Bismut Hermitian–Einstein metrics on pluriclosed deformations

Determine whether the complex-structure deformations of compact simply-connected Bismut flat Lie groups constructed in Theorem A carry genuine Bismut Hermitian–Einstein metrics, pluriclosed solitons, or neither in higher dimensions, and characterize how the outcome is governed by the geometry of the underlying toric foliation.

Background

The paper constructs smooth deformations of compact simply-connected Bismut flat Hermitian structures whose deformed complex manifolds admit pluriclosed metrics but, generically, admit no Bismut flat metric. These deformations are obtained by varying the toric foliation associated with a maximal torus while preserving the relevant transverse holomorphic and cohomological structures.

The broader motivation is to produce non-trivial Bismut Hermitian–Einstein metrics, meaning pluriclosed metrics with vanishing Bismut Ricci form that are not locally Bismut flat. The authors explain that the next step would be to analyze the linearized Bismut Ricci form along the constructed deformations. They note that analogous deformations on Hopf surfaces yield pluriclosed solitons rather than Bismut Hermitian–Einstein metrics, leaving the higher-dimensional outcome unresolved.