Dependence of Lefschetz maps on the Dolbeault class (${\varphi }_{\omega ,k}[\alpha ]=[{\omega }^k\wedge \alpha ]$)

Wedge multiplication by a closed $(1,1)$ form induces maps on Dolbeault cohomology. If ${\omega }^{\prime }-\omega =\bar{\partial }\theta$, then $({\omega }^{\prime }{)}^k-{\omega }^k=\bar{\partial }(\theta \wedge {\sum }_{j=0}^{k-1}({\omega }^{\prime }{)}^j\wedge {\omega }^{k-1-j})$. Wedging with a closed representative shows that the maps agree. For a change of Kähler metric potential, take $\theta =-i\partial f$.