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?

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