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

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    $\begingroup$ Almost nothing. You can deduce something from exact sequence $0 \to H^1({\mathcal{Hom}}(F,G)) \to Ext^1(F,G) \to H^0({\mathcal{Ext}}^1(F,G)) \to H^2({\mathcal{Hom}}(F,G))$, but this does not give much. $\endgroup$
    – Sasha
    Commented Feb 8, 2014 at 20:04

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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 $\mathrm{Ext}^1(\mathcal{F},\mathcal{G})=0$.

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  • $\begingroup$ This I know. My question is extremely vague. Now I'll edit it. $\endgroup$
    – Puzzled
    Commented Feb 9, 2014 at 12:00

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