# Generalization of the Lefschetz fixed point theorem

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!

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This is proved in SGA 5, Exposé III, page 111. The statement there is slightly more general: $\mathbb{C}$ can be replaced by any field and $g$ by any endomorphism. –  Marc Hoyois May 16 '13 at 14:39
@Marc: I don't understand your comment. The cohomology above is presumably the topological cohomology. Over an arbitrary field is it replaced by étale cohomology or something? In that case it seems like it's ambiguous what it means to interpret this statement over $\mathbb{R}$. –  Qiaochu Yuan Oct 4 '13 at 23:59
@Qiaochu: Yes, étale cohomology with coefficients in $\mathbb{Q}_\ell$. Over an arbitrary field $k$, the RHS is replaced by a trace in the derived category of étale $\mathbb{Q}_\ell$-sheaves over $Spec(k)$. But both sides of the equation are integers and hence are invariant under change of base fields, so you can always make a change of base to an algebraic closure first. –  Marc Hoyois Oct 16 '13 at 15:26
@Marc: it's not the RHS I'm worried about, it's the LHS. What does "Euler characteristic of the fixed locus" mean for a variety over $\mathbb{R}$? I would be surprised if it could be taken to mean the topological Euler characteristic of the fixed locus of $\mathbb{R}$-points (in such a way that the statement remains true). –  Qiaochu Yuan Oct 20 '13 at 5:18
@Qiaochu: Sorry for taking so long to reply; I don't log in often. The LHS is of the same nature as the RHS: the étale Euler characteristic is the trace of the identity in the derived category of étale sheaves over the base field. This is invariant under base change, so over $\mathbb{R}$ it equals the topological Euler characteristic of the $\mathbb{C}$-points. To recover the Euler characteristic of the $\mathbb{R}$-points you need something finer than étale cohomology, e.g. $C_2$-equivariant étale cohomology (traces then live in the Burnside ring of $C_2$). –  Marc Hoyois Nov 15 '13 at 16:40

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 $g = \mathrm{id}$!)