## Twisted connected sums and special Riemannian holonomy.(English)Zbl 1043.53041

The author presents a new method for constructing Riemannian metrics with holonomy $$G_2$$ on compact $$7$$-dimensional manifolds. Based on some work by Tian and Yau the author first constructs a class of noncompact asymptotically cylindrical Riemannian manifolds with holonomy $$SU(3)$$. The Riemannian product of such a manifold and a circle $$S^1$$ has again holonomy $$SU(3)$$ and an end that is asymptotic to a half-cylinder. A cross-section of this half-cylinder is the Riemannian product of two circles and a complex surface of type K3 with a hyper-Kähler metric. By truncating the ends of two such asymptotically cylindrical $$7$$-manifolds, cutting off to the cylindrical metric and identifying the boundaries by an orientation-reversing isometry that interchanges the two $$S^1$$ factors he obtains a closed compact Riemannian $$7$$-manifold. The twisting that is built into the isometry identifying the boundaries has the purpose to avoid an infinite fundamental group.
The construction is based on a gluing theorem for appropriate elliptic partial differential equations and yields compact $$7$$-manifolds with a $$G_2$$-structure. The author then discusses the topology of the resulting $$7$$-manifolds when one starts the construction with Fano manifolds, which leads to many new topological types of compact $$G_2$$-manifolds.

### MSC:

 53C29 Issues of holonomy in differential geometry 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

### Keywords:

special holonomy; $$G_2$$ holonomy; Fano manifolds
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### References:

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