Journal of the Mathematical Society of Japan

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

Mamoru DOI and Naoto YOTSUTANI

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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 $\operatorname{Spin}(7)$.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 2 (2019), 349-382.

Dates
Received: 27 December 2016
Revised: 12 October 2017
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1551322823

Digital Object Identifier
doi:10.2969/jmsj/77007700

Mathematical Reviews number (MathSciNet)
MR3943852

Zentralblatt MATH identifier
07090047

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 14J32: Calabi-Yau manifolds

Keywords
Ricci-flat metrics $\operatorname{Spin}(7)$-structures gluing doubling

Citation

DOI, Mamoru; YOTSUTANI, Naoto. Gluing construction of compact $\operatorname{Spin}(7)$-manifolds. J. Math. Soc. Japan 71 (2019), no. 2, 349--382. doi:10.2969/jmsj/77007700. https://projecteuclid.org/euclid.jmsj/1551322823


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