  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.
