Cohomology of semisimple local systems and the Decomposition theorem

TL;DR

This paper extends classical Hodge theory to semisimple local systems, establishing new bilinear relations, a canonical isomorphism, and a proof of Sabbah's Decomposition Theorem using geometric methods.

Contribution

It generalizes Hodge-Riemann relations and provides a geometric proof of Sabbah's Decomposition Theorem for semisimple local systems.

Findings

Established a generalization of Hodge-Riemann bilinear relations.

Defined a canonical isomorphism relating cohomologies of local systems.

Provided a new geometric proof of Sabbah's 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 Poincar\'e 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.