On the Hodge Structure of Projective Hypersurfaces in Toric Varieties

TL;DR

This paper extends classical Hodge theory results to hypersurfaces in toric varieties, relating cohomology to Jacobian ideals and providing new descriptions of forms and vanishing theorems.

Contribution

It generalizes Griffiths, Dolgachev, and Steenbrink's work to toric varieties, linking Hodge filtration to Jacobian ideals and describing forms on toric varieties.

Findings

Isomorphism between Hodge filtration pieces and graded Jacobian quotient

Description of forms on toric varieties in algebraic terms

Proof of Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties

Abstract

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 $P$.