Recall the following definition.
Definition 1. A pure $\mathbb{Z}$/$\mathbb{Q}$/$\mathbb{R}$-Hodge structure of weight $w$ is the data of
- a finite free $\mathbb{Z}$/$\mathbb{Q}$/$\mathbb{R}$-module $V$,
- a decomposition $V_\mathbb{C} = \bigoplus_{p+q=w} V^{p,q}$ satisfying $V^{p,q} = \bar{V}^{q,p}$.
By Hodge theory, for $X$ a compact Kähler manifold, the cohomology group $H^w(X, \mathbb{Z})/\mathrm{torsion}$ has a natural structure of a Hodge structure. We note that the information of the decomposition $V^{p,q}$ is equivalent to the information of the Hodge filtration $$ F^p V_\mathbb{C} = \bigoplus_{p^\prime \ge p} V^{p^\prime,w-p^\prime}, $$ because we can intersect $F^p V_\mathbb{C}$ with the complex conjugate of $F^{w+1-p} V_\mathbb{C}$ to recover $V^{p,w-p}$.
Variations of Hodge structures#
We now want make a definition that works in families.
Definition 2. Let $X$ be a complex manifold. A polarized $\mathbb{Q}$-variation of Hodge structure of weight $w$ consists of data of
- a $\mathbb{Q}$-local system $V$ on $X$,
- a holomorphic vector bundle with a connection $(\mathcal{V}, \nabla)$,
- a Hodge filtration $F^\bullet \mathcal{V}$ by holomorphic subbundles,
- a $\mathbb{Q}$-bilinear form $Q \colon V \otimes_\mathbb{Q} V \to \mathbb{Q}(-w)$ (where we denote $\mathbb{Q}(1) = 2 \pi i \mathbb{Q}$),
satisfying the condition that
- $\mathcal{V}^{\nabla = 0} = V \otimes_\mathbb{Q} \mathbb{C}$,
- for every $x \in X$, $(\mathcal{V}_ x, F^\bullet \mathcal{V}_ x, V, Q)$ is a pure Hodge structure of weight $w$,
- it satisfies Griffiths transversality, i.e., that $\nabla(F^p \mathcal{V}) \subseteq \Omega_X^1 \otimes F^{p-1}$.
The motivation comes from a smooth family of varieties. Let $f \colon Y \to X$ be a smooth projective morphism between smooth varieties, or more generally a proper submersion of Kähler manifolds. In this setting, we have that $$ H^k = R^k f_\ast \mathbb{Q} $$ is a local system on $X$, because Ehresmann’s theorem implies that topologically $f$ is a locally trivial fiber bundle.
Remark 3. If we use something other coefficient, this is also true algebraically with the étale topology. This is because properness implies that it is constructible and smoothness implies that it is locally acyclic.
This local system corresponds to the de Rham vector bundle, and the de Rham filtration satisfies Griffiths transversality.
Period maps#
We provide a sketch of why this defines a variation of Hodge structures.
Fix a point $0 \in U$ in a contractible open $U$. It can be shown that $b_{p,k} = \dim F^p H^k(Y_x, \mathbb{C})$ is locally constant, in particular on all of $U$.
Definition 4. The period map is defined as $$ \mathcal{P}^{p,k} \colon U \to \mathrm{Gr}(H^k(Y_0, \mathbb{C}), b_{p,k}); \quad x \mapsto F^p H^k(Y_x, \mathbb{C}). $$
Theorem 5 (Griffiths, 1968). The map $\mathcal{P}^{p,k}$ is holomorphic.
To prove this, one first checks that $\mathcal{P}^{p,k}$ is smooth, and then proves that the derivative is $\mathbb{C}$-linear. This computation will also imply the following fact.
Corollary 6. The image of $d\mathcal{P}^{p,k}$ lies in $\Hom(F^p, F^{p-1}/F^p) \subseteq \Hom(F^p, H^k/F^p)$.
We can also put the period maps together and consider $$ \mathcal{P}^k \colon U \to \mathrm{Fl}_ {b^\bullet}(H^k(Y_0, \mathbb{C})); \quad x \mapsto (\mathcal{P}^{b_1,k}(x), \dotsc, \mathcal{P}^{b_b,k}(x)), $$ where $b_\bullet = (b_1, \dotsc, b_b)$. If we impose the condition that $F^p H^k \oplus \overline{F^{k-p+1} H^k} = H^k$, this defines an open subset $$ \mathcal{D} \subseteq \mathrm{Fl}_ {b^\bullet}. $$
Definition 7. We call this $\mathcal{D}$ the (unpolarized) period domain.
Polarizations#
Assume that $(Y, \omega)$ is a compact Kähler manifold, $f \colon Y \to X$ is a proper submersion, and that $\omega \vert_{x \in X}$ is a Kähler manifold for each $x$. There is a Lefschetz map $$ L \colon R^k f_\ast \mathbb{C} \to R^{k+2} f_\ast \mathbb{C} $$ that is just wedging with $\omega$.
Definition 8. We define $$ (R^k f_\ast \mathbb{C})_ \mathrm{prim} = \ker L^{n-k+1}, $$ where $n$ is the relative dimension of $f$.
There there is the Lefschetz decomposition $$ H^k = R^k f_\ast \mathbb{C} = \bigoplus_{2k-2n \le 2r \le k} L^r H^{k-2r}_ \mathrm{prim}, $$ and also a bilinear form $$ Q(\alpha, \beta) = \langle L^{n-k} \alpha, \beta \rangle $$ on $H^k$. This has the property that $$ F^p H^k = (F^{k-p+1} H^k)^\perp, \quad H^k_\mathrm{prim} = \bigoplus_{p+q=k} H^{p,q}(Y_0, \mathbb{C})_ \mathrm{prim}.$$ On this primitive piece, we have $$ (-1)^{k(k-1)/2} i^{p-q} Q(\alpha, \bar{\alpha}) \gt 0 $$ for $0 \neq \alpha \in H^{p,q}_ \mathrm{prim}$.
Definition 9. There is a further complex submanifold $\mathcal{D}^\mathrm{pol} \subseteq \mathcal{D}$ satisfying these additional properties with respect to $Q$.
We can now define the holomorphic vector bundles $F^p \mathcal{H}_ k$ by pulling back the universal vector bundles on $\mathrm{Fl}_ {b_\bullet}$ along the holomorphic period map $\mathcal{P}$.
Theorem 10. All of this define a variation of polarized $\mathbb{Q}$-Hodge structures on $R^k f_\ast \mathbb{Q}$.