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 $$ \sum_{i=0}^{\dim X}(-i)^itr(f, H^i(X,L)) = \sum_{x\in X^f}\frac{tr(f,L_x)}{\det(1-T_xf)} $$ 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!

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!