Locus of fixed points of an involution in a surface 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 :).
 A: 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.
