---
tag: YGNXQVTWP3
kind: concept
name: Period map
msc: [32G20, 14D07]
sources:
  - url: https://en.wikipedia.org/wiki/Period_mapping
same_as:
  wikidata: Q7168619
  wikipedia: Period mapping
---

## Summary

The period map of a family of compact [[Kähler manifolds|X60JG9AEQJ]] records how the [[Hodge structure|ZSM083M97C]] on the cohomology of the fibres moves.
All fibres have the same cohomology group, and only the position of the Hodge filtration inside it changes.
So the period map goes from the base of the family to a space of flags, a period domain.
Griffiths showed that it is holomorphic and satisfies a transversality condition.
Classical periods of elliptic curves and abelian integrals are the simplest case.

## Explanations

### L1 human

When the complex structure of a manifold changes, its topology and so its cohomology stay the same.
What changes is how the cohomology splits into pieces of different types $(p, q)$.
The period map follows these pieces as subspaces of one fixed vector space.
For the elliptic curve $\C/(\Z + \tau\Z)$, the holomorphic form $dz$ has periods $1$ and $\tau$ along the two basic loops, and the period map is $\tau \mapsto [1 : \tau]$.