# Euler characteristic of open varieties as degree of Chern class of logarithmic differentials

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?

A simple explaination of the equivalent dual statement, i.e. using $T_X(- \log D)$ instead of $\Omega^1(\log D),$ can be found in the book by Burt Totaro Group cohomology and algebraic cycles, page 25.
• You are welcome. Well, in the other few books that I checked, I did not find the explicit computation of $\chi(U)$ in terms of $c_n$ of the logarithmic (co)tangent bundle. I think that the original computation is due to Deligne (maybe Theorie de Hodge II?), you should try and look there. Oct 17, 2014 at 18:34