= Holomorphic Lefschetz fixed-point formula
For a <holomorphic map> $f:X\to X$ of a <compact> <complex manifold> whose <fixed points> are isolated and satisfy $\det_{\mathbb C}(1-df_x)\ne0$, the <holomorphic Lefschetz fixed-point formula> for the structure sheaf says
$$
\sum_{q=0}^{\dim_{\mathbb C}X}(-1)^q\operatorname{tr}\!\left(f^*:H^q(X,\mathcal O_X)\to H^q(X,\mathcal O_X)\right)
=\sum_{f(x)=x}\frac1{\det_{\mathbb C}(1-df_x)}.
$$
The left side uses <sheaf cohomology> of <holomorphic functions>; the right side uses the complex <determinant>, rather than its absolute value. This is a holomorphic form of the <Atiyah-Bott fixed-point theorem>.<A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications by M. F. Atiyah and R. Bott>[section 4, holomorphic maps and their local contributions]{ref}
On the <complex projective line>, $f(z)=az$, with $a\ne0,1$, fixes zero and infinity. The two derivatives are $a$ and $a^{-1}$, so the local sum is $1/(1-a)+1/(1-a^{-1})=1$. Globally $H^0(\mathcal O)=\mathbb C$ and $H^1(\mathcal O)=0$, giving the same answer. Individual local terms need not be integers.
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