I might give a (informal) talk about Lefschetz hyperplane theorem, and its original proof, which uses very concrete description of the vanishing cycles and several explicit topological manipulations to obtain the Picard-Lefschetz formula. I learned this proof from the Lamotke's article "Topology of projective complex varieties after S. Lefschetz".
1) What would be the nice examples where everything can be computed explicitely ?
(Remark : I already know the family of elliptic curves $\{y^2 = x(x-1)(x-t)\}$ which is indeed instructive, I'm looking for other examples. I didn't find much in the second book by Voisin on Hodge theory)
Also, in the article no proof of the Hard Lefschetz theorem is given, only lot of equivalent statements.
2) Since then, did someone found a proof of the Hard Lefschetz theorem only using Lefschetz's original approach ?