Bundle-valued Dolbeault adjoint (${\bar{\partial }}_E^{\ast }=-{\ast }_{E^{\ast }}{\bar{\partial }}_{E^{\ast }}{\ast }_E$)

Using the bundle-valued conjugate-linear Hodge star, the formal adjoint of the holomorphic bundle's Dolbeault operator is ${\bar{\partial }}_E^{\ast }=-{\ast }_{E^{\ast }}{\bar{\partial }}_{E^{\ast }}{\ast }_E$. It lowers antiholomorphic degree. Stokes applied to the coefficient-contracted form $\psi \wedge {\ast }_E\eta$ proves the integrated adjoint identity on a compact complex manifold.