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Zbl 0598.14007
Donagi, Ron
Generic Torelli for projective hypersurfaces.
(English)
[J] Compos. Math. 50, 325-353 (1983). ISSN 0010-437X; ISSN 1570-5846/e

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 $X\sb f$ of degree d in $P\sp{n+1}$ 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. \par The proof is based on investigation of the Jacobian ring $R=S/I\sb f$ where S is the ring of polynomials and $I\sb f$ is the ideal generated by the partial derivatives $\frac{\partial f}{\partial x\sb i}$. This Artinian graded ring is a projective invariant of f. Since the initial terms $R\sp t$ of its gradation coincide with $S\sp t$, the smooth hypersurfaces having the same rings R are projectively equivalent. The Hodge structure of $X\sb f$ permits to recover partially the multiplication in R. Namely, $R\sp{t\sb a}$, $t\sb a=(n-a+1)d-(n+2)$ is isomorphic to the factor $F\sp a/F\sp{a+1}$ of the Hodge filtration in $H\sp n\sb p(X\sb f,{\bbfC})$. This isomorphism is established by taking the k-residue of $(n+1)$-forms holomorphic in the complement of $X\sb t$ and having a pole of order k on $X\sb f$, which turns out to be an element of $F\sp a/F\sp{a+1}\approx H\sp{a,n-a}(X)$. Moreover, $R\sp d$ is isomorphic to the space of infinitesimal variations D of the hypersurface $X\sb f$. The indicated isomorphisms are compatible also with the multiplication in R. The Kodaira map $D\times F\sp a/F\sp{a+1}\to F\sp{a-1}/F\sp a$ goes over to the multiplication $R\sp d\times R\sp{t\sb a}\to R\sp{t\sb a+d}$ and the cohomology cup product $F\sp a/F\sp{a+1}\times F\sp{n-a}/F\sp{n-a+1}$ to $R\sp{t\sb a}\times R\sp{\sigma -t\sb a}\to R\sp{\sigma}\approx {\bbfC}.$ \par The author notices that for a generic hypersurface, the multiplication $R\sp a\times R\sp b\to R\sp{a+b}$ for $a<b\le d$ determines the multiplication $R\sp{b-a}\times R\sp a\to R\sp b$ if (d-2)(n-1)$\ge 3$. The space $R\sp{b-a}$ turns out to be isomorphic to the subspace $T\subset Hom(R\sp a,R\sp b)$ where $x\in T$ if $x(u)\cdot v=u\cdot x(v)$ for any $u,v\in R\sp a$. This fact is checked explicitly for the Fermat hypersurface $f=\sum\sp{n+2}\sb{1}x\sp d\sb i$ and follows from the embedding $R\sp{b-a}\subseteq T$ in the general case and from the constancy of the dimension of $R\sp{b-a}$. Starting from $R\sp{t\sb 0}\times R\sp d\to R\sp{t\sb 0+d}$, we stop at $R\sp a\times R\sp d$, $a\le d$. If $2a<d-1$, then $R\sp a=S\sp aV$ and $R\sp{2a}=S\sp{2a}V$. The map $S\sp 2(S\sp aV)\to\sp{\mu}S\sp{2a}V$ recovers the structure of $R\sp a\approx S\sp aV$ since the zeros of all quadrics from Ker($\mu)$ define the Veronese embedding $\nu (P(V))\subset P(R\sp a)$. But if $d- 1\le 2a<2d-1$, then, under not strong restrictions, the structure of the symmetric power on $R\sp a$ is determined by consideration of quadrics of rank 4 in ker $S\sp 2R\sp a\to R\sp{2a}$. We notice that the author recovers, in fact, the hypersurface by variation of the first non-trivial term of the Hodge filtration only.
MSC 2000:
*14C30 Transcendental methods
14K30 Picard schemes, higher Jacobians
32G20 Period matrices

Keywords: variation of Hodge filtration; global Torelli theorem; period map; Hodge structure; Jacobian ring; infinitesimal variations

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