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Geometry-Topology Seminar

Thursday, March 2, 2017 - 4:30pm

Lorenzo Foscolo

Stony Brook University


University of Pennsylvania


G2 manifolds are the Riemannian 7-manifolds with G2 holonomy.  Every G2 manifold is necessarily Ricci-flat. Only four examples of complete non-compact G2 manifolds are currently known. In joint work with Mark Haskins (Imperial College London) and Johannes Nordström (University of Bath) we construct infinitely many families of new complete non-compact G2-holonomy metrics. The underlying 7-manifolds are all circle bundles over asymptotically conical (AC) Calabi-Yau manifolds of complex dimension 3. The metrics are circle-invariant and their geometry at infinity is that of a circle bundle over a Calabi-Yau cone with fibres of fixed finite length. The G2 manifolds we construct are
therefore 7-dimensional analogues of 4-dimensional ALF hyperkähler metrics.

The dimensional reduction of the equations for G2 holonomy in the presence of a Killing field was considered by Apostolov-Salamon and by several groups of physicists. We reinterpret the dimensionally-reduced equations in terms of a pair consisting of an SU(3) structure on the 6-dimensional orbit space coupled to an abelian Calabi-Yau monopole on this 6-manifold. We solve this coupled system of non-linear PDEs by considering the adiabatic limit in which the circle fibres of the associated circle-invariant G2-holonomy metrics collapse. The
G2-holonomy metrics we construct should be thought of as arising from abelian Hermitian-Yang-Mills connections on AC Calabi-Yau 3-folds, especially AC Calabi-Yau metrics on crepant resolutions of Calabi-Yau cones. All our examples provide instances of families of G2-holonomy metrics that collapse with bounded curvature to Calabi-Yau 3-folds. This is a new feature of G2-holonomy metrics compared to the 4-dimensional hyperkähler setting.