Perverse sheaves on varieties with large fundamental groups
Donu Arapura, Botong Wang
TL;DR
This work extends the Singer–Hopf conjecture to perverse sheaves on aspherical compact Kähler manifolds by proving nonnegativity of Euler characteristics under curvature and Hodge-theoretic hypotheses. The authors develop a framework based on Kashiwara’s index theorem, interpreting $\chi(X,{\mathcal P})$ as an intersection number of the characteristic cycle with the zero section of $T^*X$, and employ curvature positivity to obtain nonnegativity. They show nonnegativity for manifolds with nonpositive holomorphic bisectional curvature, with strict positivity when the cotangent bundle is ample, and extend the result to projective manifolds possessing a cohomologically rigid, almost faithful semisimple local system underlying a CVHS, via a period-map reduction. The technique blends period-domain geometry, horizontal subvarieties, and nef bundle positivity, and applies to Shimura-type varieties and other large fundamental group examples, illustrating broad applicability of the conjecture in the Kähler and algebraic contexts.
Abstract
We conjecture that any perverse sheaf on a compact aspherical Kähler manifold has non-negative Euler characteristic. This extends the Singer-Hopf conjecture in the Kähler 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.