Compact manifolds with exceptional holonomy
The author describes (without giving details) his famous construction of compact Riemannian 7-manifolds \(M\) with holonomy group \(G_2\), which is an imitation of the Kummer construction of a K3 surface as a desingularization of the orbifold \(T^4/\mathbb Z_2\). A \(G_2\)-structure on a 7-manifold \(M\) can be defined as a pair \((g,\phi)\) where \(g\) is a Riemannian metric and \(\phi\) is a 3-form of special type. The tensor \( T= \nabla \phi\) is called the torsion of the \(G_2\)-structure \((g,\phi)\). If the torsion \(T=0\) and the fundamental group of \(M\) is finite, then the metric \(g\) has the holonomy group \(\text{Hol}(g) \subset G_2 \). The construction of compact Riemannian manifolds with holonomy \(G_2\) consists of four steps: Let \(T^7\) be the 7-torus with a flat \(G_2\)-structure \((g_0,\phi_0)\). The author chooses some finite group \(\Gamma\) of automorphisms of \((g_0,\phi_0)\) and considers the orbifold \(T^7/\Gamma \). Using complex geometry and results by P. B. Kronheimer, the author resolves the singularities of the orbifold and gets a smooth compact 7-manifold \(M\) with a map \(\pi: M \to T^7/\Gamma \), the resolving map. He constructs a 1-parameter family \((g_t,\phi_t)\), \(t \in (0,\varepsilon)\), of \(G_2\)-structures on \(M\), such that the torsion \(T_t\) becomes small when \(t \to 0\). Using analysis, the author proves that a \(G_2\)-structure with sufficiently small torsion can be deformed to a \(G_2\)-structure without torsion. This implies existence of a metric with holonomy \(G_2\) on \(M\). Some information about 68 compact manifolds \(M\) with holonomy \(G_2\) which can be obtained by this method is given. In particular, a graph of their Betti numbers \(b_2, b_3\) is presented.
scientific article; zbMATH DE number 1184335