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



automorphism of Enrique surface
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 ${Trace}(\iota|H^2(S,\mathbb C)) = 2-D^2$. 


References: (1) the holomorphic fixed point formula can be found at pg.566 of M.F.Atiyah, I.M.Singer, The index of elliptic; 
(2) for the topological fixed point formula, see (30.9) of M. Greenberg, Algebraic Topology: A First Course,
W. A. Benjamin Publ., Reading, Mass. 1981.
operators: III,
 Ann. of Math. 87 (1968), 546-604.


