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



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



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> **Question 2.** If $D$ is not normal crossing, is there a similar exact sequence? Is there a suitable way to define a residue map $\Omega^1\_X(\log D) \rightarrow \nu\_\* \mathcal{O}\_D$? 
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exact sequence of logarithmic differential sheaves associated to an effective Cartier divisor on a smooth variety
Here is a way to define this sheaf algebraically over any field of characteristic zero. Let $\mathrm{T}\_{X}$ denote the tangent sheaf on $X$. Choose a local equation $\phi\_U$ for $D$ on $U$. Consider the following submodule:


$\mathrm{T}\_X(-\log\phi\_U)=\ ${$\partial\in\mathrm{Der}(\mathcal{O}\_X(U))\mid \partial\phi\_U\in (\phi\_U)$}$\ \subset \mathrm{T}\_X(U)$.


It is an easy exercise to check that this does not depend on the choice of equation and glues into a sheaf $\mathrm{T}\_X(-\log D)$. Now take the dual $\Omega^1\_X(\log D)=\mathrm{T}\_X(-\log D)^\*$.


The good thing about having normal crossings is that in this case $\Omega^1\_X(\log D)$ becomes a locally free sheaf.


For your second question, look at **Definition 2.5** in this paper by Dolgachev, arXiv:math/0508044


