Hard Lefschetz theorems for free line bundles

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.