## 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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### References:

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