Hi,


Atiyah and Bott apparently proved the following theorem:


* Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$.
Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely many fixed points $X^f$.
Then
$$
\sum\_{i=0}^{\dim X}(-1)^itr(f, H^i(X,L)) = \sum\_{x\in X^f}\frac{tr(f,L\_x)}{\det(1-T\_xf)}
$$
where $T\_xf : T\_xX\to T\_xX$ is the derivative of $f$ at $x\in X$.


Where can one find an algebraic proof of this result?


Thanks!



algebraic proof of Atiyah-Bott fixed point formula?
I am not sure this is the best place to learn the subject, but at least this book is an algebraic reference:


Riemann-Roch algebra By William Fulton, Serge Lang


more precisely VI \S 9 Lefschetz-Riemann-Roch . You can find your formula proven for an arbitrary vector bundle (not only a line bundle) under the name "fixed point formula". The machinery behind is quite heavy, tough, there is probably a more straightforward algebraic proof.


