  We establish the Hodge conjecture for some subvarieties of a class of toric
varieties. First we study quasi-smooth intersections in a projective simplicial
toric variety, which is a suitable notion to generalize smooth complete
intersection subvarieties in the toric environment, and in particular
quasi-smooth hypersurfaces. We show that under appropriate conditions, the
Hodge Conjecture holds for a very general quasi-smooth intersection subvariety,
generalizing the work on quasi-smooth hypersurfaces of the first author and
Grassi in [3]. We also show that the Hodge Conjecture holds asymptotically for
suitable quasi-smooth hypersurface in the Noether-Lefschetz locus, where
"asymptotically" means that the degree of the hypersurface is big enough. This
extendes to toric varieties Otwinowska's result in [15].
