Let $U$ be a smooth variety over a subfield $k$ of $\mathbb{C}$. Let $X$ be a smooth projective variety containing $U$ as the complement of a normal crossings divisor $D$. Denote by $\chi(U)$ the Euler characteristic of $U$, defined using either de Rham cohomology or singular cohomology of the complex analytic manifold attached to $U$. Let $\Omega^1_X(\log D)$ be the sheaf of logarithmic differentials. If $X$ has dimension $n$, then it is locally free of rank $n$ and one has the equality $$ \chi(U)=\deg c_n(\Omega^1_X(\log D)) $$

This is quite easy to prove using the Riemann-Roch theorem, but I have been unable to find a written proof. Can anybody provide a reference that I can cite in a paper?