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A mapping which assigns to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721401.png" /> of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721402.png" /> of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721403.png" /> of algebraic varieties over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721404.png" /> of complex numbers the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721405.png" /> of the fibre over this point, provided with a [[Hodge structure|Hodge structure]]. The Hodge structure thus obtained is considered as a point in the moduli variety of Hodge structures of a given type.
+
{{TEX|done}}
 +
 
 +
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|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|Abelian differential]]). However, until recently, the only period mappings that have been studied were those which correspond to families of curves.
 
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|Abelian differential]]). However, until recently, the only period mappings that have been studied were those which correspond to families of curves.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721406.png" /> be the family of fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721407.png" /> of a smooth projective morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721408.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p0721409.png" /> is a smooth variety. The cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214010.png" /> are then provided with a pure polarized Hodge structure, which is defined by a homomorphism of real algebraic groups (cf. [[Algebraic group|Algebraic group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214012.png" /> is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while
+
Let $  \{ X _{s} \} $
 
+
be the family of fibres $  X _{s} = f ^{ {\ } -1} (s) $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214013.png" /></td> </tr></table>
+
of a smooth projective morphism $  f: \  X \rightarrow S $,  
 
+
where $  S $
is the algebraic group of linear transformations of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214014.png" /> that multiply a non-singular (symmetric or skew-symmetric) bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214015.png" /> by a scalar factor; the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214017.png" /> is thus a Cartan involution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214018.png" /> lies in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214019.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214020.png" /> of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214021.png" /> which possess the above properties is naturally provided with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214022.png" />-invariant structure of a homogeneous [[Kähler manifold|Kähler manifold]] and is called a Griffiths variety, while the quotient variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214023.png" /> is the moduli space of the Hodge structures. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214024.png" /> defines the Hodge decomposition
+
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|Algebraic group]]) $  h: \  \mathbf C ^{*} \rightarrow G _{ {\mathbf R}} $,  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214025.png" /></td> </tr></table>
+
where $  \mathbf C ^{*} $
 
+
is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while $$
of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214026.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214028.png" /> is the subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214029.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214030.png" /> operates by multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214031.png" />. The assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214033.png" /> is the parabolic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214034.png" /> with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214035.png" />, defines an open dense imbedding of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214036.png" /> into the compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214037.png" />-homogeneous flag manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214038.png" />. In the tangent space
+
=   \{ {g \in  \mathop{\rm GL}\nolimits (V)} : {\psi (gx,\  gv) = \lambda (g) \psi (x,\  y)} \}
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214039.png" /></td> </tr></table>
+
is the algebraic group of linear transformations of a space $  V $
 
+
that multiply a non-singular (symmetric or skew-symmetric) bilinear form $  \psi $
to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214040.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214041.png" />, the horizontal subspace
+
by a scalar factor; the automorphism $  \mathop{\rm Ad}\nolimits \  h(i) $
 
+
of $  G _{ {\mathbf R}} $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214042.png" /></td> </tr></table>
+
is thus a Cartan involution and $  h( \mathbf R ^{*} ) $
 
+
lies in the centre of $  G _{ {\mathbf R}} $.  
is distinguished. A holomorphic mapping into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214043.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214044.png" /> is said to be horizontal if the image of its tangential mapping lies in a horizontal subbundle.
+
The set $  X _{G} $
 
+
of homomorphisms $  h: \  \mathbf C ^{*} \rightarrow G _{ {\mathbf R}} $
It has been established that the period mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214045.png" /> is horizontal (see , [[#References|[3]]]). The singularities of period mappings are described by the Schmid nilpotent orbit theorem, which, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214046.png" /> is a curve with a deleted point, asserts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214047.png" /> is the local coordinate on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214049.png" />, then when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214051.png" /> is asymptotically close to
+
which possess the above properties is naturally provided with the $  G _{ {\mathbf R}} $-
 
+
invariant structure of a homogeneous [[Kähler manifold|Kähler manifold]] and is called a Griffiths variety, while the quotient variety $  M _{G} = X _{G} /G _{ {\mathbf Z}} $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214052.png" /></td> </tr></table>
+
is the moduli space of the Hodge structures. The homomorphism $  h $
 
