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تسجيل مستخدم جديدPerverse sheaves on varieties with large fundamental groups
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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.
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Another introduction to perverse sheaves with some exercises. Expanded version of five lectures at the 2015 PCMI.
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