TITLE: Chern class of a logarithmic connection
QUESTION [8 upvotes]: Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c_1(E)$ vanishes. I am interesting in the following variant: let $D$ be a normal crossing divisor and $\nabla: E \to E \otimes \Omega^1_X(\log D)$ an integrable connection with logarithmic singularities along $D$. Denote 
$$
\mathrm{Res}_{D_i} \nabla
$$ the residue of $\nabla$ at an irreducible component $D_i$ of $D$. As $E$ has rank one, it can be identified with a complex number. 
Proposition. One has $c_1(E)=-\sum_i \mathrm{Res}_{D_i} \nabla \cdot [D_i]$
Could anybody explain me how to prove such a result?

REPLY [4 votes]: There exist a paper due to Makoto Ohtsuki about a Residue Formula for Chern Classes Associated with Logarithmic Connections:
http://projecteuclid.org/download/pdf_1/euclid.tjm/1270215030