Let $D$ be a reduced projective scheme over $\mathbb{C}$ such that $H^1(D,\mathcal{O}_D) = 0$ and $D$ is Gorenstein. There is a map

\begin{equation} r:= \frac{d \log}{2 \pi i }: H^1(D, \mathcal{O}_D^{\ast}) \otimes \mathbb{C} \rightarrow H^1(D,\Omega_D^1) \end{equation} locally defined by $f \mapsto \frac{df}{f}$.

* Question* Is this homomorphism injective?

If $D$ is smooth or a V-manifold, it is known that this is injective. If $D$ is general, I considered the following argument. If there is a mistake, please let me know about it. ;

Let $(\underline{\Omega}_D^{\bullet},F)$ be the Du bois complex on $D$. Then there is a homomorphism $H^1(D, \Omega_D^1) \rightarrow \mathbb{H}^1(D, \underline{\Omega}^1_D) $ where $\underline{\Omega}^1_D := Gr_F^1 \underline{\Omega}^{\bullet}_D[1]$.

Consider homomorphisms $ t : H^1(D, \mathcal{O}_D^{\ast}) \rightarrow H^2(D, \mathbb{C})$ which is induced by the exponential exact sequence $0 \rightarrow \mathbb{Z} \rightarrow \mathcal{O}_D \rightarrow \mathcal{O}_D^{\ast} \rightarrow 0$ and the homomorphism which is composition of the form $H^1(D, \mathcal{O}_D^{\ast}) \stackrel{r}{\rightarrow} H^1(D, \Omega^1_D) \stackrel{s}{\rightarrow} \mathbb{H}^1(D, \underline{\Omega}^1_D) \stackrel{u}{\rightarrow} H^2(D, \mathbb{C})$.

I don't know how to define $u$ in a natural way. Take a hyperresolution $f_{\bullet}: D_{\bullet} \rightarrow D $. I think that $\underline{\Omega}_D^1$

can be expressed by using the terms come from $\Omega^1_{D_{\bullet}}$ on each $D_{\bullet}$.

and there is a complex homomorphism $\underline{\Omega}^1_D \rightarrow \underline{\Omega}_D^{\bullet}$ induced by the expression above and define $u$ by this complex homomorphism.

If $t = u \circ s \circ r$, then $r$ is injective.
Is there a mistake in this argument?

Moreover, I'm not familiar with arguments using Du Bois complex. I don't know the above arguments make sense. If there are useful literatures, please let me know about it.

(add) I had a mistake in the definition of $\underline{\Omega}_D^1$. I forgot a shift by 1.

I'm readin p.174 of the book "Mixed Hodge Structures" by Peters and Steenbrink.
I thought that I can define
$\underline{\Omega}_D^1 \rightarrow \underline{\Omega}_D^{\bullet}$ whose homomorphisms on $k$-th term are defined by using the direct summand inclusions
$ (f_k)_{*} \Omega_{D_k}^1 \rightarrow \oplus_{p+q = k+1} ( f_q )_{*} \Omega_{D_q}^p$.