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On the Hodge structure of projective hypersurfaces in toric varieties. (English) Zbl 0851.14021
Let \(\Sigma\) be a simplicial complete fan for a free \(\mathbb{Z}\)-module \(N\) of rank \(d\) and denote by \(P_\Sigma\) the associated \(d\)-dimensional toric variety over the complex number field \(\mathbb{C}\). M. Audin [“The topology of torus actions on symplectic manifolds”, Prog. Math. 93 (1991; Zbl 0726.57029)] and D. Cox [J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)] showed that \(P_\Sigma\) is a geometric quotient of a Zariski open subset \(U(\Sigma)\) of an affine space \(\mathbb{A}^n\) by a linear diagonal action of an algebraic subgroup \(D(\Sigma) \subset (\mathbb{C}^*)^n\). Here \(n\) is the number of 1-dimensional cones in the fan \(\Sigma\), \(\mathbb{C}^*\) is the multiplicative group of non-zero complex numbers, and the character group of \(D(\Sigma)\) coincides with the group \(\text{Cl} (\Sigma)\) of linear equivalence classes of Weil divisors on \(P_\Sigma\).
The complement of \(U(\Sigma)\) in \(\mathbb{A}^n\) is of codimension at least 2 so that the ring of regular algebraic functions on \(U(\Sigma)\) coincides with the polynomial ring \(S(\Sigma)= \mathbb{C} [z_1, \dots, z_n]\) which carries a canonical grading with respect to the additive group \(\text{Cl} (\Sigma)\) induced by the action of \(D(\Sigma)\) on \(\mathbb{A}^n\). The second author called \(S(\Sigma)\) the homogeneous coordinate ring of the toric variety \(P_\Sigma\). It is indeed an extremely fruitful generalization of the homogeneous coordinate rings of projective spaces and weighted projective spaces. For instance, a hypersurface \(X\subset P_\Sigma\) is defined by a homogeneous polynomial \(f\in S(\Sigma)\) of degree equal to the linear equivalence class \(\beta\in \text{Cl} (\Sigma)\) of the divisor \(X\).
Here is what the authors do in this paper among other things: When \(X\subset P_\Sigma\) is a quasi-smooth ample hypersurface, the authors relate the pure Hodge structure on the primitive cohomology group \(PH^{d-1} (X, \mathbb{C})\) with the Jacobian ring \(S(\Sigma)/ (\partial f/\partial z_1, \dots, \partial f/ \partial z_n)\), generalizing the classical results for hypersurfaces in projective spaces and weighted projective spaces due to P. A. Griffiths [Ann. Math., II. Ser. 90, 460-495, 496-541 (1969; Zbl 0215.08103)], J. Steenbrink [Compos. Math. 34, 211-223 (1977; Zbl 0347.14001)] and I. Dolgachev [in: Group actions and vector fields, Proc. Pol.-North Am. Semin., Vancouver 1981, Lect. Notes Math. 956, 34-71 (1982; Zbl 0516.14014)].
Reviewer: T.Oda (Sendai)

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J70 Hypersurfaces and algebraic geometry
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