# 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]).

#### 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