Bundle-valued Dolbeault adjoint
= Bundle-valued Dolbeault adjoint
{title2=$\bar\partial_E^*=-*_{E^*}\bar\partial_{E^*}*_E$}
Using the <bundle-valued conjugate-linear Hodge star>, the <formal adjoint> of the holomorphic bundle's <Dolbeault operator> is $\bar\partial_E^*=-*_{E^*}\bar\partial_{E^*}*_E$. It lowers antiholomorphic degree. Stokes applied to the coefficient-contracted form $\psi\wedge *_E\eta$ proves the integrated adjoint identity on a compact <complex manifold>.