# Period mapping

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A mapping which assigns to a point of the base of a family of algebraic varieties over the field of complex numbers the cohomology spaces 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.

The study of period mappings dates back to the studies of N.H. Abel and C.G.J. Jacobi on integrals of algebraic functions (see Abelian differential). However, until recently, the only period mappings that have been studied were those which correspond to families of curves.

Let be the family of fibres of a smooth projective morphism , where is a smooth variety. The cohomology spaces are then provided with a pure polarized Hodge structure, which is defined by a homomorphism of real algebraic groups (cf. Algebraic group) , where is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while is the algebraic group of linear transformations of a space that multiply a non-singular (symmetric or skew-symmetric) bilinear form by a scalar factor; the automorphism of is thus a Cartan involution and lies in the centre of . The set of homomorphisms which possess the above properties is naturally provided with the -invariant structure of a homogeneous Kähler manifold and is called a Griffiths variety, while the quotient variety is the moduli space of the Hodge structures. The homomorphism defines the Hodge decomposition of the Lie algebra of the group , where is the subspace in on which operates by multiplication by . The assignment , where is the parabolic subgroup in with Lie algebra , defines an open dense imbedding of the variety into the compact -homogeneous flag manifold . In the tangent space to at the point , the horizontal subspace is distinguished. A holomorphic mapping into or 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 is horizontal (see , ). The singularities of period mappings are described by the Schmid nilpotent orbit theorem, which, when is a curve with a deleted point, asserts that if is the local coordinate on , , then when , is asymptotically close to where and is a nilpotent element (see ). The image of the monodromy group is semi-simple in every rational representation of the group , while transference of around a divisor with normal intersections in a smooth compactification of the variety generates quasi-unipotent elements (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 , then the relevant period mappings and from into coincide if and only if at a certain point , and if the homomorphisms , , coincide.

Complete results on the structure of the kernel and the image of a period mapping generally relate to the cases of curves and -surfaces (cf. -surface). If is a family of varieties of the type indicated and , then (Torelli's theorem), while for -surfaces the maximum possible image of the period mapping coincides with (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 such that the period mapping can be continued to a holomorphic mapping for every smooth compactification . Such a compactification is known (1983) only for the case where is a symmetric domain .

How to Cite This Entry:
Period mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Period_mapping&oldid=16699
This article was adapted from an original article by A.I. Ovseevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article