View source for Period mapping

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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.

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|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|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 , [[#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  $  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====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> P.A. Griffiths, "Periods of integrals on algebraic manifolds I" ''Amer. J. of Math.'' , '''90''' (1968) pp. 568–625 {{MR|0229641}} {{ZBL|0169.52303}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> P.A. Griffiths, "Periods of integrals on algebraic manifolds II" ''Amer. J. of Math.'' , '''90''' (1968) pp. 805–865 {{MR|0233825}} {{ZBL|0183.25501}} </TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> P.A. Griffiths, "Periods of integrals on algebraic manifolds III" ''Publ. Math. IHES'' , '''38''' (1970) pp. 125–180 {{MR|0282990}} {{ZBL|0212.53503}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.A. Griffiths, "A transcendental method in algebraic geometry" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars (1971) pp. 113–119 {{MR|0414551}} {{ZBL|0227.14008}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Deligne, "Travaux de Griffiths" , ''Sem. Bourbaki Exp. 376'' , ''Lect. notes in math.'' , '''180''' , Springer (1971) pp. 213–235 {{MR|}} {{ZBL|0208.48601}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|616797}} {{ZBL|0549.58038}} {{ZBL|0478.58038}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.A. Kulikov, ''Uspekhi Mat. Nauk'' , '''32''' : 4 (1977) pp. 257–258 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> D. Mumford, ''Matematika'' , '''17''' : 4 (1973) pp. 34–43 {{MR|0360082}} {{MR|0339966}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> W. Baily, A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains" ''Ann. of Math.'' , '''84''' (1966) pp. 442–528 {{MR|0216035}} {{ZBL|0154.08602}} </TD></TR></table>



====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 [[#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====
<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>