Suppose $M$ is Calabi-Yau $m$-fold with complex structure $J$, Kahler form $\omega$, metric $g$ and holomorphic $m$-form $\Omega$. What are the conditions on $M$ for the existence of a map $\sigma: M \to M$ such that $\sigma^2 = id$, $\sigma^* (J)=-J$, $\sigma^* (\omega)=-\omega$, $\sigma^*(g)=g$, $\sigma^* (\Omega) = \bar \Omega$ ? in particular, suppose we see $M$ as a point in moduli space, then what are the restrictions on the moduli space so that the above involution exists?


Simplest example : An elliptic curve admits an involution. In general, If $X$ is a curve then $X/\tau$ is $\mathbb CP^1$ , where $\tau$ is an involution.

Borcea and Voisin construct Calabi-Yau threefolds using an elliptic curve and a K3 surface with an involution. They take the product of the surface and the curve and take the quotient by the product involution

History: K3 surfaces with involutions had been studied earlier by Nikulin

Every $K3$ surface $S$ over $\mathbb C$ is homeomorphic to the surface obtained from the quotient $E^2/(±id)$ by blowing up 16 singular points coming from fixed points of the involution $x → −x$ ($E$ is an elliptic curve

Suppose, we have an involution, $τ_X : X → X,$ with $τ^2_X = id$, which is freely acting, that is, has no fixed points, and which preserves the holomorphic 3-form. The quotient space $Z$ formed by identifying points related by the involution, $Z = X/τ_X$ is then a smooth Calabi–Yau threefold with $π_1(Z) = \mathbb Z_2$.

K3 surface is Calabi-Yau manifold. On $X=K3$ surfaces the only automorphisms without fixed points are non-symplectic involutions ($i^∗ω_X=-ω_X$)

Let $X$ be a $K3$ surface with a holomorphic involution $ι$ acting nontrivially on the holomorphic 2-forms on $X$. Note that if the fixed point set $X^ι$ of $ι$ is the empty set, then the pair $(X,ι)$ defines an Enriques surface $Y$, and vice versa

Let $(N^6, g, ω, Ω, J)$ be a compact Calabi-Yau manifold admitting an antiholomorpic isometric involution $τ$ . On a quintic in $\mathbb CP^4$ with real coefficients, complex conjugation yields such an involution. There are several examples here

In page 141, 142, there are several examples https://people.maths.ox.ac.uk/joyce/theses/ChanDPhil.pdf

Let me mention the important result of Beauville: A non-symplectic involution $ι$ on an IHS 4-fold fixes Lagrangian surfaces.


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