Period map

Definition

Let $\varpi \colon \mathcal{X} \to B$ be a complex analytic family of compact complex manifolds, as in a deformation. Assume that $B$ is simply connected and that every fibre $X_t$ is a Kähler manifold. Fix $k \ge 0$, a base point $0 \in B$, and put $H = H^k(X_0, \C)$. By 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 $t$. By the Hodge decomposition, $H^k(X_t, \C)$ 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 $t$. The period map of the family is $$\mathcal{P} \colon B \to \mathcal{F}, \qquad t \mapsto \left(F^k H^k(X_t) \subseteq F^{k-1} H^k(X_t) \subseteq \cdots \subseteq F^1 H^k(X_t)\right).$$ Here each $F^p H^k(X_t)$ is viewed as a subspace of $H$, and $\mathcal{F}$ is the flag variety of flags in $H$ with dimensions $f^k \le \cdots \le f^1$. Its components $\mathcal{P}^p \colon t \mapsto F^p H^k(X_t)$ map to the Grassmannian of $f^p$-dimensional subspaces of $H$.