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Generic Torelli for projective hypersurfaces. (English) Zbl 0598.14007
The global Torelli theorem for hypersurfaces in a projective space is proved, i.e., the following assertion: Theorem. The period map, which assigns to a smooth hypersurface \(X_ f\) of degree d in \(P^{n+1}\) the type of its Hodge structure on the middle primitive cohomologies, is injective in a generic point of the variety of modules of hypersurfaces of degree d with the exception of the case \(d=3\), \(n=2\) and, possible, of the cases (1) d divides \(n+2\); (2) \(d=4,\) \(n=4m\) or \(d=6\), \(n=6m+1\). Thus, a generic hypersurface is recovered by the periods.
The proof is based on investigation of the Jacobian ring \(R=S/I_ f\) where S is the ring of polynomials and \(I_ f\) is the ideal generated by the partial derivatives \(\frac{\partial f}{\partial x_ i}\). This Artinian graded ring is a projective invariant of f. Since the initial terms \(R^ t\) of its gradation coincide with \(S^ t\), the smooth hypersurfaces having the same rings R are projectively equivalent. The Hodge structure of \(X_ f\) permits to recover partially the multiplication in R. Namely, \(R^{t_ a}\), \(t_ a=(n-a+1)d-(n+2)\) is isomorphic to the factor \(F^ a/F^{a+1}\) of the Hodge filtration in \(H^ n_ p(X_ f,{\mathbb{C}})\). This isomorphism is established by taking the k-residue of \((n+1)\)-forms holomorphic in the complement of \(X_ t\) and having a pole of order k on \(X_ f\), which turns out to be an element of \(F^ a/F^{a+1}\approx H^{a,n-a}(X)\). Moreover, \(R^ d\) is isomorphic to the space of infinitesimal variations D of the hypersurface \(X_ f\). The indicated isomorphisms are compatible also with the multiplication in R. The Kodaira map \(D\times F^ a/F^{a+1}\to F^{a-1}/F^ a\) goes over to the multiplication \(R^ d\times R^{t_ a}\to R^{t_ a+d}\) and the cohomology cup product \(F^ a/F^{a+1}\times F^{n-a}/F^{n-a+1}\) to \(R^{t_ a}\times R^{\sigma -t_ a}\to R^{\sigma}\approx {\mathbb{C}}.\)
The author notices that for a generic hypersurface, the multiplication \(R^ a\times R^ b\to R^{a+b}\) for \(a<b\leq d\) determines the multiplication \(R^{b-a}\times R^ a\to R^ b\) if (d-2)(n-1)\(\geq 3\). The space \(R^{b-a}\) turns out to be isomorphic to the subspace \(T\subset Hom(R^ a,R^ b)\) where \(x\in T\) if \(x(u)\cdot v=u\cdot x(v)\) for any \(u,v\in R^ a\). This fact is checked explicitly for the Fermat hypersurface \(f=\sum^{n+2}_{1}x^ d_ i\) and follows from the embedding \(R^{b-a}\subseteq T\) in the general case and from the constancy of the dimension of \(R^{b-a}\). Starting from \(R^{t_ 0}\times R^ d\to R^{t_ 0+d}\), we stop at \(R^ a\times R^ d\), \(a\leq d\). If \(2a<d-1\), then \(R^ a=S^ aV\) and \(R^{2a}=S^{2a}V\). The map \(S^ 2(S^ aV)\to^{\mu}S^{2a}V\) recovers the structure of \(R^ a\approx S^ aV\) since the zeros of all quadrics from Ker(\(\mu)\) define the Veronese embedding \(\nu (P(V))\subset P(R^ a)\). But if \(d- 1\leq 2a<2d-1\), then, under not strong restrictions, the structure of the symmetric power on \(R^ a\) is determined by consideration of quadrics of rank 4 in ker \(S^ 2R^ a\to R^{2a}\). We notice that the author recovers, in fact, the hypersurface by variation of the first non-trivial term of the Hodge filtration only.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K30 Picard schemes, higher Jacobians
32G20 Period matrices, variation of Hodge structure; degenerations
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