  We conjecture that any perverse sheaf on a compact aspherical K\"ahler
manifold has non-negative Euler characteristic. This extends the Singer-Hopf
conjecture in the K\"ahler 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.