+
defines the Hodge decomposition $$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214054.png" /> is a nilpotent element (see [[#References|[4]]]). The image of the monodromy group
+
\mathfrak G _{ {\mathbf C}}  =   \oplus \mathfrak G ^{p,-p}
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214055.png" /></td> </tr></table>
+
of the Lie algebra $  \mathfrak G $
 +
of the group $  G $,
 +
where  $  \mathfrak G ^{p,-p} $
 +
is the subspace in $  \mathfrak G _{ {\mathbf C}} $
 +
on which $  \mathop{\rm Ad}\nolimits \  h(z) $
 +
operates by multiplication by $  \overline{z}  {} ^{p} z ^{-p} $.  
 +
The assignment $  h \rightarrow P(h) $,  
 +
where $  P(h) $
 +
is the parabolic subgroup in $  G _{ {\mathbf C}} $
 +
with Lie algebra $  \oplus _{ {p} \geq 0} \mathfrak G ^{p,-p} $,  
 +
defines an open dense imbedding of the variety $  X _{G} $
 +
into the compact $  G _{ {\mathbf C}} $-
 +
homogeneous flag manifold $  X _{G} $.  
 +
In the tangent space $$
 +
\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.
  
is semi-simple in every rational representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214056.png" />, while transference of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214057.png" /> around a divisor with normal intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214058.png" /> in a smooth compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214059.png" /> of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214060.png" /> generates quasi-unipotent elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214061.png" /> (i.e. elements which take roots of unity as eigen values). The importance of the monodromy group is underlined by the rigidity theorem (see , [[#References|[2]]], [[#References|[4]]]): If there are two families of algebraic varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214062.png" />, then the relevant period mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214064.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214065.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214066.png" /> coincide if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214067.png" /> at a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214068.png" />, and if the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214070.png" />, coincide.
+
It has been established that the period mapping  $  \Phi : \  S \rightarrow M _{G} $
 +
is horizontal (see , [[#References|[3]]]). 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 [[#References|[4]]]). 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 , [[#References|[2]]], [[#References|[4]]]): 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214071.png" />-surfaces (cf. [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214072.png" />-surface]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214073.png" /> is a family of varieties of the type indicated and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214074.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214075.png" /> (Torelli's theorem), while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214076.png" />-surfaces the maximum possible image of the period mapping coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214077.png" /> (see [[#References|[7]]]). In the case of curves, the image of the period mapping has been described partially (Schottky–Yung relations, see [[#References|[6]]], [[#References|[8]]]). The Griffiths conjecture states that a moduli variety permits a partial analytic compactification, i.e. an open imbedding in an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214078.png" /> such that the period mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214079.png" /> can be continued to a holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214080.png" /> for every smooth compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214081.png" />. Such a compactification is known (1983) only for the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214082.png" /> is a symmetric domain [[#References|[9]]].
+
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| $  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 [[#References|[7]]]). In the case of curves, the image of the period mapping has been described partially (Schottky–Yung relations, see [[#References|[6]]], [[#References|[8]]]). 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 [[#References|[9]]].
  
 
====References====
 
====References====
Line 39: Line 113:
  
