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 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 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 $({\overline{X}}_1\setminus D_1)/⟨{\sigma }_1⟩$ and $({\overline{X}}_2\setminus D_2)/⟨{\sigma }_2⟩$ 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 $\hat{A}$-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).

24 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:1502.00208