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Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Put $n=\dim X$. It follows from the Hirzebruch-Riemann-Roch theorem that the degree of the top Chern class of the tangent bundle (i.e. $c_n(TX)$) is the Euler characteristic of $X$. Look at mathoverflow for a nice explanation.

Now let $D$ be a normal crossing divisor on $X$. One has the sheaves of logarithmic differentials $\Omega^i_X(\log D)$ and one can define the $TX(\log D)$ as the dual of $\Omega^1_X(\log D)$.

Question: Is it true that the top Chern class of $TX(\log D)$ is the Euler characteristics $\chi(X - D)$?

I guess one has to try to imitate the first proof using moreover the exact sequence

$$ 0 \to \Omega^1_X \to \Omega^1_X(\log D) \to \oplus \mathcal{O}_{D_i} \to 0 $$

where $D_i$ are the irreducible components of $D$. But I am a bit lost with the details... Could anybody help me?

I am also interested in the degree of $c_1(\Omega^1_X(\log D))\cdot c_{n-1}(\Omega^1_X(\log D)$. Does anybody know one to compute it?

Thank you very much!

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