We conjecture that any perverse sheaf on a compact aspherical Kahler manifold has non-negative Euler characteristic. This extends the Singer-Hopf conjecture in the Kahler setting. We verify the stronger conjecture when the manifold X has non-positive
holomorphic bisectional curvature. We also show that the conjecture holds when X is projective and in possession of a faithful semi-simple rigid local system. The first result is proved by expressing the Euler characteristic as an intersection number involving the characteristic cycle, and then using the curvature conditions to deduce non-negativity. For the second result, we have that the local system underlies a complex variation of Hodge structure. We then deduce the desired inequality from the curvature properties of the image of the period map.
This is a sequel to Kodaira-Saito vanishing via Higgs bundles in positive characteristic (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for sem
istable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.
