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Let $D$ be a normal crossings divisor on a smooth projective algebraic variety $X$. Put $U=X -D$ and $j: U \hookrightarrow X$.

One usually defines the sheaf of logarithmic differentials along $D$ as the subsheaf $\Omega^p_X(\log D)$ of $j_\ast \Omega^p_U$ such that, for any open $V \subset X$, $\Omega^p_X(\log D)(V)$ consists of those $\omega \in j_\ast \Omega^P_U(V)$ such that both $\omega$ and $d\omega$ have at worst simple poles along $D \cap V$.

My question concerns the precise, rigurous definition of "having simple poles". I've seen the following:

Definition. $\omega \in j_\ast \Omega^p_U(V)$ has at worst simple poles along $D \cap V$ if, for any affine open $W \subseteq V$ such that the ideal of $D \cap W$ in $\mathcal{O}_X(W)$ is principal and for any generator $f$ of this ideal,

the section $f \cdot \omega_{| U \cap W}$ lies in the image of the restriction $\Omega^p_X(W) \to \Omega^p_X(U \cap W)$.

Q1: Can anybody explain to me why this definition agrees with the intuitive idea of simple poles.

Q2: Is there a simpler definition?

For instance, if we see $j_\ast \Omega^p_U$ as the direct limit $$ \varinjlim_{n} \Omega^p_X(n D) $$ can one somehow say that $\omega \in j_\ast \Omega^p_U(V)$ has at worst simple poles if it "belongs" to $\Omega^p_X(D)(V)$?

Thanks for your help :)

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1 Answer

Your definition is the same as that in complex analysis.

Let $X = P^1(\mathbb{C})$, $D \colon z=0$. Then a meromorphic differential with a simple pole along $D$ is precisely of the form $\omega = f \frac{dz}{z}$, $f$ holomorphic. This agrees with your definition: $z$ is a generator of $\mathcal{I}_D$, the ideal of holomorphic functions with a zero of order one at $z=0$, and $z \cdot \omega$ is a holomorphic differential.

This generalizes to normal crossing divisors on complex manifolds in several variables, where locally, $X = \mathbb{C}^n$, $D\colon z_1 \cdots z_r = 0$ for some $r < n$, $\mathcal{I}_D = (z_1 \cdots z_r)$, and $\mathcal{O}^\cdot_X(\log(D))$ is generated over $\mathcal{O}_X$ by $\frac{dz_1}{z_1},..., \frac{dz_r}{z_r}, dz_{r+1}, ..., dz_n$.

I hope this helps!

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