Belyi's theorem for complete intersections of general type

TL;DR

This paper characterizes which smooth complete intersections of general type over complex numbers can be defined over algebraic numbers, extending Belyi's theorem to higher dimensions using advanced geometric and arithmetic tools.

Contribution

It provides a Belyi-type criterion for complete intersections of general type, employing higher-dimensional boundedness, finiteness, and rigidity results.

Findings

Characterization of complete intersections definable over algebraic numbers

Extension of Belyi's theorem to higher-dimensional varieties

Use of advanced geometric and arithmetic techniques

Abstract

We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture for families of canonically polarized varieties, finiteness results for maps to varieties of general type, and rigidity theorems for Lefschetz pencils of complete intersections.