Let $X$ be a smooth variety over $\mathbb{C}$. Let $D \subset X$ be an effective Cartier divisor.

Question 1 What is the definition of the logarithmic differential sheaf $\Omega^1_X (\log D)$ ?

I saw the definition in the book by Esnault-Viehweg (meromorphic form $\alpha$ such that $\alpha, d \alpha$ has pole of order 1 along $D$ ?). If there is another definition, I would be happy.

And I have another question. If $D$ is normal crossing, there is an sequence

$0\rightarrow \Omega^1_X \rightarrow \Omega^1_X(\log D) \rightarrow \nu_* \mathcal{O}_{\tilde{D}} \rightarrow 0 $

where $\nu: \tilde{D} \rightarrow D$ is the normalization.

Question 2 If $D$ is not normal crossing, is there a similar exact sequence? Is there a suitable way to define a residue map $\Omega^1_X(\log D) \rightarrow \nu_* \mathcal{O}_D$?

Let $X$ be a smooth variety over $\mathbb{C}$. Let $D \subset X$ be an effective Cartier divisor.

Question 1. What is the definition of the logarithmic differential sheaf $\Omega^1_X (\log D)$ ?

I saw the definition in the book by Esnault-Viehweg (meromorphic form $\alpha$ such that $\alpha, d \alpha$ has pole of order 1 along $D$ ?). If there is another definition, I would be happy.

And I have another question. If $D$ is normal crossing, there is an sequence $$0\rightarrow \Omega^1_X \rightarrow \Omega^1_X(\log D) \rightarrow \nu_* \mathcal{O}_{\tilde{D}} \rightarrow 0, $$ where $\nu: \tilde{D} \rightarrow D$ is the normalization.

Question 2. If $D$ is not normal crossing, is there a similar exact sequence? Is there a suitable way to define a residue map $\Omega^1_X(\log D) \rightarrow \nu_* \mathcal{O}_D$?