[proofplan] We construct the connection locally in a holomorphic frame and then prove that the resulting local connection matrices transform correctly on overlaps. In a holomorphic frame, the condition $D^{0,1}=\bar{\partial}_E$ forces the connection matrix to have type $(1,0)$, while metric compatibility forces the explicit formula $A=H^{-1}\partial H$, where $H$ is the Hermitian metric matrix. This formula gives uniqueness locally, and the holomorphic transition law shows that the local constructions glue to a global connection. [/proofplan]

[step:Write an arbitrary local connection in a holomorphic frame] Let $U \subset X$ be an [open set](/page/Open%20Set) admitting a holomorphic frame \begin{align*} e=(e_1,\dots,e_r) \end{align*} for $E|_U$. Every smooth section $s \in \Gamma(U,E)$ can be written uniquely as $s=e a$, where \begin{align*} a:U \to \mathbb{C}^r \end{align*} is a smooth coefficient map. A smooth connection $D$ on $E|_U$ is determined in this frame by a matrix-valued one-form \begin{align*} A \in \Omega^1(U;\operatorname{End}(\mathbb{C}^r)) \end{align*} through the formula \begin{align*} D(ea)=e(da+Aa). \end{align*} The decomposition of complex one-forms gives \begin{align*} A=A^{1,0}+A^{0,1}. \end{align*} Since the frame $e$ is holomorphic, the Dolbeault operator is \begin{align*} \bar{\partial}_E(ea)=e(\bar{\partial}a). \end{align*} Thus the condition $D^{0,1}=\bar{\partial}_E$ is equivalent to \begin{align*} e(\bar{\partial}a+A^{0,1}a)=e(\bar{\partial}a) \end{align*} for every smooth map $a:U\to\mathbb{C}^r$, hence \begin{align*} A^{0,1}=0. \end{align*} Therefore any such connection has a local connection matrix of type $(1,0)$. [/step]

[step:Use metric compatibility to determine the local matrix]Define the Hermitian metric matrix \begin{align*} H:U &\to \operatorname{Herm}^+_r \\ x &\mapsto (h_x(e_i(x),e_j(x)))_{i,j}. \end{align*} For smooth coefficient maps $a,b:U\to\mathbb{C}^r$, the metric is \begin{align*} h(ea,eb)=a^*Hb, \end{align*} where $a^*$ denotes the conjugate transpose of the column vector $a$. Assume $D^{0,1}=\bar{\partial}_E$ and $D$ is compatible with $h$. By the previous step, $A$ has type $(1,0)$. Apply metric compatibility to constant coefficient sections $s=ea$ and $t=eb$, where $a,b\in\mathbb{C}^r$ are constant vectors. Since $da=db=0$, \begin{align*} D(ea)=e(Aa), \qquad D(eb)=e(Ab). \end{align*} Metric compatibility gives \begin{align*} d(a^*Hb)=h(eAa,eb)+h(ea,eAb). \end{align*} Using conjugate-linearity in the first argument and linearity in the second, \begin{align*} h(eAa,eb)=a^*A^*Hb, \qquad h(ea,eAb)=a^*HAb. \end{align*} Hence \begin{align*} a^*(dH)b=a^*(A^*H+HA)b \end{align*} for all constant vectors $a,b\in\mathbb{C}^r$. Therefore \begin{align*} dH=A^*H+HA. \end{align*} Taking the $(1,0)$-part of both sides, and using that $A$ has type $(1,0)$ while $A^*$ has type $(0,1)$, gives \begin{align*} \partial H=HA. \end{align*} Because $H(x)$ is positive definite for every $x\in U$, it is invertible, so \begin{align*} A=H^{-1}\partial H. \end{align*} Thus any connection satisfying the two required conditions has this local matrix in the holomorphic frame $e$.[/step]

