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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.

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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. 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 a the relation for $D^2$.{Trace}(\iota|H^2(S,\mathbb C)) = 2-D^2$. 1 In general, the fixed locus of an involution$\iota$on a complex surface$S$is the union of a smooth curve$D$and of$k$isolated points. There are trace formulae that relate these and the action of$\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 a relation for$D^2\$.