PomeProposition 11.14.3.7215288BDF53 On the smooth bundle underlying $E_0$ there is a holomorphic one-parameter family of Dolbeault operators $\bar\partial _ t=\bar\partial _0+tA$, $t\in \mathbb C$, such that for $t\ne 0$ the holomorphic bundle defined by $\bar\partial _ t$ is $\mu _ P$-stable and prioritary, but not ample. As $t\to 0$, these operators converge to $\bar\partial _0$ in the $C^ m$ norm for every integer $m\geq 0$.
Proposition 11.14.37215288BDF53
Proof. Choose a nonzero extension class and a smooth Dolbeault representative $\eta \in \Omega ^{0,1}(X,L_2^*\otimes L_1)$ of it, so that $\bar\partial \eta =0$. On the smooth direct sum, set
\[ A=\begin{pmatrix} 0
& \eta
\\ 0
& 0
\end{pmatrix}. \]
The covariant Dolbeault derivative of $A$ is zero and $A\wedge A=0$, so $\bar\partial _ t^2=0$. Adding the Dolbeault operator of the base $\mathbb C$ of the parameter $t$ also gives an integrable operator on $X\times \mathbb C$, because the family depends holomorphically on $t$. This defines a holomorphic family of bundles whose extension class is $t[\eta ]$. It is nonsplit for $t\neq 0$, so the two preceding propositions apply. For a fixed smooth background connection and fixed metrics,
\[ \| \bar\partial _ t-\bar\partial _0\| _{C^ m} =|t|\, \| A\| _{C^ m}\longrightarrow 0, \]
and the norm $\| A\| _{C^ m}$ is finite for every $m$ because $X$ is compact. $\square$
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