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Coassociative K3 fibrations of compact G_2-manifolds
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A class of examples of Riemannian metrics with holonomy G_2 on compact 7-manifolds was constructed by the author in arXiv:math.DG/0012189 and later in a joint work with N.-H. Lee in arXiv:0810.0957, using a certain `generalized connected sum' of two asymptotically cylindrical manifolds with holonomy SU(3). We consider, on each of the two initial SU(3)-manifolds, a fibration arising from a Lefschetz pencil of K3 surfaces. The gluing of the two K3 fibrations yields a coassociative fibration of the connected sum G_2-manifold over a 3-dimensional sphere. The singular fibres of this fibration are diffeomorphic to K3 orbifolds with ordinary double points and are parameterized by a Hopf-type link. We believe that these are the first examples of fibrations of compact manifolds of holonomy G_2 by coassociative minimal submanifolds.
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Complete Calabi-Yau metrics on noncompact K3 fibered threefolds
Constructs complete Calabi-Yau metrics on Lefschetz K3 fibrations over C with unbounded sectional curvature at infinity using gluing and perturbation.
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