& 
{\rm {\mathbb E}xt } ^2 (0 \rightarrow \Omega ^1 _C , {\mathcal O}_C)}$$

We can now rewrite many of these terms. First of all, the stability condition is equivalent to ${\rm Hom } \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) = 0$. Also, remembering the fact that all the complexes are concentrated in degrees $-1$ and 0, and using the fact that $f^* \Omega _X ^1$ is locally free, that its dual is $f^* {\mathcal T}_X$ and the isomorphisms ${\rm Ext} ^i (f^* \Omega _X ^1 , {\mathcal O}_C) \simeq 
{\rm H} ^i (C, {\mathcal T}_X)$ we obtain the sequence $$\label{preserra}
\xymatrix@C=20pt @R=3pt {0 \ar[r] & {\rm Hom } (\Omega ^1 _C , {\mathcal O}_C) \ar[r] &
{\rm H}^0 (C, f^* {\mathcal T}_X) \ar[r] & 
{\rm {\mathbb E}xt } ^1 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) \ar[r] &\\
\ar[r] & {\rm Ext } ^1 (\Omega ^1 _C , {\mathcal O}_C) \ar[r] &
{\rm H}^1 (C, f^* {\mathcal T}_X) \ar[r] & 
{\rm {\mathbb E}xt } ^2 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) \ar[r] & 0}$$

In particular we see that if ${\rm H}^1 (C, f^* {\mathcal T}_X)=0$, then the obstruction group ${\rm {\mathbb E}xt } 