I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it is discussed or proved. The statement of the theorem should look vaguely like this:


Let $X$ be a smooth, complete variety over $\mathbb{C}$ and $g$ an automorphism of $X$. Suppose also that the fixed locus of $g$ is smooth and complete (but not necessarily discrete), and let $e(g)$ denote the Euler characteristic of this fixed locus. Then
$$
e(g)=\sum \_{i=0}^{\dim L} (-1)^i Tr (g | H^i(X)). 
$$


Any help or advice you can offer is much appreciated!



Generalization of the Lefschetz fixed point theorem
One of the standard proofs of the Lefschetz formula proceeds by showing that the RHS of your equation computes the intersection number between the graph of $f$ and the diagonal inside $X \times X$. The usual case is when this intersection has expected dimension. The general case requires the excess intersection formula. Now, the normal bundle of the diagonal $\Delta$ in $X \times X$ is the tangent bundle of $\Delta = X$. After dividing by the normal bundle of $X^g$ in $X$, the excess intersection formula shows that the intersection number is given by the top Chern class of the tangent bundle of $X^g$. But this is just the topological Euler characteristic of $X^g$, $e(g)$. (This last fact is just the special case of your formula when $g = \mathrm{id}$!)


