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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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