 
====Comments====
 
====Comments====
In the one-variable case, W. Schmid also proved a very precise description of the asymptotics of the period mapping, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072140/p07214083.png" />-orbit theorem. This was generalized to the several-variables case by E. Cattani and A. Kaplan [[#References|[a1]]]. The period mapping and period domain also have been considered in the case of singularities [[#References|[a1]]], [[#References|[a2]]]. See also [[Variation of Hodge structure|Variation of Hodge structure]].
+
In the one-variable case, W. Schmid also proved a very precise description of the asymptotics of the period mapping, the $  \mathop{\rm SL}\nolimits _{2} $-
 +
orbit theorem. This was generalized to the several-variables case by E. Cattani and A. Kaplan [[#References|[a1]]]. The period mapping and period domain also have been considered in the case of singularities [[#References|[a1]]], [[#References|[a2]]]. See also [[Variation of Hodge structure|Variation of Hodge structure]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Cattani, A. Kaplan, "Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure" ''Invent. Math.'' , '''67''' (1982) pp. 101–115 {{MR|0664326}} {{ZBL|0516.14005}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Looijenga, "A period mapping for certain semi-universal deformations" ''Compos. Math.'' , '''30''' (1975) pp. 299–316 {{MR|0376666}} {{ZBL|0312.14006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.N. Varchenko, "Mapping of periods and intersection form" ''Funct. Anal. Appl.'' , '''16''' : 2 (1982) pp. 83–93 ''Funkts. Anal. Prilozhen.'' , '''16''' : 2 (1982) pp. 7–20 {{MR|}} {{ZBL|0513.32018}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Peters, J.H.M. Steenbrink, "Infinitesimal variations of Hodge structures" K. Ueno (ed.) , ''Classical Algebraic and Analytic Manifolds'' , Birkhäuser (1984) pp. 399–463 {{MR|0728615}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Griffiths, "Variation of Hodge structures" P. Griffiths (ed.) , ''Topics in Transcendental Algebraic geometry'' , Princeton Univ. Press (1984) pp. 3–28 {{MR|0756854}} {{MR|0756846}} {{MR|0756845}} {{MR|0756843}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Griffiths, W. Schmid, "Recent developments in Hodge theory" , ''Discrete Subgroups of Lie Groups and Applications to Moduli'' , Oxford Univ. Press (1975) pp. 31–128 {{MR|0419850}} {{ZBL|0355.14003}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Griffiths, "Some transcendental aspects of algebraic geometry" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1975) pp. 3–110 {{MR|0419438}} {{ZBL|0309.14007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Carlson, N.Green, P. Griffiths, J. Harris, "Infinitesimal variations of Hodge structure I" ''Compos. Math.'' , '''50''' (1983) pp. 109–205 {{MR|0720288}} {{MR|0720289}} {{MR|0720290}} {{ZBL|0531.14006}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Donagi, "Generic Torelli for hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353 {{MR|720291}} {{ZBL|0598.14007}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Green, "The period maps for hypersurface sections of high degree of an algebraic variety" ''Compos. Math.'' , '''55''' (1984) pp. 135–156 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> D.A. Cox, "Generic Torelli and infinitesimal variation of Hodge structure" S.J. Bloch (ed.) , ''Algebraic geometry (Bowdoin, 1985)'' , ''Proc. Symp. Pure Math.'' , '''46''' , Amer. Math. Soc. (1987) pp. 235–246 {{MR|0927983}} {{ZBL|0637.14008}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Cattani, A. Kaplan, "Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure" ''Invent. Math.'' , '''67''' (1982) pp. 101–115 {{MR|0664326}} {{ZBL|0516.14005}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Looijenga, "A period mapping for certain semi-universal deformations" ''Compos. Math.'' , '''30''' (1975) pp. 299–316 {{MR|0376666}} {{ZBL|0312.14006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.N. Varchenko, "Mapping of periods and intersection form" ''Funct. Anal. Appl.'' , '''16''' : 2 (1982) pp. 83–93 ''Funkts. Anal. Prilozhen.'' , '''16''' : 2 (1982) pp. 7–20 {{MR|}} {{ZBL|0513.32018}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Peters, J.H.M. Steenbrink, "Infinitesimal variations of Hodge structures" K. Ueno (ed.) , ''Classical Algebraic and Analytic Manifolds'' , Birkhäuser (1984) pp. 399–463 {{MR|0728615}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Griffiths, "Variation of Hodge structures" P. Griffiths (ed.) , ''Topics in Transcendental Algebraic geometry'' , Princeton Univ. Press (1984) pp. 3–28 {{MR|0756854}} {{MR|0756846}} {{MR|0756845}} {{MR|0756843}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Griffiths, W. Schmid, "Recent developments in Hodge theory" , ''Discrete Subgroups of Lie Groups and Applications to Moduli'' , Oxford Univ. Press (1975) pp. 31–128 {{MR|0419850}} {{ZBL|0355.14003}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Griffiths, "Some transcendental aspects of algebraic geometry" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1975) pp. 3–110 {{MR|0419438}} {{ZBL|0309.14007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Carlson, N.Green, P. Griffiths, J. Harris, "Infinitesimal variations of Hodge structure I" ''Compos. Math.'' , '''50''' (1983) pp. 109–205 {{MR|0720288}} {{MR|0720289}} {{MR|0720290}} {{ZBL|0531.14006}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Donagi, "Generic Torelli for hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353 {{MR|720291}} {{ZBL|0598.14007}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Green, "The period maps for hypersurface sections of high degree of an algebraic variety" ''Compos. Math.'' , '''55''' (1984) pp. 135–156 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> D.A. Cox, "Generic Torelli and infinitesimal variation of Hodge structure" S.J. Bloch (ed.) , ''Algebraic geometry (Bowdoin, 1985)'' , ''Proc. Symp. Pure Math.'' , '''46''' , Amer. Math. Soc. (1987) pp. 235–246 {{MR|0927983}} {{ZBL|0637.14008}} </TD></TR></table>

Latest revision as of 10:49, 20 December 2019


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.

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 $ \{ 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 , [3]). 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 [4]). 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 , [2], [4]): 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 [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 $ \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 [9].

