Namespaces
Variants
Actions

Difference between revisions of "Torelli theorems"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
Theorems stating that the [[Hodge structure|Hodge structure]] (period matrix) in the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932601.png" /> of an algebraic or Kähler variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932602.png" /> completely characterizes the polarized Jacobi variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932603.png" />.
+
<!--
 +
t0932601.png
 +
$#A+1 = 79 n = 0
 +
$#C+1 = 79 : ~/encyclopedia/old_files/data/T093/T.0903260 Torelli theorems
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The classical Torelli theorem relates to the case of curves (see [[#References|[1]]], [[#References|[2]]]) and states that a curve is defined up to an isomorphism by its periods. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932604.png" /> be a curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932605.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932606.png" /> be a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932607.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932608.png" /> be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) and let the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t0932609.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326010.png" /> be the period matrix, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326011.png" />. The intersection of cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326012.png" /> defines a skew-symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326014.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326016.png" /> be two curves. If bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326018.png" /> can be chosen with respect to which the period matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326019.png" /> and the intersection matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326020.png" /> of the curves are the same, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326022.png" /> are isomorphic. In other words, if the canonically polarized Jacobians of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326024.png" /> are isomorphic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326025.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326026.png" /> be a projective variety (or, more generally, a compact [[Kähler manifold|Kähler manifold]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326027.png" /> be the Griffiths variety associated with the primitive cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326028.png" /> (see [[Period mapping|Period mapping]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326029.png" /> contains the period matrices of primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326030.png" />-forms on all varieties homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326031.png" />. The periods depend on the choice of the isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326032.png" /> into a fixed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326033.png" />. There is a naturally defined group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326034.png" /> of analytic automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326036.png" /> is an analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326037.png" /> determines a unique point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326038.png" />. In this situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326039.png" /> is called the modular space or the moduli space of Hodge structures.
+
Theorems stating that the [[Hodge structure|Hodge structure]] (period matrix) in the cohomology spaces $  H  ^ {*} ( X, \mathbf C ) $
 +
of an algebraic or Kähler variety  $  X $
 +
completely characterizes the polarized Jacobi variety of $  X $.
  
The global Torelli problem consists in the elucidation of the question whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326040.png" /> uniquely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326041.png" /> up to an isomorphism. In the case of an affirmative solution, the problem corresponds to the statement of the so-called (generalized) Torelli theorem. Torelli's theorem holds trivially for Abelian varieties in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326042.png" />-forms and in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326043.png" />-forms (see [[#References|[3]]]). Essentially, the only non-trivial case of a solution of the global Torelli problem (1984) is the case of a [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326044.png" />-surface]]. The Torelli theorem has also been generalized to the case of Kähler <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326045.png" />-surfaces.
+
The classical Torelli theorem relates to the case of curves (see [[#References|[1]]], [[#References|[2]]]) and states that a curve is defined up to an isomorphism by its periods. Let  $  X $
 +
be a curve of genus  $  g $,  
 +
let  $  \gamma _ {1} \dots \gamma _ {2g} $
 +
be a basis of $  H _ {1} ( X, \mathbf Z ) $,
 +
let  $  \omega _ {1} \dots \omega _ {g} \in H  ^ {0} ( X, \Omega _ {X}  ^ {1} ) = H  ^ {1,0} \subset  H  ^ {1} ( X, \mathbf C ) $
 +
be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) and let the ( g \times 2g) $-
 +
matrix  $  \Omega = \| \pi _ {ij} \| $
 +
be the period matrix, where  $  \pi _ {ij} = \int _ {\gamma _ {j}  } \omega _ {i} $.
 +
The intersection of cycles  $  \gamma _ {i} \gamma _ {j} = q _ {ij} $
 +
defines a skew-symmetric bilinear form  $  Q $
 +
in  $  H _ {1} ( X, \mathbf Z ) $.  
 +
Let  $  X $
 +
and  $  \widetilde{X}  $
 +
be two curves. If bases  $  \gamma $
 +
and  $  \omega $
 +
can be chosen with respect to which the period matrices  $  \Omega $
 +
and the intersection matrices  $  Q $
 +
of the curves are the same, then  $  X $
 +
and  $  \widetilde{X}  $
 +
are isomorphic. In other words, if the canonically polarized Jacobians of the curves  $  X $
 +
and  $  \widetilde{X}  $
 +
are isomorphic, then  $  X \simeq \widetilde{X}  $.
  
