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
,
[3]).
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
[4]).
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
,
[2],
[4]):
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
[7]).
In the case of curves, the image of
the period mapping has been described partially (Schottky–Yung relations, see
[6],
[8]).
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
[9].