5
$\begingroup$

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?

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Thank you Francesco! Do you know of any more standard reference? $\endgroup$
    – reef
    Oct 17, 2014 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. $\endgroup$ Oct 17, 2014 at 18:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.