We know that for any pair of holomorphic vector bundles $\mathcal{E}_1$ and $\mathcal{E}_2$ over complex surface $X$(i.e. complex dimension 2), $\mathrm{Ext}^1(\mathcal{E}_1, \mathcal{E}_2)$ can be discribed as $\mathrm{H^{0,1}}(X, \mathrm{Hom}((\mathcal{E}_1, \mathcal{E}_2))$, the $\mathrm{Hom}^1(\mathcal{E}_1, \mathcal{E}_2)$-valued holomorphic 1-forms on $X$.
I wonder what will happen if we replace $\mathrm{Ext}^1(\mathcal{E}_1, \mathcal{E}_2)$ to line bundle $\mathcal{L}$ and ideal sheaf $\mathcal{I}_Z$ for some subvariety $Z$ with dimension $0$? I mean, can we give an analytic describption of $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{I}_Z, \mathcal{L})$?
I know the geometric description of $\mathrm{Ext}^1(\mathcal{E}_1, \mathcal{E}_2)$ can be understood as extending $\mathcal{E}_1$ to a higher holomorphic vector bundle by $\mathcal{E}_2$, i.e. finding all isomorphism class of short exact sequence:$0\rightarrow \mathcal{E}_2\rightarrow \mathcal{E}\rightarrow \mathcal{E}_1\rightarrow 0$, where the differential $\bar{\partial}_E$ of $\mathcal{E}$ is given by uppertriangle matrix: $$ \begin{bmatrix} \bar{\partial}_1 & \psi\\ 0&\bar{\partial}_2 \end{bmatrix} $$ Where $\psi\in\mathrm{H^{0,1}}(X, \mathrm{Hom}(\mathcal{E}_1 , \mathcal{E}_2))$ by integrability $\bar{\partial}_E\cdot\bar{\partial}_E=0$ and $\bar{\partial}_1$ the differential on $\mathcal{E}_1$, $\bar{\partial}_2$ the differential on $\mathcal{E}_2$. So for the case $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{I}_Z, \mathcal{L})$, as $\mathcal{I}_Z|_{X-Z}=\mathcal{O}_{X-Z}$, which is the trivial line bundle over $X-Z$, I think we can obtain similar $\bar{\partial}$ matrix: $$ \begin{bmatrix} \bar{\partial} & \phi\\ 0&\bar{\partial}_L \end{bmatrix} $$ on $X-Z$, here $\bar{\partial}$ is the standard one on $\mathcal{O}_{X-Z}$, and $\bar{\partial}_L$ the differential on $\mathcal{L}$. Since $\phi\in\mathrm{H^{0,1}}(X-Z, \mathrm{Hom}(\mathcal{O}_{X-Z}, \mathcal{L}))$, then $\bar{\partial}\phi=0$ on $X-Z$ and $\bar{\partial}\phi$ should be some 2-form valued functions surpported on $Z$, But do all 2-form valued functions surpported on $Z$ represent some elements in $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{I}_Z, \mathcal{L})$?
Answers and comments are both welcomed!