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$.



Question about arguments using Du Bois complex  
Let me convert my obscure comment into an (obscure?) answer, with some corrections.


The problem with your argument as it stands is that a map is not well defined:
$H^1(D,\underline{\Omega}\_D^1)$ is only a subquotient of $H^2(D,\mathbb{C})$.
But the problem is minor. To fix things observe that $H^2(D)$ carries a mixed Hodge structure. 
Also the image $im(c\_1)$ of the Chern class map (what you call $t$) lies in $F^1\cap \bar F^1$ because it lies in
$$ker[H^2(D,\mathbb{C})\to H^2(D,\mathcal{O}\_D)]\subseteq F^1$$
and is invariant under conjugation.
The space $F^1\cap \bar F^1$ maps injectively to 
$H^1(D,\underline{\Omega}\_D^1)=Gr\_F^1H^2(D,\mathbb{C})$. Now by your assumption $c\_1$ is injective, so the map to $H^1(D,\underline{\Omega}\_D^1)$ is also injective, and as (the normalized) $dlog$ factors through it, it is also injective.


References:
I'm using the standard facts from the papers of Deligne and Du Bois (which should be in
Peters-Steenbrink). For more information about $c\_1$ in this setting,
see Barbieri Viale and Srinivas "The Neron Severi groups and the mixed Hodge structure 
on $H^2$" Crelles (1994); for higher Chern classes, see my paper with Kang "Kaehler-de Rham cohomology and Chern classes" Commun. Alg. 2011.


