TITLE: Examples of surface automorphisms with no periodic points
QUESTION [5 upvotes]: Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$.

What are some examples of surface automorphisms $g$ with no periodic points?

For example, $S$ can be an abelian variety and $g$ could be translation by some general point $\tau\in S$. Are there others, say on a rational surface, or a K3 surface?

REPLY [5 votes]: This is just the collection of my comments above.  First of all, for every automorphism $g$ (finite order or not) of every proper, smooth, separably rationally connected variety $X$ over any field $k$, the fixed scheme $X^g$ is nonempty.  So there will be no examples with $X$ a rational surface.
Second, by the holomorphic Lefschetz fixed point theorem, if $h^{1,0}(X) = 0$ or $1$ and if $h^{2,0}(X)$ equals $1$, then $g$ or $g\circ g$ has a fixed point.  If $h^{1,0}(X)$ equals $0$ and $h^{2,0}(X)$ equals $0$, then $g$ has a fixed point.
Finally, if $g$ preserves an effective divisor class, then $g$ preserves an effective divisor in that class.  Thus the effective divisor is genus $1$.  This rules out Kobayashi hyperbolic examples.