Gluing construction of compact Spin(7)-manifolds

Abstract

We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\mathrm{Spin}(7)$-manifolds, at least one of which is \emph{new}. Ingredients in our construction are \emph{orbifold admissible pairs with} a compatible antiholomorphic involution. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a four-dimensional compact K\"{a}hler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $\sigma$ on $\overline{X}$ which fixes the singular points in $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ with $\dim_{\mathbb{C}} \overline{X}_i = 4$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ satisfy the \emph{gluing condition}, we can glue $(\overline{X}_1 \setminus D_1)/\braket{\sigma_1}$ and $(\overline{X}_2 \setminus D_2)/\braket{\sigma_2}$ together to obtain a compact Riemannian $8$-manifold $(M, g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals $1$, then $M$ is a $\mathrm{Spin}(7)$-manifold, i.e., a compact Riemannian manifold with holonomy $\mathrm{Spin}(7)$. We shall investigate our gluing construction using $(\overline{X}_i,D_i)$ with $i=1,2$ when $D_1=D_2=D$ and $D$ is a complete intersection in a weighted projective space, as well as when $(\overline{X}_1,D_1)=(\overline{X}_2,D_2)$ and $\sigma_1=\sigma_2$ (the \emph{doubling} case).