Gluing construction of compact \(\mathrm{Spin}(7)\)-manifolds. (English) Zbl 1422.53039

Doi, Mamoru; Yotsutani, Naoto

J. Math. Soc. Japan 71, No. 2, 349-382 (2019).

Summary: 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 and Kovalev’s gluing construction of compact \(G_2\)-manifolds. We provide several examples of compact \(\mathrm{Spin}(7)\)-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 \((\overline{X}_1 \backslash D_1)/\langle\sigma_1\rangle\) and \((\overline{X}_2 \backslash D_2)/\langle\sigma_2\rangle\) 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 compact \(\mathrm{Spin}(7)\)-manifold, i.e. a compact Riemannian manifold with holonomy \(\mathrm{Spin}(7)\).

MSC:

53C29	Issues of holonomy in differential geometry
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
14J32	Calabi-Yau manifolds (algebro-geometric aspects)

Keywords:

Ricci-flat metrics; \(\mathrm{Spin}(7)\)-structures; gluing; doubling

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