Twisted Connected Sums and Special Riemannian Holonomy

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7-manifolds with holonomy G2 are constructed using Fano 3-folds. The purpose of this paper is to give a new construction of compact 7-dimensional Riemannian manifolds with holonomy group G2. The holonomy group of a Riemannian manifold is the group of isometries of a tangent space generated by parallel transport using the Levi–Civita connection over closed paths based at a point. For an oriented n-dimensional manifold the holonomy group may be identified as a subgroup of SO(n). If there is a structure on a manifold defined by a tensor field and parallel with respect to the Levi–Civita connection then the holonomy may be a proper subgroup of SO(n) (it is just the subgroup leaving invariant the corresponding tensor on Rn). There is essentially just one possibility for such holonomy reduction in odd dimensions, as follows from the well-known Berger classification theorem. This ‘special holonomy’ group is the exceptional Lie group G2 and it occurs in dimension n = 7, when a metric is ‘compatible’ with the non-degenerate cross-product on R7, see §1 for the precise definitions. The only previously known examples of compact Riemannian 7-manifolds with holonomy G2 are due to Joyce who used a generalized Kummer construction and resolution of singularities [J1, J2] (non-compact examples were known earlier [B, Br, BS]). The compact 7-manifolds with holonomy G2 in this paper are obtained by a different, connectedsum-like construction for a pair of non-compact manifolds with asymptotically cylindrical ends. It is by now understood that the existence problem for the metrics with holonomy G2 can be expressed as a non-linear system of PDEs for a ‘non-degenerate’ differential 3-form φ on a 7-manifold. More precisely, a solution 3-form defines a torsion-free G2-structure and thus a metric, from the inclusion G2 ⊂ SO(7). The holonomy of this metric is in general only contained in G2. In fact, we shall obtain the solutions, torsion-free G2-structures, by proving a gluing theorem for pairs of manifolds with holonomy SU(3), a maximal subgroup of G2. To claim that the holonomy on the resulting compact 7-manifold is exactly G2 it suffices to verify a topological condition: that the 7-manifold has finite fundamental group. To implement a connected sum strategy we require, in the first place, a suitable class of noncompact, asymptotically cylindrical Riemannian manifolds with holonomy SU(3). The point is that while the task of constructing asymptotically cylindrical metrics of holonomy G2 is not likely to be any easier than the search for holonomy G2 on compact manifolds, the metrics with Date: 11th October 2000. 2000 Mathematics Subject Classification. 53C25 (Primary); 53C21, 58J60 (Secondary).