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

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

2009-11-19

arxiv.org/abs/0911.3845v2

Mathematics

Algebraic Geometry

11 pages; uses xy-pic; a few details, references, and a proof of Goldman-Millson quasi-abelianity theorem added

Scientific paper

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.

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