The local Torelli problem consists in solving the question when the Hodge structures on the cohomology spaces separate points in the local moduli space (the Kuranishi space) for a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326046.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326047.png" /> be a family of polarized algebraic varieties, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326048.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326049.png" /> be the Griffiths variety associated with the periods of primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326050.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326051.png" />. The period mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326052.png" /> associates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326053.png" /> with the period matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326054.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326055.png" />. This mapping is holomorphic; the corresponding tangent mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326056.png" /> has been calculated (see [[#References|[3]]]). The local Torelli problem is equivalent to the question: When is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326057.png" /> an imbedding? By considering the mapping dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326058.png" /> one obtains a cohomological criterion for the validity of the local Torelli theorem: If the mapping
+
Let $  X $
 +
be a projective variety (or, more generally, a compact [[Kähler manifold|Kähler manifold]]), and let $  D = D _ {k} $
 +
be the Griffiths variety associated with the primitive cohomology spaces  $  H  ^ {k} ( X, \mathbf C ) _ {0} $(
 +
see [[Period mapping|Period mapping]]). Then  $  D $
 +
contains the period matrices of primitive  $  k $-
 +
forms on all varieties homeomorphic to  $  X $.  
 +
The periods depend on the choice of the isomorphism of $  H  ^ {k} ( X, \mathbf C ) _ {0} $
 +
into a fixed space  $  H $.  
 +
There is a naturally defined group  $  \Gamma $
 +
of analytic automorphisms of  $  D $
 +
such that  $  M = D/ \Gamma $
 +
is an analytic space and  $  X $
 +
determines a unique point  $  \Phi ( X) \in M $.  
 +
In this situation,  $  M $
 +
is called the modular space or the moduli space of Hodge structures.
  
<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/t/t093/t093260/t09326059.png" /></td> </tr></table>
+
The global Torelli problem consists in the elucidation of the question whether  $  \Phi ( X) $
 +
uniquely determines  $  X $
 +
up to an isomorphism. In the case of an affirmative solution, the problem corresponds to the statement of the so-called (generalized) Torelli theorem. Torelli's theorem holds trivially for Abelian varieties in the case of  $  1 $-
 +
forms and in the case of  $  2 $-
 +
forms (see [[#References|[3]]]). Essentially, the only non-trivial case of a solution of the global Torelli problem (1984) is the case of a [[K3-surface| $  K3 $-
 +
surface]]. The Torelli theorem has also been generalized to the case of Kähler  $  K3 $-
 +
surfaces.
  
<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/t/t093/t093260/t09326060.png" /></td> </tr></table>
+
The local Torelli problem consists in solving the question when the Hodge structures on the cohomology spaces separate points in the local moduli space (the Kuranishi space) for a variety  $  X $.
 +
Let  $  \pi : \mathfrak X \rightarrow B $
 +
be a family of polarized algebraic varieties,  $  \pi  ^ {-} 1 ( 0) = X $,
 +
and let  $  M = D/ \Gamma $
 +
be the Griffiths variety associated with the periods of primitive  $  k $-
 +
forms on  $  X $.  
 +
The period mapping  $  \Phi :  B \rightarrow M $
 +
associates  $  t \in B $
 +
with the period matrix of  $  k $-
 +
forms on  $  \pi  ^ {-} 1 ( t) $.  
 +
This mapping is holomorphic; the corresponding tangent mapping  $  d \Phi $
 +
has been calculated (see [[#References|[3]]]). The local Torelli problem is equivalent to the question: When is  $  d \Phi $
 +
an imbedding? By considering the mapping dual to  $  d \Phi $
 +
one obtains a cohomological criterion for the validity of the local Torelli theorem: If the mapping
  
is an epimorphism, then the periods of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326061.png" />-forms give local moduli for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326062.png" />. The local Torelli theorem for curves is equivalent to the fact that quadratic differentials are generated by Abelian differentials. Noether's theorem states that this is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326063.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326065.png" /> is not hyper-elliptic. The local Torelli theorem clearly holds in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326066.png" /> if the canonical class is trivial. Such varieties include the Abelian varieties, hypersurfaces of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326069.png" />-surfaces. The validity of the local Torelli theorem has been established for various classes of higher-dimensional varieties. For non-singular hypersurfaces of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326070.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326071.png" /> it has been proved that the period mapping is an imbedding at a generic point except for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326073.png" /> and, possibly, the cases: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326074.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326077.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093260/t09326079.png" /> (see [[#References|[4]]]).
+
$$
 +
\mu : \
 +
\oplus _ {0 \leq  r \leq  [( k - 1)/2] }
 +
H ^ {n - r - 1 }
 +
( X, \Omega ^ {n - k + r + 1 } )
 +
\otimes H  ^ {r}
 +
( X, \Omega ^ {k - r } ) \rightarrow
 +
$$
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Torelli,  ''Rend. Accad. Lincei V'' , '''22'''  (1913)  pp. 98–103</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,   "Zum Beweis der Torellischen Satzes" ''Nachr. Akad. Wiss. Göttingen'' (1957) pp. 33–53</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.A. Griffiths,  "Periods of integrals on algebraic manifolds I, II"  ''Amer. J. Math.'' , '''90'''  (1968)  pp. 568–626; 805–865</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Donagi,  "Generic Torelli for projective hypersurfaces"  ''Compos. Math.'' , '''50'''  (1983)  pp. 325–353</TD></TR></table>
+
\rightarrow \
 +
H ^ {n - 1 } ( X, \Omega ^ {1} \otimes \Omega ^ {n} )
 +
$$
  
