Talk 6: Intersection cohomology 3: Poincar´e duality (Klaus Mattis)

Prove the support and cosupport conditions for perverse sheaves [Dim04,
Proposition 5.1.16], [KS90, Proposition 10.2.4 and 10.2.5]. Recall Verdier duality for the six functor formalism of sheaves on topological spaces and the duality functor DX = hom(−, ωX). Explain how duality interchanges j∗ and j!. Use the support and cosupport conditions to show how DX interchanges the p-connective and q-coconnective part of the perverse t-structure for complementary perversities p and q(n) = −n − p(n). From this, prove that DXICp(L) ≃ ICq(L∨), where L∨ is the dual local system of L and deduce Poincar´e duality for intersection cohomology. Also see [Ban07, Theorem 4.4.1].