Torelli theorems

Theorems stating that the Hodge structure (period matrix) in the cohomology spaces $H^{\ast }(X,\mathbf{C})$ 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 ${\omega }_1\dots {\omega }_g\in H^0(X,{\mathrm{\Omega }}_X^1)=H^{1,0}\subset H^1(X,\mathbf{C})$ be a basis of the Abelian differentials (cf. Abelian differential) and let the $(g\times 2g)$- matrix $\mathrm{\Omega }=\Vert {\pi }_{ij}\Vert$ 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 $\tilde{X}$ be two curves. If bases $\gamma$ and $\omega$ can be chosen with respect to which the period matrices $\mathrm{\Omega }$ and the intersection matrices $Q$ of the curves are the same, then $X$ and $\tilde{X}$ are isomorphic. In other words, if the canonically polarized Jacobians of the curves $X$ and $\tilde{X}$ are isomorphic, then $X\simeq \tilde{X}$.

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

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

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

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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