TITLE: algebraic proof of Atiyah-Bott fixed point formula?
QUESTION [8 upvotes]: 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!

REPLY [5 votes]: 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.

REPLY [4 votes]: Notice that you must assume that the graph of $f$ intersects the diagonal tranversally (otherwise some determinants in the formula might vanish). This transversality condition is automatic if $f$ has finite order. With that assumption, the above formula is a special case of the "Woods hole" formula, which is proven using Grothendieck duality in SGA 5 (Springer Lecture Notes in mathematics 589), Appendix to Exp. III, Cor. 6.12, p. 131.