Gluing construction of compact Spin(7)-manifolds

摘要

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 G_2-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 (X, D) consisting of a compact Kahler orbifold X with isolated singular points modelled on C~4/Z_4, and a smooth anticanonical divisor D on X. Also, we need a compatible antiholomorphic involution σ on X which fixes the singular points on X and acts freely on the anticanoncial divisor D. If two orbifold admissible pairs (X_1, D_1), (X_2, D_2) and compatible antiholomorphic involutions σ_i on X_i for i = 1, 2 satisfy the gluing condition, we can glue (X_1 D_1)/<σ_1> and (X_2 D_2)/<σ_2> together to obtain a compact Riemannian 8-manifold (M, g) whose holonomy group Hol(g) is contained in Spin(7). Furthermore, if the A-genus of M equals 1, then M is a compact Spin(7)-manifold, i.e. a compact Riemannian manifold with holonomy Spin(7).