Seminar on perverse sheaves and intersection cohomology

Talk 10: Proof part 2: The weight filtration and defect of semismallness (Marco Volpe)

Introduce the weight filtration [CM05, Theorem 4.5–4.6]. Define the defect of semismallness. Construct the universal hyperplane section and prove that it has smaller defect than the original map. Use this to prove the relative weak Lefschetz theorem [CM05, Section 4.7]. In the end, briefly explain the inductive procedure of the proof [CM05, Section 2.6] and for-mulate the other main Hodge-theoretic results [CM05, Theorems 2.1.4 – 2.1.8] that will be needed in the inductive proof.