Lefschetz morphisms on singular cohomology and local cohomological dimension of toric varieties

Given a proper toric variety and a line bundle on it, we describe the morphism on singular cohomology given by the cup product with the Chern class of that line bundle in terms of the data of the associated fan. Using that, we relate the local cohomological dimension of an affine toric variety with the Lefschetz morphism on the singular cohomology of a projective toric variety of one dimension lower. As a corollary, we show that the local cohomological defect is not a combinatorial invariant. We also produce numerous examples of toric varieties in every dimension with any possible local cohomological defect, by showing that the local cohomological defect remains unchanged under taking a pyramid.