Compact 8-manifolds with holonomy \(\text{Spin}(7)\)
The author constructs many examples of Riemannian metrics \(g\) with the exceptional holonomy group Spin(7) on some compact 8-manifolds \(M\). A metric \(g\) with the holonomy group \(H \subset \text{Spin}(7)\) is canonically defined by a closed 4-form \(\Omega\) such that the stabilizer of \(\Omega\) in the group SO(8) at each point is equal to Spin(7). Such form \(\Omega\) is called a Spin(7)-structure or, more generally, if it is not necessarily closed, an almost Spin(7)-structure. The manifold \(M\) is constructed as a desingularization of the Spin(7)-orbifold \(T^8/\Gamma\), where \(\Gamma\) is a finite group of automorphisms of the flat Spin(7)-structure on an 8-torus \(T^8\). The author proves the existence of an almost Spin(7)-structure \(\Omega\) on \(M\) with sufficiently small differential \(d\Omega\). Then, using the theory of nonlinear elliptic differential equations, he proves that \(\Omega\) can be deformed into a Spin(7)-structure \(\Omega_0\). To prove that the metric associated with \(\Omega_0\) has holonomy group exactly Spin(7) (and not smaller), he uses a characterisation of the holonomy group in terms of existence of parallel spinor fields. Then, using the Atiyah-Singer index theorem for the Dirac operator, he proves that for simply connected manifolds the holonomy group is Spin(7) if and only if the \(\widehat{A}\)-genus of \(M\) is equal to 1.
nonlinear elliptic differential equations
scientific article; zbMATH DE number 922461