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Compact manifolds with exceptional holonomy. (English) Zbl 0911.53021
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.
53C20 Global Riemannian geometry, including pinching
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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