Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $(E,h)$ be a Hermitian holomorphic vector bundle on $X$. For integers $p,q$ with $0\le p,q\le n$, let $A^{p,q}(X,E)$ denote the [Fréchet space](/page/Fr%C3%A9chet%20Space) of smooth $E$-valued $(p,q)$-forms, let
be the Dolbeault Laplacian, where $\bar\partial_E^*$ is the formal adjoint determined by $\omega$ and $h$. Define the space of harmonic $E$-valued $(p,q)$-forms by
which sends a harmonic form to its Dolbeault cohomology class, identified with sheaf cohomology by the Dolbeault resolution, is an isomorphism of complex vector spaces.