smooth projective variety, we define a canonical isomorphism from the complex conjugate of its cohomology to the cohomology of the dual local system, which is a generalization of the classical Weil operator for pure Hodge structures. This isomorphism establishes a relation between the twisted Poincaré pairing, a purely topological object, and a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems.As an application, we give a new and geometric proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, without using the category of polarizable twistor D-modules.

$L^2$-complexes and twistor complexes of tame harmonic bundles

“…Though there are many related important works, we just mention two recent stimulating studies [43] and [49], where the previous works and the backgrounds are explained in detail. (See also an excellent survey paper [52] for the decomposition theorem of perverse sheaves of geometric origin, in particular the proof due to de Cataldo and Migliorini [6,7].…”

“…In [49], Wei and Yang obtained another proof of the Hard Lefschetz Theorem for semisimple local systems on projective manifolds with respect to a proper morphism between projective manifolds, which was originally proved by Sabbah [35]. They generalized the argument of de Cataldo and Migliorini in the Hodge case [6,7] to the twistor case, by following the idea of Simpson's Meta Theorem [48], instead using the theory of pure twistor D-modules.…”

“…I thank Maxim Kontsevich for a related stimulating question. My interest in Hard Lefschetz Theorem for pure twistor D-modules has been renewed by recent works due to Junchao Shentu and Chen Zhao [43] and Chuanhao Wei and Ruijie Yang [49] though this study is mathematically independent from theirs. I thank Jürgen Jost for answering my question.…”