The local cohomological defect of an affine toric variety is characterized by Lefschetz cup-product maps on a projective toric variety of one dimension lower, which shows it is not a combinatorial invariant and allows examples in every dimension.
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Lefschetz morphisms on singular cohomology and local cohomological dimension of toric varieties
The local cohomological defect of an affine toric variety is characterized by Lefschetz cup-product maps on a projective toric variety of one dimension lower, which shows it is not a combinatorial invariant and allows examples in every dimension.