Toric differential forms and periods of complete intersections

TL;DR

This paper computes periods of certain algebraic cycles in toric varieties, generalizes classical theorems to the toric setting, and determines cycle classes for complete intersections.

Contribution

It introduces a toric generalization of Macaulay's theorem and extends the algebraic cup formula for residue forms to toric varieties, providing new tools for algebraic geometry.

Findings

Computed periods of complete intersection cycles in toric varieties

Determined the cycle class of algebraic cycles in this setting

Developed a toric generalization of Macaulay's theorem

Abstract

Let $n$ be an even natural number. We compute the periods of any $\frac{n}{2}$-dimensional complete intersection algebraic cycle inside an $n$-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.