Mobile Product and Zariski decomposition

We explain the relationship between ${\alpha }_1\mathrm{.}\mathrm{.}\mathrm{.}{\alpha }_q$ (standard cohomology product) and $({\alpha }_1\mathrm{.}\mathrm{.}\mathrm{.}{\alpha }_q)$ (mobile intersection product) of pseudo-effective classes ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_q$ on a compact K$\ddot{a}$hler manifold. We also show how to use this relationship for proving some holomorphic Morse inequalities. Then we prove a result concerning the direct image of Lelong numbers under a modification in dimension 3, deriving a continuity property for the Lelong numbers of the wedge of $(1,1)-$currents.