Shafarevich's conjecture for families of hypersurfaces over function fields

Abstract: Shafarevich's conjecture suggests that over a fixed base scheme $B$, whether it is the $S$-integers of a number field or a quasiprojective variety, there should be only finitely many nonisotrivial families of projective varieties of a given type over $B$. For example, in proving the Mordell Conjecture, Faltings proved that there are only finitely many families of principally polarized abelian schemes of a given dimension over the $S$-integers of a number field. We prove a Shafarevich conjecture for Hodge-generic families of hypersurfaces for sufficiently large degree and dimension over a complex quasiprojective base. The argument follows a "boundedness and rigidity" structure to show that the space of such families is finite. For boundedness, the key input is a new result of Bakker, Brunebarbe, and Tsimerman about the ampleness of the Griffiths line bundle for quasifinite period mappings. For rigidity, we use a Hodge-theoretic formulation due to Peters.

This is joint work with Philip Engel and Salim Tayou.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar