Shafarevich's conjecture for families of hypersurfaces over function fields

TL;DR

This paper proves finiteness results for families of hypersurfaces over complex algebraic varieties, extending Shafarevich's conjecture to higher-dimensional cases and more general moduli spaces.

Contribution

It establishes the finiteness of Hodge-generic non-isotrivial families of hypersurfaces over complex varieties, including uniform bounds and examples of sharpness, and extends to complete intersections and varieties with certain period maps.

Findings

Finiteness of families over complex varieties.

Uniform bounds in the base variety.

Examples demonstrating sharpness of results.

Abstract

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}_{\mathbb C}^{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}_{\mathbb C}^{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem.