algebraic proof of Atiyah-Bott fixed point formula? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:51:44Z http://mathoverflow.net/feeds/question/80951 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80951/algebraic-proof-of-atiyah-bott-fixed-point-formula algebraic proof of Atiyah-Bott fixed point formula? unknown 2011-11-15T03:58:05Z 2011-11-15T13:27:10Z <p>Hi,</p> <p>Atiyah and Bott apparently proved the following theorem:</p> <ul> <li>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 $x\in X$.</li> </ul> <p>Where can one find an algebraic proof of this result?</p> <p>Thanks!</p> http://mathoverflow.net/questions/80951/algebraic-proof-of-atiyah-bott-fixed-point-formula/80963#80963 Answer by Niels for algebraic proof of Atiyah-Bott fixed point formula? Niels 2011-11-15T08:18:45Z 2011-11-15T08:18:45Z <p>I am not sure this is the best place to learn the subject, but at least this book is an algebraic reference:</p> <p>Riemann-Roch algebra By William Fulton, Serge Lang</p> <p>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.</p> http://mathoverflow.net/questions/80951/algebraic-proof-of-atiyah-bott-fixed-point-formula/80964#80964 Answer by Damian Rössler for algebraic proof of Atiyah-Bott fixed point formula? Damian Rössler 2011-11-15T08:27:16Z 2011-11-15T08:27:16Z <p>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.</p>