2021/09/23 by Wei, Chuanhao, Yang, Ruijie
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 14C30 #Representation Theory (math.RT) #Secondary: 32S60
paper · doi:10.48550/arxiv.2109.11578
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.