Geometry & Topology

In this talk, we discuss compact complex 3-dimensional non-Kahler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special Kahler geometry in complex dimension 2. The reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much-studied Kahler metrics of constant scalar curvature (cscK). We use this to associate to the reduced manifold or orbifold Mabuchi and Calabi functionals, as well as to obtain obstructions for the existence of solutions in terms of the automorphism group, paralleling results by Futaki and Calabi--Lichnerowicz--Matsushima in the cscK case. Through the obstruction theorem, we show that the quotients of SU(2)x SU(2) or SU(2)x R^3 as the only non-Kahler BHE 3-folds with 2-dimensional Bott--Chern (1,1)-cohomology group, for which the reduced space is a smooth Kahler surface. Lastly, we discuss explicit solutions of the PDE on orthotoric Kahler orbifold surfaces which yield infinitely many non-Kahler BHE structures on S^3\times S^3 and S^1\times S^2 \times S^3.