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.

The corresponding link on googlebooks is here.

I am sure that it can be found in many other places, anyway this is just the first explicit reference I can remember now.

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  • $\begingroup$ Thank you Francesco! Do you know of any more standard reference? $\endgroup$ – reef Oct 17 '14 at 17:43
  • $\begingroup$ 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. $\endgroup$ – Francesco Polizzi Oct 17 '14 at 18:34

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