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The Chern connection of a Hermitian metric on a holomorphic vector bundle is the unique connection on a vector bundle compatible with that metric and whose $(0,1)$ part is the bundle's Dolbeault operator ${\bar{\partial }}_E$. Fix a holomorphic local frame and column coefficients for sections. Write the metric as $h(s,t)={\bar{s}}^{T}Ht$, conjugate-linear in the first argument, and write the connection as $Ds=ds+As$. The condition $D^{0,1}={\bar{\partial }}_E$ forces $A$ to have type $(1,0)$. Metric compatibility requireswhere the dagger conjugates the differential-form coefficients as well as transposing the matrix. Taking the $(1,0)$ part gives $\boxed{A=H^{-1}\partial H}$. Its conjugate-transpose supplies the $(0,1)$ metric equation because $H$ is Hermitian. This proves uniqueness and local existence.

Under a holomorphic change of frame $e^{\prime }=eg$, the metric matrix becomes $H^{\prime }=g^{†}Hg$. The local formula for the Chern connection on a vector bundle then givesThis is precisely the transformation rule for a connection on a vector bundle, so the local connections glue and establish global existence. No Kähler hypothesis is needed for this part.

Extend $D$ to vector-bundle-valued differential forms by the graded Leibniz rule. The curvature form of a connection is the tensorial square $F_h=D^2$, acting by exterior multiplication. In the chosen frame,Indeed $\partial A+A\wedge A=0$ by differentiating $H^{-1}H=I$. It follows that $F_h$ has type $(1,1)$; the gauge change is $F_h^{\prime }=g^{-1}{F}_{h}g$. Thus it is a global smooth two-form with values in $\mathrm{End}(E)$, namely an element of ${\mathcal{A}}^2(\mathrm{End}(E))$.

Fix the convention that the Hermitian metric is conjugate-linear in its first argument. In a holomorphic local frame $e=(e_1,\dots ,e_r)$, let $H_{ij}=h(e_i,e_j)$, so $h(ev,ew)=v^{†}Hw$. Write $De=e\theta$, or equivalently $D(ev)=e(dv+\theta v)$.

Compatibility with the holomorphic vector bundle structure means $D^{0,1}={\bar{\partial }}_E$. Since the frame is holomorphic, this forces ${\theta }^{0,1}=0$. Metric compatibility meansor, in the chosen frame, $dH={\theta }^{†}H+H\theta$. Conjugate transpose here also conjugates the one-form factors. Taking the $(1,0)$ component gives $\partial H=H\theta$, and henceThis proves uniqueness and suggests existence. The proposed $\theta$ has type $(1,0)$, and its conjugate transpose obeys ${\theta }^{†}H=\bar{\partial }H$, since $H=H^{†}$. Therefore $dH={\theta }^{†}H+H\theta$, proving metric compatibility as well.

Under a change of holomorphic local frame $e^{\prime }=eG$, we have $H^{\prime }=G^{†}HG$ and $\partial G^{†}=0$. A direct computation yieldsBecause $G$ is holomorphic, $\partial G=dG$; this is exactly the transformation rule for a connection one-form. The locally defined connections therefore glue to a global connection on a vector bundle. The Chern connection exists uniquely, and its local formula for the Chern connection on a vector bundle is

In a holomorphic local frame $e$, let $H=(h(e_i,e_j))$ and write $De=eA$. Holomorphic compatibility forces $A^{0,1}=0$, while metric compatibility forces the local formula for the Chern connection on a vector bundleThis proves uniqueness. The formula transforms by the connection one-form law under a holomorphic frame change, so the local definitions glue and satisfy both conditions. This proves existence of the unique Chern connection.