Torelli theorems

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]).

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