Remark 1.3 . [03G2]
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Remark 1.3.
The Riemannian geometry of the regular collapsing regions is actually completely understood. For each , is a -dimensional Heisenberg nilmanifold if the -bundle is nontrivial, and is diffeomorphic to otherwise. Furthermore, the universal cover of a regular preimage converges to a hyperkähler manifold with a Heisenberg or Euclidean group of isometries according to whether the -bundle is nontrivial or trivial. An explicit expression for in the Heisenberg case may be found in Section 2.2. In particular, our construction gives a concrete example of Lott’s recent work classifying the regular regions in collapsing -manifolds with almost Ricci-flat metrics (see [Lot17] for more details).