Fixed-point free holomorphic involutions Here is the new version of the question which is more explicit. The older version is below.
I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free holomorphic involution that sit as a hypersurface in variety with lots of rational curves (say $\mathbb{P}^n$). Fermat quartic surface in $\mathbb{P}^3$ is an example. What are the examples and restrictions? e.g., for which $d\geq 3$, there is a hypersurface of degree $d$ in $\mathbb{P}^n$ admitting such an involution?

In complex dimension 1, (some) Riemann surfaces of odd genus admit a holomorphic involution without fixed-point. In complex dimension 2, abelian surfaces and K3 surfaces are the first examples that come to mind (for admitting a holomorphic involution without fixed-point). There are possibly other example that are elliptic fibrations over a curve.
I am looking for a larger pool of such varieties in complex dimensions 2 and 3.

*

*What are the restrictions imposed by the existence of a holomorphic involution without fixed-point? (For instance, rationally connected varieties do not admit such an involution)


*Are all examples in dim 2 and 3 abelian or K3 fibered varieties?
 A: For $n\geq 4$, a smooth hypersurface  $X\subset\mathbb{P}^n$ never admits a fixed point free involution. By the Lefschetz theorem $\operatorname{Pic}(X) $ is cyclic, so any automorphism of $X$ preserves $\mathscr{O}_X(1)$, hence is induced by an automorphism of $\mathbb{P}^n$. Now an involution of $\mathbb{P}^n$ has two fixed subspaces of dimension $p$ and $q$ with $p+q=n-1$, so one (at least) of these subspaces intersect $X$.
The same result holds for $n=3$ and $\deg (X)\neq 4$. In this case $\mathscr{O}_X(1)$ is the unique line bundle $L$ on $X$ such that $L^{d-4}=K_X$, hence it is again preserved by any automorphism of $X$, and the same argument applies.
A: The general recipe for constructing such varieties is the following.
Start with your favorite smooth variety $Y$ such that $\operatorname{Pic}^0(Y) \neq 0$ (for instance, this condition is automatically satisfied if $H^0(Y, \, \Omega_Y^1) \neq 0$), and choose a non-zero, $2$-torsion divisor $\mathcal{L}$ in $\operatorname{Pic}^0(Y)$. Correspondingly, there is an étale double cover $$\pi \colon X \to Y, \quad \pi_*\mathcal{O}_X= \mathcal{O}_Y \oplus \mathcal{L},$$ and the generator of the deck transformations of $\pi$ is a fixed-point free holomorphic involution on $X$.
The Kodaira dimension can only increase under this procedure, namely $\operatorname{kod}(X) \geq \operatorname{kod}(Y)$. In particular, if we start with $Y$ a variety of general type (for example, a product of curves of genus $\geq 2$), it follows that $X$ is of general type, too. Thus, the answer to your second question is no.
