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I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I do not know why the following is true.

If S is a minimal smooth projective surface with an involution i. Why the locus of fixed points of the involution i is either

1) a smooth curve (possibly reducible) and a finite collection of points, or

2) empty.

Maybe someone can give any advice about it :).

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  • $\begingroup$ The sentence beginning "If S is a smooth projective surface..." seems to be missing a word after minimal (if not, what does minimal refer to?). You may also wish to avail yourself of this MathJax tutorial to help format your post. Finally, is this surface assumed to be over $\Bbb C$ or over a more general field? Skimming the paper, it seems the author switches between contexts with some freedom, and I don't see the exact statement from your post. $\endgroup$
    – KReiser
    Mar 9, 2020 at 3:48
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    $\begingroup$ Dear KReiser sorry for my mess. A minimal surface is when it does not contain a (-1)-curve. $\endgroup$
    – Roxana
    Mar 10, 2020 at 17:01
  • $\begingroup$ Thank you for clarifying - I still would really recommend formatting your post with MathJax. $\endgroup$
    – KReiser
    Mar 10, 2020 at 17:56

1 Answer 1

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This is a case of a more general statement. For a finite order automorphism of a complex manifold the fixed point set is a collection of complex sub-manifolds (of possibly different dimensions). Firstly choose a $\mathbb{Z}_{2}$-invariant Hermitian metric on $X$, just by choosing any Hermitian metric and by averaging.

Now, consider a fixed point $p$. Then $\mathbb{Z}_{2}$ acts complex linearly on $T_{p}X$. The fixed point set of this representation is the Eigenspace for the Eigenvalue $1$, hence a complex subspace $V \subset T_{p}X$.

Then by considering the exponential map of the corresponding Riemannian metric, which is equivariant since the metric is invariant. The image of $V$ is precisely the fixed point set (in some neighbourhood of $p$), hence it is a complex submanifold.

Note that if the original complex manifold was a projective variety then the components of the fixed point set are smooth subvarieties by Chow's theorem.

Note that the fixed point set does not necessarily have to contain a curve i.e. the fixed point set can be just a finite collection of points. Consider the minimal surface $\mathbb{P}^1 \times \mathbb{P}^{1}$ with the involution $$([a_{1},a_{2}],[b_{1},b_{2}]) \mapsto ([-a_{1},a_{2}],[-b_{1},b_{2}]) .$$ Note that this involution has $4$ fixed points.

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  • $\begingroup$ Thank you Nick L! $\endgroup$
    – Roxana
    Mar 10, 2020 at 16:57
  • $\begingroup$ Dear Nick L, you help me a lot thanks!!!... Maybe you could give me the reference where you learned all this things (about fixed set of involutions :) Thank you. $\endgroup$
    – Roxana
    Mar 13, 2020 at 15:15
  • $\begingroup$ Hi, sorry for my late reply. I learnt this in a slightly different context. In fact, this result holds in many different categories smooth, symplectic, almost complex, complex, etc... It is a well known argument but I don't know a place where the complex case is written explicitly. $\endgroup$
    – Nick L
    Mar 16, 2020 at 7:06

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