 +
is an epimorphism, then the periods of the  $  k $-
 +
forms give local moduli for  $  X $.
 +
The local Torelli theorem for curves is equivalent to the fact that quadratic differentials are generated by Abelian differentials. Noether's theorem states that this is true if  $  g = 2 $
 +
or if  $  g > 2 $
 +
and  $  X $
 +
is not hyper-elliptic. The local Torelli theorem clearly holds in the case  $  k = n $
 +
if the canonical class is trivial. Such varieties include the Abelian varieties, hypersurfaces of degree  $  n + 2 $
 +
in  $  P ^ {n + 1 } $
 +
and  $  K3 $-
 +
surfaces. The validity of the local Torelli theorem has been established for various classes of higher-dimensional varieties. For non-singular hypersurfaces of degree  $  d $
 +
in  $  P ^ {n + 1 } $
 +
it has been proved that the period mapping is an imbedding at a generic point except for the case  $  n = 2 $,
 +
$  d = 3 $
 +
and, possibly, the cases:  $  d $
 +
divides  $  n + 2 $,
 +
$  d = 4 $
 +
and  $  n = 4m $,
 +
or  $  d = 6 $
 +
and  $  n = 6m + 1 $(
 +
see [[#References|[4]]]).
  
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Torelli, ''Rend. Accad. Lincei V'' , '''22''' (1913) pp. 98–103</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Weil, "Zum Beweis der Torellischen Satzes" ''Nachr. Akad. Wiss. Göttingen'' (1957) pp. 33–53 {{MR|89483}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths, "Periods of integrals on algebraic manifolds I, II" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626; 805–865</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Donagi, "Generic Torelli for projective hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353 {{MR|0720291}} {{ZBL|0598.14007}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths (ed.) , ''Topics in transcendental algebraic geometry'' , Princeton Univ. Press (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. van de Ven,   "Compact complex surfaces" , Springer (1984)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths (ed.) , ''Topics in transcendental algebraic geometry'' , Princeton Univ. Press (1984) {{MR|0756842}} {{ZBL|0528.00004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR></table>

Latest revision as of 08:26, 6 June 2020


Theorems stating that the Hodge structure (period matrix) in the cohomology spaces $ H ^ {*} ( X, \mathbf C ) $ of an algebraic or Kähler variety $ X $ completely characterizes the polarized Jacobi variety of $ X $.

The classical Torelli theorem relates to the case of curves (see [1], [2]) and states that a curve is defined up to an isomorphism by its periods. Let $ X $ be a curve of genus $ g $, let $ \gamma _ {1} \dots \gamma _ {2g} $ be a basis of $ H _ {1} ( X, \mathbf Z ) $, let $ \omega _ {1} \dots \omega _ {g} \in H ^ {0} ( X, \Omega _ {X} ^ {1} ) = H ^ {1,0} \subset H ^ {1} ( X, \mathbf C ) $ be a basis of the Abelian differentials (cf. Abelian differential) and let the $ ( g \times 2g) $- matrix $ \Omega = \| \pi _ {ij} \| $ be the period matrix, where $ \pi _ {ij} = \int _ {\gamma _ {j} } \omega _ {i} $. The intersection of cycles $ \gamma _ {i} \gamma _ {j} = q _ {ij} $ defines a skew-symmetric bilinear form $ Q $ in $ H _ {1} ( X, \mathbf Z ) $. Let $ X $ and $ \widetilde{X} $ be two curves. If bases $ \gamma $ and $ \omega $ can be chosen with respect to which the period matrices $ \Omega $ and the intersection matrices $ Q $ of the curves are the same, then $ X $ and $ \widetilde{X} $ are isomorphic. In other words, if the canonically polarized Jacobians of the curves $ X $ and $ \widetilde{X} $ are isomorphic, then $ X \simeq \widetilde{X} $.

