# Period mapping

A mapping which assigns to a point $s$ of the base $S$ of a family $\{ X _{s} \}$ of algebraic varieties over the field $\mathbf C$ of complex numbers the cohomology spaces $H ^{*} (X _{s} )$ of the fibre over this point, provided with a Hodge structure. The Hodge structure thus obtained is considered as a point in the moduli variety of Hodge structures of a given type.
Let $\{ X _{s} \}$ be the family of fibres $X _{s} = f ^{ {\ } -1} (s)$ of a smooth projective morphism $f: \ X \rightarrow S$, where $S$ is a smooth variety. The cohomology spaces $H ^{*} (X _{s} ,\ \mathbf Z ) = V _{ {\mathbf Z}}$ are then provided with a pure polarized Hodge structure, which is defined by a homomorphism of real algebraic groups (cf. Algebraic group) $h: \ \mathbf C ^{*} \rightarrow G _{ {\mathbf R}}$, where $\mathbf C ^{*}$ is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while $$G = \{ {g \in \mathop{\rm GL}\nolimits (V)} : {\psi (gx,\ gv) = \lambda (g) \psi (x,\ y)} \}$$ is the algebraic group of linear transformations of a space $V$ that multiply a non-singular (symmetric or skew-symmetric) bilinear form $\psi$ by a scalar factor; the automorphism $\mathop{\rm Ad}\nolimits \ h(i)$ of $G _{ {\mathbf R}}$ is thus a Cartan involution and $h( \mathbf R ^{*} )$ lies in the centre of $G _{ {\mathbf R}}$. The set $X _{G}$ of homomorphisms $h: \ \mathbf C ^{*} \rightarrow G _{ {\mathbf R}}$ which possess the above properties is naturally provided with the $G _{ {\mathbf R}}$- invariant structure of a homogeneous Kähler manifold and is called a Griffiths variety, while the quotient variety $M _{G} = X _{G} /G _{ {\mathbf Z}}$ is the moduli space of the Hodge structures. The homomorphism $h$ defines the Hodge decomposition $$\mathfrak G _{ {\mathbf C}} = \oplus \mathfrak G ^{p,-p}$$ of the Lie algebra $\mathfrak G$ of the group $G$, where $\mathfrak G ^{p,-p}$ is the subspace in $\mathfrak G _{ {\mathbf C}}$ on which $\mathop{\rm Ad}\nolimits \ h(z)$ operates by multiplication by $\overline{z} {} ^{p} z ^{-p}$. The assignment $h \rightarrow P(h)$, where $P(h)$ is the parabolic subgroup in $G _{ {\mathbf C}}$ with Lie algebra $\oplus _{ {p} \geq 0} \mathfrak G ^{p,-p}$, defines an open dense imbedding of the variety $X _{G}$ into the compact $G _{ {\mathbf C}}$- homogeneous flag manifold $X _{G}$. In the tangent space $$\mathfrak G _{ {\mathbf G}} / \oplus _ {p \geq 0} \mathfrak G ^{p,-p}$$ to $X _{G}$ at the point $h$, the horizontal subspace $$\oplus _ {p \geq -1} \mathfrak G ^{p,-p} / \oplus _ {p \geq 0} \mathfrak G ^{p,-p}$$ is distinguished. A holomorphic mapping into $X _{G}$ or $M _{G}$ is said to be horizontal if the image of its tangential mapping lies in a horizontal subbundle.
It has been established that the period mapping $\Phi : \ S \rightarrow M _{G}$ is horizontal (see , ). The singularities of period mappings are described by the Schmid nilpotent orbit theorem, which, when $S = \overline{S} \setminus \{ 0 \}$ is a curve with a deleted point, asserts that if $z$ is the local coordinate on $S$, $z(0) = 0$, then when $z \rightarrow 0$, $\Phi (z)$ is asymptotically close to $$\mathop{\rm exp}\nolimits \left ( \frac{ \mathop{\rm log}\nolimits \ z}{2 \pi i} N \ \right ) a,$$ where $a \in X _{G}$ and $N \in \mathfrak G _{ {\mathbf Q}}$ is a nilpotent element (see ). The image of the monodromy group $$\Phi _{*} ( \pi _{1} (S,\ s)) \subset G _{ {\mathbf Z}}$$ is semi-simple in every rational representation of the group $G$, while transference of $T$ around a divisor with normal intersections $\overline{S} \setminus S$ in a smooth compactification $\overline{S}$ of the variety $S$ generates quasi-unipotent elements $\Phi _{*} (T) \in G _{ {\mathbf Z}}$( i.e. elements which take roots of unity as eigen values). The importance of the monodromy group is underlined by the rigidity theorem (see , , ): If there are two families of algebraic varieties over $S$, then the relevant period mappings $\Phi _{1}$ and $\Phi _{2}$ from $S$ into $M _{G}$ coincide if and only if $\Phi _{1} (s _{0} ) = \Phi _{2} (s _{0} )$ at a certain point $s _{0} \in S$, and if the homomorphisms $\Phi _{i\star} : \ \pi _{1} (S,\ s _{0} ) \rightarrow G _{ {\mathbf Z}}$, $i = 1,\ 2$, coincide.
Complete results on the structure of the kernel and the image of a period mapping generally relate to the cases of curves and $K3$- surfaces (cf. $K3$- surface). If $\{ X _{s} \}$ is a family of varieties of the type indicated and $\Phi (s) = \Phi (s ^ \prime )$, then $X _{s} \widetilde \rightarrow X _{ {s} ^ \prime }$( Torelli's theorem), while for $K3$- surfaces the maximum possible image of the period mapping coincides with $M _{G}$( see ). In the case of curves, the image of the period mapping has been described partially (Schottky–Yung relations, see , ). The Griffiths conjecture states that a moduli variety permits a partial analytic compactification, i.e. an open imbedding in an analytic space $\overline{M} _{G}$ such that the period mapping $S \rightarrow M _{G}$ can be continued to a holomorphic mapping $\overline{S} \supset S$ for every smooth compactification $\overline{S} \supset S$. Such a compactification is known (1983) only for the case where $X _{G}$ is a symmetric domain .