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.

Chern curvature defines $\alpha (E)=[\Theta ]\in H^1(M,{\Omega }_M^1\otimes \mathrm{End}E)$. If two metrics give connections differing by $a$, then ${\Theta }_1-{\Theta }_0={\bar{\partial }}_{\mathrm{End}E}a$. Thus this vector-bundle curvature class depends on the holomorphic bundle rather than the chosen metric.

In frames with $e^{(\beta )}=g_{\alpha \beta }{e}^{(\alpha )}$ and input-index connection matrices, the Chern connection has ${\theta }_{\beta }-g_{\alpha \beta }{\theta }_{\alpha }{g}_{\alpha \beta }^{-1}={\sigma }_{\alpha \beta }$. These holomorphic one-forms, interpreted in the beta frame, satisfy the transported Čech cocycle condition because $g_{\alpha \gamma }=g_{\beta \gamma }{g}_{\alpha \beta }$. Local primitives of curvature differ by this cocycle. A holomorphic frame change $e_{\alpha }^{\prime }=u_{\alpha }{e}_{\alpha }$ adds the Čech coboundary of $u_{\alpha }^{-1}\partial u_{\alpha }$, so the resulting curvature Dolbeault class is intrinsic.

Composition of endomorphisms and wedge product of differential forms give the class $[{\Theta }^k]\in H^k(M,{\Omega }_M^k\otimes \mathrm{End}E)$. Even total degree of vector-bundle curvature and the noncommutative telescoping identity give ${\Theta }_1^k-{\Theta }_0^k=\bar{\partial }{\sum }_{j=0}^{k-1}{\Theta }_1^{j}a{\Theta }_0^{k-1-j}$ whenever ${\Theta }_1-{\Theta }_0=\bar{\partial }a$. Thus all powers are metric independent; no commutativity of endomorphisms is assumed.