A new construction of compact 8-manifolds with holonomy Spin(7).
A Spin(7)-structure on an \(8\)-manifold \(M\) is a principal subbundle \(Q\) of the frame bundle \({\mathcal F}\) with structure group Spin(7). Each Spin(7)-structure \(Q\) induces a \(4\)-form \(\Omega \), a metric \(g\) and an orientation on \(M\). A Spin(7)-manifold is a triple (\(M,\Omega ,g\)) where \(M\) is an \(8\)-manifold and (\(\Omega \), \(g\)) is a torsion-free Spin(7)-structure. The author has already given [Invent. Math. 123, 507--552 (1996; Zbl 0858.53037)] a method for constructing compact \(8\)-manifolds starting with a \(8\)-torus \(T^{8}\) factorized by a finite group of isometries of \(T^{8}\) preserving the standard flat Spin(7)-structure on \(T^{8}\). In this paper the author describes a new method for constructing compact \(8\)-manifolds starting with a Calabi-Yau \(4\)-orbifold \(Y\) with isolated singular points \(p_{1},p_{2}, \dots ,p_{k}\). An antiholomorphic isometric involution \(\sigma \) \(:Y\rightarrow Y\) whose only fixed points are \(p_{1},p_{2}, \dots , p_{k}\) is assumed to exist. The main result consists in proving that given a Calabi-Yau \(4\)-orbifold \(Y\) and an antiholomorphic isometric involution \(\sigma : Y\rightarrow Y\) satisfying certain conditions, a compact \(8\)-manifold \(M\) with a Spin(7)-structure can be constructed. Using algebraic geometry the author gives plenty of suitable complex orbifolds \(Y\). Moreover, several ways of finding orbifolds \(Y\) and involutions \(\sigma :Y\rightarrow Y\) satisfying suitable conditions are show. Also, it is indicated how to calculate the Betti numbers of the resulting \(8\)-manifolds \(M\) with holonomy Spin(7). The last three sections of this paper are concerned with important examples of compact \(8\)-manifolds with Spin(7)-structures, whose Betti numbers are determined.
asymptotically locally Euclidean manifold
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