A short note on infinity-groupoids and the period map for projective manifolds

TL;DR

This paper demonstrates how the concept of infinity-groupoids can clarify classical results in algebraic geometry, particularly regarding the infinitesimal behavior of the periods map of projective manifolds, making complex ideas more intuitive.

Contribution

It shows that a naive understanding of infinity-groupoids provides natural insights into classical theorems in algebraic geometry, simplifying their conceptual foundations.

Findings

Reveals the natural role of Cartan homotopy in the context of infinity-groupoids.

Provides a new perspective on Griffiths' differential of the periods map.

Simplifies proofs of the Kodaira principle, Bogomolov-Tian-Todorov theorem, and Goldman-Millson theorem.

Abstract

A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths' expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.