Non-Kähler Bismut Hermite Einstein Manifolds

• Non-Kähler Bismut Hermite Einstein manifolds are compact complex spaces with pluriclosed Hermitian metrics where the Bismut Ricci form vanishes, extending the Ricci-flat Kähler paradigm.
• They leverage advanced techniques such as pluriclosed flow, variational analysis, and symmetry reduction to reveal scalar nonlinear PDE formulations and stability properties.
• These manifolds underpin classification results, rigidity theorems, and explicit geometric constructions that connect complex geometry and mathematical physics.

A non-Kähler Bismut Hermite Einstein manifold is a compact complex manifold equipped with a pluriclosed Hermitian metric whose Bismut Ricci form vanishes. These metrics, known in the literature as Calabi–Yau with torsion (CYT) or Bismut Hermitian–Einstein (BHE) metrics, generalize the notion of Ricci-flat Kähler metrics to the non-Kähler category and have deep connections with pluriclosed flow, generalized geometry, and mathematical physics. Their study encompasses structure theorems, variational analysis, rigidity results, existence/nonexistence theories, and explicit geometric constructions.

1. Bismut Connection, Pluriclosed Metrics, and the Einstein Condition

Let $(M^{2n}, J, g)$ be a compact complex manifold. The metric $g$ is Hermitian with respect to $J$, and the associated $(1,1)$-form is $\omega(X,Y) = g(JX,Y)$. The Bismut connection $\nabla^B$ is uniquely characterized by

$$\nabla^B g = 0,\quad \nabla^B J = 0,\quad T^{B}(X,Y,Z) = d^c\omega(X,Y,Z)$$

or equivalently

$$\nabla^B_X Y = \nabla^{\text{LC}}_X Y + \frac{1}{2} H(X,Y,\cdot)^{\sharp},\quad H = d^c\omega.$$

A metric is pluriclosed (SKT) if $dd^c\omega = 0$, which means the torsion $H$ is closed.

The curvature $R^B$ of $\nabla^B$ yields the Bismut Ricci form $\rho^B$, given by

$\rho^B = -i\partial\bar{\partial} \log\det g$

in local holomorphic coordinates. A Hermitian metric is called Bismut Hermite Einstein if

$\rho^B = 0,$

equivalently, the $(1,1)$-component of the Bismut Ricci curvature vanishes and the metric is pluriclosed. Any Bismut Hermite Einstein metric must satisfy the pluriclosed condition ($dd^c\omega=0$) as a necessary consequence in the compact setting (Ye, 2022).

2. General Structure, Symmetry Reduction, and the Bismut Flat/Non-Flat Dichotomy

Any compact six-dimensional (complex threefold) non-Kähler BHE manifold $(M^6,J,g)$ with trivial canonical bundle has the following canonical structure (Apostolov et al., 2024):

• $M^6$ is a principal $T^2$-bundle over a compact complex fourfold $X^4$ (possibly an orbifold), equipped with a conformally Kähler metric.
• The Hermitian form takes the form

$$\omega = \pi^*(e^{-u}\omega_X) + \frac{1}{2} \theta^1 \wedge \theta^2,$$

where $u$ is a conformal factor, and $\theta^1$, $\theta^2$ are connection forms with curvature forms $\alpha_1$, $\alpha_2$ primitive on $X$.

• The Bismut Hermite Einstein equation for $\omega$ is equivalent to a single scalar nonlinear PDE on $X$ (generalizing the Monge–Ampère equation for Kähler–Einstein metrics):

$$\det(\omega_X + i\partial\bar{\partial}\varphi) = e^{\lambda\varphi - F} \det(\omega_0) + |\alpha_1 + i\alpha_2|^2_{\omega_X + i\partial\bar{\partial}\varphi}.$$

• In the case where the soliton potential is constant, the Bott–Chern $(1,1)$-cohomology $h^{1,1}_{BC}(M) \geq 2$; equality holds if and only if the manifold is Bismut-flat, i.e., $R^B = 0$ (Apostolov et al., 2024).

Thus, any compact, non-Kähler, Bismut Hermite Einstein threefold with $h^{1,1}_{BC} = 2$ must be a quotient of either $\mathrm{SU}(2)\times\mathrm{SU}(2)$ (Calabi–Eckmann) or $\mathrm{SU}(2)\times\mathbb{R}\times\mathbb{C}$ (Vaisman), up to finite quotients, and carries a left-invariant complex structure and Hermitian metric with vanishing Bismut curvature.

