On the Hodge Structure of Projective Hypersurfaces in Toric Varieties
| dc.creator | Batyrev, Victor V. | |
| dc.creator | Cox, David A. | |
| dc.date | 1993-06-25 | |
| dc.date.accessioned | 2026-07-07T09:05:51Z | |
| dc.date.available | 2026-07-07T09:05:51Z | |
| dc.description | This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$. | |
| dc.description | 43 pages, LaTeX Version 2.09 | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9306011 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9306011 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149820 | |
| dc.subject | Algebraic Geometry | |
| dc.title | On the Hodge Structure of Projective Hypersurfaces in Toric Varieties | |
| dc.type | text |