Gluing construction of compact Spin(7)-manifolds

Abstract

We give a differential-geometric construction of compact manifolds with holonomy Spin(7) which is based on Joyce's second construction of compact Spin(7)-manifolds and Kovalev's gluing construction of compact 𝐺₂-manifolds. We provide several examples of compact 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 ℂ⁴/ℤ₄, and a smooth anticanonical divisor 𝐷 on \overline{𝑋}. Also, we need a compatible antiholomorphic involution 𝜎 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, 𝐷₁), (\overline{X}₂, 𝐷₂) and compatible antiholomorphic involutions 𝜎_𝑖 on \overline{𝑋}_𝑖 for 𝑖 = 1, 2 satisfy the gluing condition, we can glue (\overline{𝑋}₁ ∖ 𝐷₁)/⟨𝜎₁⟩ and (\overline{𝑋}₂ ∖ 𝐷₂)/⟨𝜎₂⟩ together to obtain a compact Riemannian 8-manifold (𝑀, 𝑔) whose holonomy group Hol(𝑔) is contained in Spin(7). Furthermore, if the \widehat{𝐴}-genus of 𝑀 equals 1, then 𝑀 is a compact Spin(7)-manifold, i.e. a compact Riemannian manifold with holonomy Spin(7).