Gluing construction of compact $\operatorname{Spin}(7)$-manifolds

Abstract

We give a differential-geometric construction of compact manifolds with holonomy $\operatorname{Spin}(7)$ which is based on Joyce's second construction of compact $\operatorname{Spin}(7)$-manifolds and Kovalev's gluing construction of compact $G_2$-manifolds. We provide several examples of compact $\operatorname{Spin}(7)$-manifolds, at least one of which is new. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a compact Kähler 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 on $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ for $i=1,2$ satisfy the gluing condition, we can glue $(\overline{X}_1 \setminus D_1)/\langle\sigma_1\rangle$ and $(\overline{X}_2 \setminus D_2)/\langle\sigma_2\rangle$ together to obtain a compact Riemannian 8-manifold $(M, g)$ whose holonomy group $\operatorname{Hol}(g)$ is contained in $\operatorname{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals 1, then $M$ is a compact $\operatorname{Spin}(7)$-manifold, i.e. a compact Riemannian manifold with holonomy $\operatorname{Spin}(7)$.