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 
$$c\_1(\Omega\_X^1(\log D))= c\_1(\Omega\_X^1)+D.$$



Chern classes of the sheaf of LOG differentials
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.


