euler characteristic + normal crossings divisor Let $D$ be a normal crossing divisors on a projective smooth algebraic variety $X$ over a field $k$ of characteristic zero. Put $n=\dim X$ and denote by $D_i$ the irreducible components of $D$. 
I'm trying to understand why the degree of the (n-1)-th Chern class of the restriction of the sheaf of logarithmic differential forms to a component $D_i$, that is 
$\deg c_{n-1}(\Omega^1_X(\log D)_{| D_i})$  
equals the Euler characteristic of $D_i - \cup_{j \neq i} D_j$. 
Can anybody help me?
Thanks! 
 A: Consider the following commutative diagram:
$$\begin{aligned} 
&&&& 0\qquad \qquad&\to  &0\qquad\qquad && \\
&&&&\downarrow\qquad\qquad && \downarrow\qquad\qquad \\
0 &\to &\Omega_X(\log D)(-D_i)\quad &\to &\Omega_X(\log (D-D_i))\quad &\to &\Omega_{D_i}(\log (D-D_i)|_{D_i})\quad &\to &0\\
&& \downarrow\,\simeq \qquad\qquad  &&\downarrow\qquad\qquad && \downarrow\qquad\qquad \\
0 &\to &\Omega_X(\log D)(-D_i)\quad &\to &\Omega_X(\log D)\qquad &\to &\Omega_X(\log D)|_{D_i}\quad &\to &0\\
&&&&\downarrow\qquad\qquad && \downarrow\qquad\qquad \\
&&&& \mathscr O_{D_i}\quad \qquad&\overset\simeq\to  & \mathscr O_{D_i}\quad\qquad && \\
\end{aligned}$$
The rows and the middle column are standard short exact sequences for logarithmic differentials. The 9-lemma implies that then the last column is also exact. Then the usual formula for Chern classes of short exact sequences (of locally free sheaves!) shows that 
$$
c_{n-1}(\Omega_X(\log D)|_{D_i})=c_{n-1}(\Omega_{D_i}(\log (D-D_i)|_{D_i})).
$$
This looks very close to what you probably need, but it seems to me that there might be a sign difference of $(-1)^{n-1}$. It is possible that I made a mistake somewhere, but in any case, this method should work. 
