For a Kähler manifold, 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*. Am I correct in understanding that one gets the same algebra for all Kähler manifolds?