|Location:||MSRI: Simons Auditorium|
Though Shafarevich conjecture is not a conjecture (anymore), we are still using this name for the theorem proven by
Parshin (1968) and Arakelov (1971). It includes two statments. First, that over a given smooth curve there are only finitely many
non-isotrivial families of smooth projective curves of general type with fixed genus. Second, that if the base curve has positive
dimensional automorphism group then there are no such families at all. In the last decade couple of higher dimensional generalizations have been shown. My plan is twofold: I present parts of the proof of the original statement for compact base curves, emphasizing the ideas which survived to the higher dimensional cases and I also review the recent developments focusing on the rigidity of higher dimensional families.