Dolbeault cohomology with values in a holomorphic vector bundle ($H_{\bar{\partial }}^{p,q}(X,E)$)

This is the cohomology of ${\mathcal{A}}^{p,\bullet }(X,E)$ with the bundle Dolbeault operator. The Dolbeault theorem identifies it with $H^q(X,{\Omega }_X^p\otimes \mathcal{O}(E))$. For compact Hermitian $X$, Dolbeault Hodge decomposition gives unique harmonic representatives.