For a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the product operator on (1,1)-classes is exactly the span of the prime divisors annihilated by the collection.
Hard Lefschetz theorems for free line bundles
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abstract
We introduce a partial positivity notion for algebraic maps via the defect of semismallness. This positivity notion is modeled on $m$-positivity in the analytic setting and $m$-ampleness in the geometric setting. Using this positivity condition for algebraic maps, we establish K\"ahler packages, that is, Hard Lefschetz theorems and Hodge-Riemann bilinear relations, for the complete intersections of Chern classes of free line bundles.
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Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
For a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the product operator on (1,1)-classes is exactly the span of the prime divisors annihilated by the collection.