# Rigidity of Spreadings and Fields of Definition

## TL;DR

This paper explores the rigidity of algebraic varieties, especially those without deformations, over number fields, linking geometric properties with Hodge theory and arithmetic aspects, and providing new insights and proofs.

## Contribution

It offers a largely self-contained exposition connecting rigidity, Hodge structures, and fields of definition, with new proofs of key results.

## Key findings

- Varieties without deformations are defined over number fields.
- Rigidity relates to maximal Higgs fields from variations of Hodge structure.
- Provides new proofs and a comprehensive overview of related concepts.

## Abstract

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u \i\ curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. Andr\'e and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results.