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

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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
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1 Answer

up vote 12 down vote accepted

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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Dear Rita,Thanks. I will be pleased if you could tell me where I can find the proof of the above formulas and that the fixed point locus are isolated curves and points? –  user13559 Sep 21 '11 at 18:53
    
I've edited the answer and inserted the references. –  rita Sep 21 '11 at 20:21
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