Talk 8: Statement and some consequences of the decomposition theorem

State [CM05, Theorem 2.1.1] and explain its generalisation to the singular case in [CM05, Section 2.3]. Also compare with the classical Hard Lef-schetz theorem. Use the decomposition theorem to show that intersection cohomology is a direct summand of the cohomology of a resolution [Dim04, Corollary 5.4.11]. Also discuss the concrete example of a threefold [CM05, Section 2.4].