Generalized Kahler BHE threefolds

Abstract

We study the notion of generalized Kahler Bismut--Hermitian--Einstein (GKBHE) manifolds, namely generalized K\"ahler manifolds for which both Bismut Ricci forms vanish, and we develop a theory for such structures on compact threefolds. We organize the analysis according to the horizontal rank, an invariant taking values in $\{0,1,2\}$ that measures the relative position of the canonical foliations associated with the two Hermitian structures. We characterize the case of horizontal rank $0$ (which corresponds to K\"ahler Ricci flat geometry) and describe the case of horizontal rank $1$, revealing its close interplay with the steady $\nu$-soliton equation in K\"ahler geometry. This description also provides a tool for constructing new compact GKBHE threefolds.