[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 $e$.
[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]
[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]
