TITLE: Chern classes of the sheaf of LOG differentials
QUESTION [6 upvotes]: 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 
$$c_1(\Omega_X^1(\log D))= c_1(\Omega_X^1)+D.$$

REPLY [3 votes]: 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.