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Kael Dixon: Lectures

September 16, 2020
TITLE: Toric ALC G_2 manifolds

GIven an asymptotically conical Calabi-Yau 3-fold, certain circle bundles over it each admit a family of ALC torsion-free G_2 structures by the work of Foscolo-Haskins-Nordstrom. We show that when the Calabi-Yau manifold is toric, then so are the G_2 structures. We also describe the corresponding multi-moment diagrams, which are certain liftings of the pq-web of the toric Calabi-Yau into 3 dimensions.

September 14, 2018
TITLE: Asymptotic properties of toric G2 manifolds

A toric G_2 manifold is a 7-manifold M equipped with a torsion-free G_2 structure, which is invariant under the action of a 3-torus T in such a way that there exist multi-moment maps associated to the G_2 3-form and its Hodge dual. These are introduced and studied in a recent paper by Madsen and Swann, where they show that these multi-moment maps induce a local homeomorphism from the space of orbits M/T into R4. In other words, the multi-moment maps provide geometrically motivated local coordinates for M/T. In all of the known examples, this local homeomorphism is a global homeomorphism onto R4. I will describe some partial results toward showing that this is true in general.

Slides of Lecture