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Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$. Let $0 \in D=(x=f=0) \subset V$ be a divisor with only isolated singularity $0$ where $x$ is one of local coordinate.
Let $T^1_V, T^1_{(V,D)}$ be the set of 1st order deformations of $V$ and a pair $(V,D)$. We can consider the forgetting map $T^1_{(V,D)} \rightarrow T^1_V$. I want to consider the other way, that is, $T^1_V \rightarrow T^1_{(V,D)}$.

For $\eta \in T^1_V$, we can consider a small deformation $\mathcal{V} \subset \mathbb{C}^4 \times \Delta^1 \rightarrow \Delta^1$ of $V$ over a small disc $\Delta^1 \subset \mathbb{C}$ induced by $\eta$. Set $\Gamma := (x=0)$. I can consider $\mathcal{D} := \Gamma \times \Delta^1 \cap \mathcal{V} \subset \mathcal{V}$ and this defines a deformation of $D$.

Question 1 Does this construct $T^1_V \rightarrow T^1_{(V,D)}$? If yes, is it $\mathbb{C}$-linear? (Added) This construction seems to be not well-defined.

Moreover, I want to know how this is induced by a sheaf homomorphism. By $n=\dim V \ge 3$, we have

$T^1_V \simeq H^1(V', \Theta_{V'}) \simeq H^1(V', \Omega^2_{V'}(-K_V'))$,

$T^1_{(V,D)} \simeq H^1(V', \Theta_{V'}(\log D')) \simeq H^1(V', \Omega_{V'}^2 (\log D') (-K_{V'} -D')).$

We can consider an inclusion $\Omega^2_{V'} \hookrightarrow \Omega^2_{V'}(\log D'))$ and an isomorphism $O_{V'} \rightarrow O_{V'}(- D')$. Hence we have a homomorphism

$\Omega_{V'}^2 (-K_{V'}) \rightarrow \Omega^2_{V'}(\log D')(-K_V' -D')$.

Question 2 Does (Changed) Is the above homomorphism induce the homomorphism $T^1_{V} \rightarrow T^1_{(V,D)}$zero?

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