Toric differential forms and periods of complete intersections

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.