Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$ is supported on $Sing(X)$.


Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ has finite quotient singularities and $codim(Sing(X))\geq 3$ then $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X}) = 0$. However I can not find a reference for this. A reference will help.


Now, let say that there is a component $Y$ of $Sing(X)$ in codimension two. So $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$ should be supported on $Y$. If, in this particular setting, we know that $Ext^{1}(\Omega\_{X},\mathcal{O}\_{X}) = 0$ can we say something about $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$?



Global to local for Ext groups and Sheaves
What do you expect, $\mathcal{E}xt^1(\mathcal{F},\mathcal{G})=0$? This is not true. Take for $X$ a smooth surface, $\mathcal{G}=\mathcal{O}\_X$ and $\mathcal{F}=\mathcal{O}\_C$ for $C$ a smooth irreducible curve in $X$ with $C^2<0$. Then $\mathcal{E}xt^1(\mathcal{F},\mathcal{G})=N\_{C/X}$, the normal bundle of $C$ in $X$, but $\mathrm{Ext}^1(\mathcal{F},\mathcal{G})=0$.


