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Let $X$ be a normal projective variety which has only terminal singularities. Let $\Omega^1_X$ be the Kahler differential sheaf on $X$ and $\omega_X$ be the dualizing sheaf on $X$. For a coherent sheaf $F$ on $X$, let $F^{* *}$ be its double dual.

Question Are ${\rm Ext}^2 ((\Omega^1_X)^{* *}, \mathcal{O}_X)$ and ${\rm Ext}^2 ((\Omega^1_X)^{* *} \otimes \omega_X, \omega_X) $ isomorphic?

If $X$ is Gorenstein, they are isomorphic. I want to know the situation when $X$ is non-Gorenstein.

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I don't know any example, but I should strongly suspect that the answer is no, they are not isomorphic. Is there a reason you expect that they might be isomorphic? The $\text{Hom} = \text{Ext}^0$ is certainly an isomorphism. Is there a particular terminal singularity you have in mind? You should be able to check this question explicitly in Macaulay2. –  Karl Schwede Aug 5 '11 at 13:24
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In fact, I want a counterexample, particularly when $X$ is a $\mathbb{Q}$-Fano 3-fold. –  tarosano Aug 5 '11 at 14:19
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