Let $X$ be a complex manifold. For every pair of integers $p,q \geq 0$, the Dolbeault operators on complex-valued differential forms satisfy, as operator identities on $A^{p,q}(X)$,
Here $\partial\bar\partial$ denotes the composition $\partial\circ\bar\partial$, and $\bar\partial\partial$ denotes the composition $\bar\partial\circ\partial$.