# automorphism of Enrique surface

What is the fixed point set of an order two automorhism group of an Enriques surface.

• Are there two questions here? 'Why?' is perhaps a bit too broad for MO. Would you accept "it follows from the axioms of ZFC"? :P – David Roberts Sep 14 '11 at 3:24
• Ill edit it, if that bothers you. I was wondering if someone could analyze this if he/she doesnt have an answer. – user13559 Sep 14 '11 at 5:05
• What do you want to know about the fixed point set? – J.C. Ottem Sep 14 '11 at 9:32
• That what it is? Curves?points? how many? – user13559 Sep 15 '11 at 1:40

In general, the fixed locus of an involution $\iota$ on a smooth complex surface $S$ is the union of a smooth curve $D$ and of $k$ isolated points. This follows by Cartan's Lemma that says that in suitable holomorphic coordinates near a fixed point the action is linear.
There are trace formulae that relate these and the action of $\iota$ on the cohomology of $S$. (Holomorphic Fixed Point Formula): $$\sum_{i=0}^2(-1)^i\text{Trace}(\iota|H^i(S,{\mathcal O}_S)) = \frac{k-D\cdot K_S}{4}$$
(Topological Fixed Point Formula): $$\sum_{i=0}^4(-1)^i\text{Trace}(\iota|H^i(S,\mathbb C)) = k+e(D)$$ where $e(D) = -D^2-D\cdot K_S$ is the topological Euler characteristic of $D$.
In the case of Enriques surface, $h^i({\mathcal O}_S)=0$ for $i=1,2$ and $h^1(S,{\mathbb Z})=0$, $h^2(S,{\mathbb Z})=10$. So the formulae above give you $k=4$ and the relation ${Trace}(\iota|H^2(S,\mathbb C)) = 2-D^2$.