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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 $$\sum_{i=0}^{\dim X}(-i)^itr(fX}(-1)^itr(f, H^i(X,L)) = \sum_{x\in X^f}\frac{tr(f,L_x)}{\det(1-T_xf)}$$ where $T_xf : T_xX\to T_xX$ is the derivative of $f$ at $x\in X$.

Where can one find an (algebraic) algebraic proof of this result?

Thanks!

1

# algebraic proof of Atiyah-Bott fixed point formula?

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!