I am looking for a complete intersection Calabi-Yau manifold $X$ of complex dimension $3$ that admits an anti-holomorphic involution $\sigma: X \rightarrow X$ such that $L:=\operatorname{fix}(\sigma)$ is smooth and each of its connected components has first Betti number $b^1 \neq 0$.
By "complete intersection Calabi-Yau" I mean an algebraic variety of $\mathbb{CP}^{k+3}$ cut out by $k$ polynomials that is smooth and has trivial canonical bundle. I think the only possibilities for such complete intersection Calabi-Yau manifolds are:
- A quintic in $\mathbb{CP}^4$
- Intersection of a quadric and quartic in $\mathbb{CP}^5$
- Intersection of two cubics in $\mathbb{CP}^5$
- Intersection of two quadrics and a cubic in $\mathbb{CP}^6$
- Intersection of four quadrics $\mathbb{CP}^7$
For example, take the Fermat quintic $F=\{[x_0:x_1:x_2:x_3:x_4] \in \mathbb{CP}^4: x_0^5+x_1^5+x_2^5+x_3^5+x_4^5\}$ in $\mathbb{CP}^4$. It admits many anti-holomorphic involutions, such as $\sigma: [x_0:x_1:x_2:x_3:x_4] \mapsto [\overline{x_0}:\overline{x_1}:\overline{x_2}:\overline{x_3}:\overline{x_4}]$ and $\sigma': [x_0:x_1:x_2:x_3:x_4] \mapsto [\overline{x_1}:\overline{x_0}:\overline{x_2}:\overline{x_3}:\overline{x_4}]$ (the map $\sigma'$ interchanges the first two coordinates). Unfortunately, I was unable to compute their fixed point sets. (In a previous version of the question I wrote that $L:=\operatorname{fix}(\sigma)$ seems to be singular, but I realised it probably is smooth.)
Comment: often "complete intersection Calabi-Yau" means an algebraic variety in a product of projective spaces. My definition in this question is more restrictive. If I allowed products of projective spaces, then an example in $\mathbb{CP}^1 \times \mathbb{CP}^1 \times \mathbb{CP}^1 \times \mathbb{CP}^1$ would be given in Example 7.6 of Joyce, Karigiannis: A new construction of compact torsion-free G2-manifolds by gluing families of Eguchi–Hanson spaces. This question is motivated by their article.
Here are two examples of intersections of quartics and quadrics.
Example 1:
Let $Q(z_1,z_2,z_3)=z_1^4+z_2^4+z_3^4-1$ and $C(z_4, z_5)=z_4^2+z_5^2-1$, i.e. the real locus of the zero set $Z(Q)$ is a squished sphere, and the real locus of $Z(C)$ is a circle. Then $Y:=Z(Q(z_1,z_2,z_3), C(z_4, z_5)) \subset \mathbb{C}^5$ is an affine complex variety whose real locus is diffeomorphic to $S^2 \times S^1$. The projectivation $Y^{proj} \subset \mathbb{CP}^5$ is singular at infinity, but the real locus of $Y$ is compact, so no points at infinity are added and the real locus of $Y$ is smooth. One checks that a small, generic perturbation of $Y^{proj}$ is smooth, and because the real locus was smooth to begin with, the real locus of the perturbation is homotopic to the real locus of $Y$.
Using the code from https://www.juliahomotopycontinuation.org/examples/sampling_bottlenecks/ I checked that the following example has zero-th Betti number $1$ and first Betti number $1$: the zero set of the two polynomials
$z_1^2 + z_2^2 + 1/10z_5^2 + 1/10z_41 - 1^2$
$1/10z_1z_2^2z_3 + 11/10z_3^4 + 1/10z_2z_3z_4^2 + z_4^4 + 1/10z_1z_2^2z_5 + 1/10z_2z_3^2z_5 + 1/10z_3^2z_4z_5 + 1/10z_2z_4z_5^2 + 1/10z_4^2z_5^2 + 1/10z_2z_5^3 + z_5^4 + 1/10z_2^31 + 1/10z_1z_3z_41 + 1/10z_4z_5^21 + 1/10z_1z_41^2 + 1/10z_3z_51^2 - 1^4$
Example 2:
Let $Q(z_1,z_2,z_3,z_4,z_5)=z_1^4+z_2^4+z_3^4+z_4^4+z_5^4-1$ and $C(z_4, z_5)=z_4^2+z_5^2-1/2$. $Q$ is a squished sphere, for easier calculation we pretend that $Q$ is precisely a sphere, i.e. $\tilde{Q}(z_1,z_2,z_3,z_4,z_5)=z_1^2+z_2^2+z_3^2+z_4^2+z_5^2-1$. Let $z=(z_1, \dots, z_5)$ be a point on $Z(Q,C)$. Subtracting $C$ from $\tilde{Q}$ we get $z_1^2+z_2^2+z_3^2-1/2=0$, which is the equation for a $2$-sphere. The equation $z_4^2+z_5^2-1/2=0$, which is the equation for a $1$-sphere, has all different variables from the previous equation, so $Z(Q,C)=S^2 \times S^1$. As in example 1, the example is singular, and we have to perturb to get a smooth projective Calabi-Yau manifold.