For a Kähler manifolds, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the supersymmetric algebra, or in physics parlance the $N=(2,2)$ SUSY algebra. I am correct in understanding that one gets the same algebra for all Kähler manifolds?