Ample subvarieties and q-ample divisors

We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti–Grauert vanishing theorem.

► We introduce a notion of ampleness for subschemes of any codimension. ► These subschemes are shown to have properties similar to those of ample divisors. ► Ample subschemes satisfy a generalized Lefschetz hyperplane theorem. ► We give counterexamples to the converse of the Andreotti–Grauert vanishing theorem.