Perverse sheaves on varieties with large fundamental groups

TL;DR

This paper explores a conjecture that perverse sheaves on certain K"ahler manifolds have non-negative Euler characteristic, extending classical conjectures and verifying it under specific geometric conditions.

Contribution

It extends the Singer-Hopf conjecture to the K"ahler setting and proves the non-negativity of Euler characteristic for manifolds with certain curvature and local system conditions.

Findings

Euler characteristic expressed as an intersection number.

Non-negativity deduced from curvature conditions.

Conjecture verified for projective manifolds with specific local systems.

Abstract

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.