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This question is related to this one and this one.

Let $X$ be a normal variety over algebraically closed field $k$ and $D$ be an effective Cartier divisor. Let $\zeta := (\mathcal{D} \subset \mathcal{X} \rightarrow {\rm Spec} A)$ be a deformation of the pair $(X,D)$ over Artin local $k$-algebra $A$. Let $ e := (0 \rightarrow (t) \rightarrow \tilde{A} \rightarrow A \rightarrow 0)$ be a small extension where $(t) \simeq k$.

Question Can we define an obstruction element

$o_{\zeta}(e) \in {\rm Ext}^2( \Omega^1_X( \log D), \mathcal{O}_X)$

with the property that, if $o_{\zeta}(e) =0$, then there exists a deformation $\tilde{\zeta} = (\tilde{\mathcal{D}} \subset \tilde{\mathcal{X}}\rightarrow {\rm Spec} \tilde{A})$ which is a lifting of $\zeta$?

If $X$ is smooth, the answer is in this post. If $X$ has only l.c.i. singularities and $D =0$, this is Proposition 2.4.8 in Sernesi's book. Is there a similar interpretation? If you know about some reference around this topic, please let me know about it.

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