Let $E$ be a rank two holomorphic vector bundle, consider the following extension of two holomorphic line bundles $$ \mathbb{E}:\ 0\rightarrow S\stackrel{j}{\rightarrow}E\stackrel{g}{\rightarrow} Q\rightarrow 0 $$ on a compact Riemann surface $X$ of genus $g\ge 1$. Assume that $E$ is a simple vector bundle and a hermitian metric $K$ on $E$ is given. $S$ and $Q$ can be endowed with the induced and quotient metrics respectively. Let $D_{E}$, $D_{S}$, $D_{Q}$ be the corresponding Chern connections, $D_{E}$ can be written $$ D_{E} = \begin{pmatrix} D_{S} & -\beta \\ \beta^{*} & D_{Q} \end{pmatrix}. $$ We have that $\beta^{*}=(g\otimes \rm{id})\circ \partial_{E}\circ j$, where $\partial_{E}$ is the $(1,0)$-part of $D_{E}$. So for each $\lambda\in \mathbb{C}\backslash \{0\}$, consider the extension $$ \mathbb{\lambda E}:\ 0\rightarrow S\stackrel{j/\lambda}{\longrightarrow}E\stackrel{g}{\rightarrow} Q\rightarrow 0, $$ then we have the corresponding $\tilde{\beta}^{*}=(g\otimes \rm{id})\circ \partial_{E}\circ (j/\lambda)=\frac{\beta^{*}}{\lambda}.$
Let $\Gamma\big({\rm Hom}(S,\,E)\big)$ be the space of holomorphic sections of ${\rm Hom}(S,\,E)$ and $\Gamma^{*}\big({\rm Hom}(S,\,E)\big)$ the set of nowhere vanishing holomorphic sections of ${\rm Hom}(S, E)$. Then $\Gamma^{*}\big({\rm Hom}(S,\,E)\big)$ is a Zariski open subset of $\Gamma\big({\rm Hom}(S,\,E)\big)$.
Then we can define a map \begin{align*} \phi \colon {\Bbb P}\Bigl(\Gamma^*\big({\rm Hom}(S,\,E)\big)\Bigr) &\to {\Bbb P}\Bigl(\Omega^{1,0}\big(X, {\rm Hom}(S, Q)\big)\Bigr) \\ j &\mapsto \beta^{*}. \end{align*} where ${\Bbb P}\Bigl(\Gamma^*\big({\rm Hom}(S,\,E)\big)\Bigr)$ and ${\Bbb P}\Bigl(\Omega^{1,0}\big(X, {\rm Hom}(S, Q)\big)\Bigr)$ are the projective space of $\Gamma^*\big({\rm Hom}(S,\,E)\big)$ and $\Omega^{1,0}\big(X, {\rm Hom}(S, Q)\big)$ respectively, $\beta^{*}$ is the second fundamental form of $S$ in $E$. My question is: Is this map injective?
Suppose that $j_{1}\ne cj_{2}$, where $c\in \mathbb{C}\backslash \{0\}$ is a constant. Let $\beta_{1}$ and $\beta_{2}$ be the adjoint of second fundamental forms corresponding to extensions $$ 0\rightarrow S\stackrel{j_{1}}{\rightarrow}E\stackrel{g_{1}}{\rightarrow} Q\rightarrow 0 $$ and $$ 0\rightarrow S\stackrel{j_{2}}{\rightarrow}E\stackrel{g_{2}}{\rightarrow} Q\rightarrow 0 $$ respectively. Since $E$ is simple, the above extensions are not isomorphic. (See page 275 of J. P. Demailly's book ``Complex analytic and differential geometry''). Then by the bijective correspondence from ${\rm Ext}_{X}^{1}(Q,S)$ to $H^{0,1}(X,{\rm Hom}(Q, S))$, we know that $[\beta_{1}]\ne [\beta_{2}]$, so $\beta_{1}\ne \beta_{2}$, where ${\rm Ext}_{X}^{1}(Q,S)$ is the isomorphism classes of extensions and $[\beta]\in H^{0,1}(X,{\rm Hom}(Q, S))$. Actually $\beta_{1}\ne c\beta_{2}$ by our condition, where $c\in \mathbb{C}\backslash \{0\}$ is a constant. I guess that $\beta_{1}^{*}\ne \lambda\beta_{2}^{*}$, where $\lambda\in \mathbb{C}\backslash \{0\}$ is a constant. Is this true? Thanks a lot!
