variation on an exact sequence of logarithmic differentials

Question

Let $X$ be a smooth projective complex variety and $D$ a divisor with simple normal crossings on $X$, with irreducible components $D_i$. If $D_1$ is one of these (smooth) irreducible components, then one has an exact sequence of sheaves $$ 0 \to \Omega^p_X(\log D)(-D_1) \to \Omega^p_X(\log(D-D_1)) \to \Omega^p_{D_1}(\log (D-D_1)_{|D_1}) \to 0 $$ (This is explained e.g. in section 2 of the book by Esnault and Viehweg).

I'm wondering what happens if you remove several irreducible components at the same time. That is, let $H$ a reduced divisor with irreducible components some of the $D_i$. Is there a similar exact sequence relating $\Omega^p_X(\log D)(-H)$ and $\Omega^p_X(\log(D-H))$? Of course the problem is that $H$ is not smooth anymore, so I don't know how to make sense of the last term... Thanks for your help!

Answer 1

For simplicity let's say $X = \mathbb C^2$ and $D = D_1 + D_2$ with $D_1 = \{ x = 0 \}$ and $D_2 = \{ y = 0 \}$. Then you have a short exact sequence $$ 0 \to \Omega_X^1(\log D)(-D) \to \Omega_X^1 \to \Omega_{D_1}^1 \oplus \Omega_{D_2}^1 \to 0. $$ Exactness can be checked by a local computation. The sheaves are freely generated by $\{ y \,\mathrm dx, x \,\mathrm dy \}$, $\{ \mathrm dx, \mathrm dy \}$, and $\{ (\mathrm dy, 0), (0, \mathrm dx) \}$, respectively. The second map is given by $\mathrm dx \mapsto (0, \mathrm dx)$ and $\mathrm dy \mapsto (\mathrm dy, 0)$.

This should work the same in general.