Differential Geometry Seminar: Boundedness problems in algebraic geometry and their consequences

In 1960’s Shafarevich asked a simple question: Do families of curves of genus at least 2 (over a fixed base) have finite number of deformation classes?

Soon after Parshin gave an affirmative answer to this question and furthermore showed that this finiteness property is precisely the geometric underpinning of Mordell’s famous conjecture regarding finiteness of rational points for projective curves of genus at least 2 (over number fields). It was this very observation that led Faltings to his complete proof of Mordell’s conjecture in 1980s.

For higher dimensional analogues of curves of high genus, Shafarevich’s problem was settled in 2010 by Kovács and Lieblich, partially thanks to multiple advances in moduli theory of the so-called stable varieties. In this talk I will present this circle of ideas and then focus on our recent solution to Shafarevich’s question for other higher dimensional (non-stable) varieties, e.g. the Calabi-Yau case. This is based on joint work with Kenneth Ascher (UC Irvine).

School of Computer and Mathematical Sciences