Let $ X $ be a projective variety (or, more generally, a compact Kähler manifold), and let $ D = D _ {k} $ be the Griffiths variety associated with the primitive cohomology spaces $ H ^ {k} ( X, \mathbf C ) _ {0} $( see Period mapping). Then $ D $ contains the period matrices of primitive $ k $- forms on all varieties homeomorphic to $ X $. The periods depend on the choice of the isomorphism of $ H ^ {k} ( X, \mathbf C ) _ {0} $ into a fixed space $ H $. There is a naturally defined group $ \Gamma $ of analytic automorphisms of $ D $ such that $ M = D/ \Gamma $ is an analytic space and $ X $ determines a unique point $ \Phi ( X) \in M $. In this situation, $ M $ is called the modular space or the moduli space of Hodge structures.

The global Torelli problem consists in the elucidation of the question whether $ \Phi ( X) $ uniquely determines $ X $ up to an isomorphism. In the case of an affirmative solution, the problem corresponds to the statement of the so-called (generalized) Torelli theorem. Torelli's theorem holds trivially for Abelian varieties in the case of $ 1 $- forms and in the case of $ 2 $- forms (see [3]). Essentially, the only non-trivial case of a solution of the global Torelli problem (1984) is the case of a $ K3 $- surface. The Torelli theorem has also been generalized to the case of Kähler $ K3 $- surfaces.

The local Torelli problem consists in solving the question when the Hodge structures on the cohomology spaces separate points in the local moduli space (the Kuranishi space) for a variety $ X $. Let $ \pi : \mathfrak X \rightarrow B $ be a family of polarized algebraic varieties, $ \pi ^ {-} 1 ( 0) = X $, and let $ M = D/ \Gamma $ be the Griffiths variety associated with the periods of primitive $ k $- forms on $ X $. The period mapping $ \Phi : B \rightarrow M $ associates $ t \in B $ with the period matrix of $ k $- forms on $ \pi ^ {-} 1 ( t) $. This mapping is holomorphic; the corresponding tangent mapping $ d \Phi $ has been calculated (see [3]). The local Torelli problem is equivalent to the question: When is $ d \Phi $ an imbedding? By considering the mapping dual to $ d \Phi $ one obtains a cohomological criterion for the validity of the local Torelli theorem: If the mapping

$$ \mu : \ \oplus _ {0 \leq r \leq [( k - 1)/2] } H ^ {n - r - 1 } ( X, \Omega ^ {n - k + r + 1 } ) \otimes H ^ {r} ( X, \Omega ^ {k - r } ) \rightarrow $$

$$ \rightarrow \ H ^ {n - 1 } ( X, \Omega ^ {1} \otimes \Omega ^ {n} ) $$

is an epimorphism, then the periods of the $ k $- forms give local moduli for $ X $. The local Torelli theorem for curves is equivalent to the fact that quadratic differentials are generated by Abelian differentials. Noether's theorem states that this is true if $ g = 2 $ or if $ g > 2 $ and $ X $ is not hyper-elliptic. The local Torelli theorem clearly holds in the case $ k = n $ if the canonical class is trivial. Such varieties include the Abelian varieties, hypersurfaces of degree $ n + 2 $ in $ P ^ {n + 1 } $ and $ K3 $- surfaces. The validity of the local Torelli theorem has been established for various classes of higher-dimensional varieties. For non-singular hypersurfaces of degree $ d $ in $ P ^ {n + 1 } $ it has been proved that the period mapping is an imbedding at a generic point except for the case $ n = 2 $, $ d = 3 $ and, possibly, the cases: $ d $ divides $ n + 2 $, $ d = 4 $ and $ n = 4m $, or $ d = 6 $ and $ n = 6m + 1 $( see [4]).

References

[1] R. Torelli, Rend. Accad. Lincei V , 22 (1913) pp. 98–103
[2] A. Weil, "Zum Beweis der Torellischen Satzes" Nachr. Akad. Wiss. Göttingen (1957) pp. 33–53 MR89483
[3] P.A. Griffiths, "Periods of integrals on algebraic manifolds I, II" Amer. J. Math. , 90 (1968) pp. 568–626; 805–865
[4] R. Donagi, "Generic Torelli for projective hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353 MR0720291 Zbl 0598.14007

Comments

References

[a1] P.A. Griffiths (ed.) , Topics in transcendental algebraic geometry , Princeton Univ. Press (1984) MR0756842 Zbl 0528.00004
[a2] A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023
How to Cite This Entry:
Torelli theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torelli_theorems&oldid=12863
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article