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  "body": "\n## Definition\n\nLet $\\varpi \\colon \\mathcal{X} \\to B$ be a complex analytic family of compact complex manifolds, as in a [[deformation|VKQSM3BCE4]].\nAssume that $B$ is simply connected and that every fibre $X_t$ is a [[Kähler manifold|X60JG9AEQJ]].\nFix $k \\ge 0$, a base point $0 \\in B$, and put $H = H^k(X_0, \\C)$.\nBy Ehresmann's theorem the family is smoothly locally trivial, and since $B$ is simply connected this gives canonical isomorphisms $H^k(X_t, \\C) \\cong H$ for all $t$.\nBy the [[Hodge decomposition|HABCJ4Y0SR]], $H^k(X_t, \\C)$ has the Hodge filtration $F^p H^k(X_t) = \\bigoplus_{r \\ge p} H^{r, k-r}(X_t)$, whose dimensions $f^p$ do not depend on $t$.\nThe *period map* of the family is\n$$\\mathcal{P} \\colon B \\to \\mathcal{F}, \\qquad t \\mapsto \\left(F^k H^k(X_t) \\subseteq F^{k-1} H^k(X_t) \\subseteq \\cdots \\subseteq F^1 H^k(X_t)\\right).$$\nHere each $F^p H^k(X_t)$ is viewed as a subspace of $H$, and $\\mathcal{F}$ is the [[flag variety|NHRWCQQKTH]] of flags in $H$ with dimensions $f^k \\le \\cdots \\le f^1$.\nIts components $\\mathcal{P}^p \\colon t \\mapsto F^p H^k(X_t)$ map to the [[Grassmannian|CYPA882RB9]] of $f^p$-dimensional subspaces of $H$.\n\n## Remarks\n\nFor a base that is not simply connected, one defines the period map on the universal cover.\nAlternatively, it is a map to the quotient of a period domain by the monodromy group.\nWhen the fibres are projective with a fixed polarization, the period map on primitive cohomology takes values in a period domain of polarized [[Hodge structures|ZSM083M97C]].\n\nExample: the family $\\C/(\\Z + \\tau\\Z)$ over the upper half-plane, with $k = 1$.\nIdentify $H = H^1(X_\\tau, \\C)$ with $\\C^2$ by integrating over the loops $[0, 1]$ and $[0, \\tau]$.\nThen $F^1 = H^{1,0}$ is spanned by the class of $dz$, with periods $(1, \\tau)$, so $\\mathcal{P}(\\tau) = [1 : \\tau] \\in \\mathbb{P}^1$, which is injective.\n\nExample: a trivial family $X \\times B \\to B$ has a constant period map.\n",
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