References

[1a] P.A. Griffiths, "Periods of integrals on algebraic manifolds I" Amer. J. of Math. , 90 (1968) pp. 568–625 MR0229641 Zbl 0169.52303
[1b] P.A. Griffiths, "Periods of integrals on algebraic manifolds II" Amer. J. of Math. , 90 (1968) pp. 805–865 MR0233825 Zbl 0183.25501
[1c] P.A. Griffiths, "Periods of integrals on algebraic manifolds III" Publ. Math. IHES , 38 (1970) pp. 125–180 MR0282990 Zbl 0212.53503
[2] P.A. Griffiths, "A transcendental method in algebraic geometry" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 113–119 MR0414551 Zbl 0227.14008
[3] P. Deligne, "Travaux de Griffiths" , Sem. Bourbaki Exp. 376 , Lect. notes in math. , 180 , Springer (1971) pp. 213–235 Zbl 0208.48601
[4] W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" Invent. Math. , 22 (1973) pp. 211–319 MR0382272 Zbl 0278.14003
[5] E.H. Cattani, A.G. Kaplan, "Existence of period mappings for Hodge structures of weight two" Duke Math. J. , 4 : 1 (1977) pp. 1–43
[6] B.A. Dubrovin, "Theta-functions and non-linear equations" Russian Math. Surveys , 36 : 2 (1981) pp. 11–92 Uspekhi Mat. Nauk , 36 : 2 (1981) pp. 11–80 MR616797 Zbl 0549.58038 Zbl 0478.58038
[7] V.A. Kulikov, Uspekhi Mat. Nauk , 32 : 4 (1977) pp. 257–258
[8] D. Mumford, Matematika , 17 : 4 (1973) pp. 34–43 MR0360082 MR0339966
[9] W. Baily, A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains" Ann. of Math. , 84 (1966) pp. 442–528 MR0216035 Zbl 0154.08602


Comments

In the one-variable case, W. Schmid also proved a very precise description of the asymptotics of the period mapping, the $ \mathop{\rm SL}\nolimits _{2} $- orbit theorem. This was generalized to the several-variables case by E. Cattani and A. Kaplan [a1]. The period mapping and period domain also have been considered in the case of singularities [a1], [a2]. See also Variation of Hodge structure.

References

[a1] E. Cattani, A. Kaplan, "Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure" Invent. Math. , 67 (1982) pp. 101–115 MR0664326 Zbl 0516.14005
[a2] E. Looijenga, "A period mapping for certain semi-universal deformations" Compos. Math. , 30 (1975) pp. 299–316 MR0376666 Zbl 0312.14006
[a3] A.N. Varchenko, "Mapping of periods and intersection form" Funct. Anal. Appl. , 16 : 2 (1982) pp. 83–93 Funkts. Anal. Prilozhen. , 16 : 2 (1982) pp. 7–20 Zbl 0513.32018
[a4] C. Peters, J.H.M. Steenbrink, "Infinitesimal variations of Hodge structures" K. Ueno (ed.) , Classical Algebraic and Analytic Manifolds , Birkhäuser (1984) pp. 399–463 MR0728615
[a5] P. Griffiths, "Variation of Hodge structures" P. Griffiths (ed.) , Topics in Transcendental Algebraic geometry , Princeton Univ. Press (1984) pp. 3–28 MR0756854 MR0756846 MR0756845 MR0756843
[a6] P. Griffiths, W. Schmid, "Recent developments in Hodge theory" , Discrete Subgroups of Lie Groups and Applications to Moduli , Oxford Univ. Press (1975) pp. 31–128 MR0419850 Zbl 0355.14003
[a7] P. Griffiths, "Some transcendental aspects of algebraic geometry" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 3–110 MR0419438 Zbl 0309.14007
[a8] J. Carlson, N.Green, P. Griffiths, J. Harris, "Infinitesimal variations of Hodge structure I" Compos. Math. , 50 (1983) pp. 109–205 MR0720288 MR0720289 MR0720290 Zbl 0531.14006
[a9] R. Donagi, "Generic Torelli for hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353 MR720291 Zbl 0598.14007
[a10] M. Green, "The period maps for hypersurface sections of high degree of an algebraic variety" Compos. Math. , 55 (1984) pp. 135–156
[a11] D.A. Cox, "Generic Torelli and infinitesimal variation of Hodge structure" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 , Amer. Math. Soc. (1987) pp. 235–246 MR0927983 Zbl 0637.14008
How to Cite This Entry:
Period mapping. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Period_mapping&oldid=44308
This article was adapted from an original article by A.I. Ovseevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article