View source for Torelli theorems

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" />.

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" />.

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.

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 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

<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>

<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>

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]]]).

====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>



====Comments====


====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>