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Associatives in the generalised Kummer construction

Daniel Platt

Associatives are a class of 3-dimensional submanifolds of G2-manifolds. They are examples of minimal surfaces and interesting in their own right, but there are also ideas to count them to define numerical invariants of G2-manifolds. For this, it is important to understand all possible degenerations of associatives in 1-parameter families, but all previously known families of associatives in 1-parameter families in compact G2-manifolds are roughly constant. I will explain some new associatives in the Generalised Kummer Construction whose volume tends to zero as the ambient G2-manifolds degenerate. The construction starts from a family of obstructed, non-rigid associatives, finitely many of which survive a perturbation of the ambient metric. I will explain the original family and the perturbation step. This is joint work with Shubham Dwivedi and Thomas Walpuski.


Simons- Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems