Let $\Omega_X^1(\log D)$ be the (locally free) of logarithmic differentials on a smooth projective variety $X$ with respect to a simple normal crossing divisor $D$.

What are the Chern classes of $\Omega_X^1(\log D)$ in terms of the Chern classes of $\Omega_X^1$ and $D$? I think, the first one is $$c_1(\Omega_X^1(\log D))= c_1(\Omega_X^1)+D.$$


2 Answers 2


Let $D_1,\ldots,D_m$ be the irreducible components of $D$. The reasoning in Prp 2.3 in MR1240599 gives an exact sequence $$0\longrightarrow \Omega_X^1\longrightarrow \Omega_X^1(\log D)\longrightarrow\bigoplus_{i=1}^m{\mathcal O}_{D_i}\longrightarrow 0.$$ Thus, the total Chern class is given by $$c\big(\Omega_X^1(\log D)\big)=c(\Omega_X^1)\Big/\prod_{i=1}^m\big(1-[D_i]\big).$$ Mohammad's formula for the first Chern class is correct.


From the exact sequence $$0\rightarrow \Omega ^1_X\rightarrow \Omega ^1_X(\log D)\rightarrow \mathcal{O}_D\rightarrow 0$$you get $\ c(\Omega ^1_X(\log D))=c(\Omega ^1_X).c(\mathcal{O}_D)=c(\Omega ^1_X).(1-[D])^{-1}$, hence $$c_p(\Omega ^1_X(\log D))=c_p(\Omega ^1_X)+c_{p-1}(\Omega ^1_X).[D]+\ldots +[D]^p\ .$$

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    $\begingroup$ I think this is correct only if D is smooth. Take a look at Prp 2.3/2.4 in MR1240599. Both formulas there extend to arbitrary simple normal crossings divisors. $\endgroup$
    – Aleksey
    Jul 31, 2016 at 22:11

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