# Geometry & Topology Seminar

Friday, April 5, 2019 — 2:30 PM EDT

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"A curious system of second order nonlinear PDEs for $\mathrm{U}(m)$-structures on manifolds"

Compact Kähler manifolds possess a number of remarkable properties, such as the Kähler identities, the $\partial \bar{\partial}$-lemma, and the relation between Betti numbers and Hodge numbers. I will discuss an attempt in progress to generalize some of these ideas to more general compact $\mathrm{U}(m)$-manifolds, where we do not assume integrability of the almost complex structure nor closedness of the associated real $(1,1)$-form. I will present a system of second order nonlinear PDEs for such a structure, of which the Kähler structures form a trivial class of solutions. Any compact non-Kähler solutions to this second order system would have properties that are formally similar to the above-mentioned properties of compact Kähler manifolds, including relations between cohomological (albeit non-topological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (King's College London and ICTP).

MC 5403

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