Pluriclosed 3-folds with vanishing Bismut Ricci form: General theory in the quasi-regular case

Abstract

We study compact complex $3$-dimensional non-K\"ahler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special K\"ahler geometry in complex dimension $2$, recently obtained by Barbaro, Streets and the first and third authors. We show that in the quasi-regular case, the reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much studied K\"ahler 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 authomorphism group, paralleling results by Futaki and Calabi-Lichnerowicz-Matsushima in the cscK case. This is used to characterize the Samelson locally homogeneous BHE geometries in complex dimension 3 as the only non-K\"ahler BHE $3$-folds with $2$-dimensional Bott-Chern $(1,1)$-cohomology group, for which the reduced space is a smooth K\"ahler surface. We also discuss explicit solutions of the PDE on orthotoric K\"ahler orbifold surfaces, extending examples found by Couzens-Gauntlett-Martelli-Sparks in the framework of supersymmetric ${\rm AdS}_3 \times Y_7$ type IIB supergravity. Our construction yields infinitely many non-K\"ahler BHE structures on $S^3\times S^3$ and $S^1\times S^2 \times S^3$, which are not locally isometric to a Samelson geometry. These appear to be the first such examples.