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  "body": "\n## Summary\n\nThe period map of a family of compact [[Kähler manifolds|X60JG9AEQJ]] records how the [[Hodge structure|ZSM083M97C]] on the cohomology of the fibres moves.\nAll fibres have the same cohomology group, and only the position of the Hodge filtration inside it changes.\nSo the period map goes from the base of the family to a space of flags, a period domain.\nGriffiths showed that it is holomorphic and satisfies a transversality condition.\nClassical periods of elliptic curves and abelian integrals are the simplest case.\n\n## Explanations\n\n### L1 human\n\nWhen the complex structure of a manifold changes, its topology and so its cohomology stay the same.\nWhat changes is how the cohomology splits into pieces of different types $(p, q)$.\nThe period map follows these pieces as subspaces of one fixed vector space.\nFor 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 $\\tau \\mapsto [1 : \\tau]$.\n",
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