exact sequence of logarithmic differential sheaves associated to an effective Cartier divisor on a smooth variety

Question

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 $\Omega^1_X(\log D) \rightarrow \nu_* \mathcal{O}_D$?

Answer 1

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