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?


  • 1
    $\begingroup$ Maybe you could try to restrict the exact sequence $$0 \to \Omega^1_X \to \Omega^1_X(\log D) \to \bigoplus \mathcal{O}_{D_j} \to 0$$ to the divisor $D_i$. $\endgroup$ – Francesco Polizzi Jan 29 '13 at 15:15
  • $\begingroup$ I don't manage to get it from here. Could you please give more details? Thanks! $\endgroup$ – ncd Jan 29 '13 at 16:41

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.


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