[guided]The point of this step is to extract the connection matrix from the metric. Define \begin{align*} H:U &\to \operatorname{Herm}^+_r \\ x &\mapsto (h_x(e_i(x),e_j(x)))_{i,j}. \end{align*} Since $h_x$ is positive definite, each matrix $H(x)$ is invertible. If $s=ea$ and $t=eb$ for smooth maps $a,b:U\to\mathbb{C}^r$, then \begin{align*} h(s,t)=h(ea,eb)=a^*Hb. \end{align*} We now test metric compatibility on the simplest possible sections: those with constant coefficient vectors. Let $a,b\in\mathbb{C}^r$ be constant and set $s=ea$, $t=eb$. Because $da=db=0$, the connection formula gives \begin{align*} D(ea)=e(Aa), \qquad D(eb)=e(Ab). \end{align*} Metric compatibility says \begin{align*} d(h(ea,eb))=h(D(ea),eb)+h(ea,D(eb)). \end{align*} Substituting the local expressions gives \begin{align*} d(a^*Hb)=h(eAa,eb)+h(ea,eAb). \end{align*} Since $h$ is conjugate-linear in the first argument and linear in the second, the two terms on the right become \begin{align*} h(eAa,eb)=a^*A^*Hb, \qquad h(ea,eAb)=a^*HAb. \end{align*} Therefore \begin{align*} a^*(dH)b=a^*(A^*H+HA)b \end{align*} for every pair of constant vectors $a,b\in\mathbb{C}^r$. Since equality of all matrix coefficients implies equality of matrices, \begin{align*} dH=A^*H+HA. \end{align*} Now use the type decomposition. From $D^{0,1}=\bar{\partial}_E$, the preceding step proved that $A$ has type $(1,0)$. Consequently $A^*$ has type $(0,1)$. Taking the $(1,0)$-part of \begin{align*} dH=A^*H+HA \end{align*} therefore removes the $A^*H$ term and yields \begin{align*} \partial H=HA. \end{align*} Multiplying on the left by $H^{-1}$ gives the forced formula \begin{align*} A=H^{-1}\partial H. \end{align*} Thus the local connection matrix is completely determined by the metric and the holomorphic structure.[/guided]

[step:Construct the local connection and verify the two defining properties] On every holomorphic trivializing open set $U\subset X$ with frame $e$, define \begin{align*} A_e:=H^{-1}\partial H \in \Omega^{1,0}(U;\operatorname{End}(\mathbb{C}^r)). \end{align*} For a smooth section $s=ea$, define \begin{align*} D_e(ea):=e(da+A_ea). \end{align*} Since $A_e$ has type $(1,0)$, \begin{align*} D_e^{0,1}(ea)=e(\bar{\partial}a)=\bar{\partial}_E(ea). \end{align*} It remains locally to check metric compatibility. Since $A_e=H^{-1}\partial H$, we have \begin{align*} \partial H=HA_e. \end{align*} Taking the conjugate transpose gives \begin{align*} \bar{\partial}H=A_e^*H, \end{align*} because $H^*=H$. Therefore \begin{align*} dH=\partial H+\bar{\partial}H=HA_e+A_e^*H. \end{align*} For arbitrary smooth maps $a,b:U\to\mathbb{C}^r$, \begin{align*} d(a^*Hb) &=(da)^*Hb+a^*(dH)b+a^*H\,db \\ &=(da+A_ea)^*Hb+a^*H(db+A_eb) \\ &=h(D_e(ea),eb)+h(ea,D_e(eb)). \end{align*} Thus $D_e$ has the required local properties. [/step]

[step:Check that the local connection matrices glue on overlaps] Let $U,V\subset X$ be holomorphic trivializing open sets with holomorphic frames $e$ and $\tilde e$. On $U\cap V$, write \begin{align*} \tilde e=e g, \end{align*} where \begin{align*} g:U\cap V \to GL(r,\mathbb{C}) \end{align*} is a holomorphic transition map. Let $H$ and $\tilde H$ be the metric matrices in the frames $e$ and $\tilde e$, respectively. Since $\tilde e=e g$, \begin{align*} \tilde H=g^*Hg. \end{align*} Because $g$ is holomorphic, $\bar{\partial}g=0$, and \begin{align*} \partial \tilde H &=\partial(g^*Hg) \\ &=g^*(\partial H)g+g^*H\,\partial g. \end{align*} Using $\tilde H^{-1}=g^{-1}H^{-1}(g^*)^{-1}$, we obtain \begin{align*} \tilde A &=\tilde H^{-1}\partial \tilde H \\ &=g^{-1}H^{-1}(g^*)^{-1}\bigl(g^*(\partial H)g+g^*H\,\partial g\bigr) \\ &=g^{-1}H^{-1}(\partial H)g+g^{-1}\partial g \\ &=g^{-1}Ag+g^{-1}\partial g. \end{align*} This is exactly the transformation law for connection matrices under the change of holomorphic frame $\tilde e=e g$. Hence the local operators $D_e$ agree on overlaps and define a global smooth connection $D_h$ on $E$. [/step]

[step:Conclude existence and uniqueness] The preceding gluing step produces a global smooth connection $D_h$ satisfying \begin{align*} D_h^{0,1}=\bar{\partial}_E \end{align*} and compatible with $h$, so existence holds. For uniqueness, let $D$ be any smooth connection on $E$ satisfying the same two conditions. On every holomorphic trivializing open set with frame $e$, its connection matrix $A$ must have type $(1,0)$ and, by metric compatibility, must satisfy \begin{align*} A=H^{-1}\partial H. \end{align*} This is precisely the local matrix used to define $D_h$. Therefore $D$ and $D_h$ have the same local connection matrices in every holomorphic frame, so $D=D_h$ globally. This proves both existence and uniqueness. [/step]