Let $X$ be a projective variety and let $\mathcal{F},\mathcal{G}$ be coherent sheaves on $X$.

If $Ext^{1}(\mathcal{F},\mathcal{G}) = 0$ can one say something about $\mathcal{E}xt^{1}(\mathcal{F},\mathcal{G})$?

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 $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$?