3. Variational Theory, Pluriclosed Flow, and Stability

The relevant geometric variational principle is the generalized Einstein–Hilbert functional

$$\mathcal{F}(g, b, f) = \int_M \left(R(g) - \frac{1}{12}|H|^2 + |\nabla f|^2\right) e^{-f} dV_g,$$

whose critical points are pluriclosed steady solitons. For steady solitons $(g, b, f)$, necessary equations are \begin{align*} 0 &= \operatorname{Rc} - \frac{1}{4} H^2 + \nabla^2 f, \ 0 &= d^*H + i_{\nabla f} H. \end{align*} A compact Bismut Hermite Einstein metric is a pluriclosed steady soliton with $f$ constant and vice versa (Lee, 2024).

The pluriclosed flow is defined by

$\partial_t \omega_t = -\rho^B(\omega_t),$

with supplementary evolution to ensure $H_t = -d\omega_t$ remains closed. On compact Bismut-flat manifolds, the flow exists for all time and converges to a Bismut-flat metric. Any pluriclosed metric in the same torsion class evolves under the flow to a Bismut-flat limit, and such Bismut-flat metrics are globally stable fixed points for the flow (Garcia-Fernandez et al., 2021, Barbaro, 2023, Lee, 2024).

Stability analysis mirrors the classical Koiso/Tian–Zhu theory but includes torsion: all Bismut Hermite Einstein metrics are linearly stable (the second variation is non-positive), and in the Bismut-flat scenario with additional positivity conditions on the Riemannian Ricci tensor, strict (linear) stability is achieved (Lee, 2024).

4. Classification Results, Rigidity, and Nonexistence

The structure and rigidity theory yield a dichotomy:

• If the Einstein constant $\lambda\neq0$, any Bismut Hermitian–Einstein metric is necessarily Kähler–Einstein; non-Kähler examples with $\rho^B = \lambda\omega, \lambda\neq 0$ do not exist (Ye, 2022).
• For $\lambda = 0$, non-Kähler Bismut-Ricci-flat Hermitian metrics exist. However, on simply-connected compact homogeneous spaces (C-spaces, Samelson spaces), all such Bismut Hermite Einstein metrics are necessarily flat with respect to the Bismut connection—no non-flat homogeneous examples can occur (Barbaro, 2023, Barbaro, 2021, Brienza et al., 2024).

Obstructions to existence of non-flat Bismut Hermitian–Einstein metrics arise from a slope stability criterion: the associated holomorphic Courant algebroid must be $[\omega^{n-1}]$-polystable. Every proper holomorphic isotropic subsheaf must have slope $\leq 0$, with equality only for holomorphic splittings (Garcia-Fernandez et al., 2021). Infinitely many topologically distinct complex manifolds with $c_1(M)=0$ are known to admit no Bismut Hermitian–Einstein metrics, typically constructed as non-trivial torus bundles over Kähler–Einstein bases with $c_1<0$.

Tables summarizing key cases for compact non-Kähler Bismut Hermite Einstein 3-folds:

Geometry	Bott–Chern $(1,1)$	Bismut Ricci	Group Quotient
Calabi–Eckmann	2	Flat	$\mathrm{SU}(2)\times\mathrm{SU}(2)$
Vaisman Solvmanifold	2	Flat	$\mathrm{SU}(2)\times\mathbb{R}\times\mathbb{C}$
Non-homogeneous	$\geq 2$	Flat/Non-flat	$S^3\times S^3$, $S^1\times S^2\times S^3$, etc.

5. Explicit Constructions and Generalized Kähler Structures

Bismut–Ricci-flat metrics can be constructed via principal torus bundles over Hermitian (often Kähler–Einstein) manifolds. For a principal $T^2$-bundle $\pi:M\to X$ with connections $\theta^i$ and base metric $\omega_X$,

$\omega = \pi^*\omega_X + f\,\theta^1\wedge\theta^2$

is Bismut–Ricci-flat if $\omega_X$ is Bismut–Ricci-flat and curvature forms $w_i$ are primitive (Barbaro, 2021). On class $C$ manifolds (homogeneous torus bundles over products of flag manifolds), the standard metric is unique among homogeneous CYT metrics up to homothety.

Mapping torus constructions over Ricci-flat Kähler manifolds produce non-Bismut-flat SKT–CYT metrics with parallel Bismut torsion (Brienza et al., 2024). In particular, mapping tori of K3 surfaces and Samelson spaces yield explicit non-flat Bismut Hermite Einstein manifolds, each possessing natural generalized Kähler structures.

6. Analytical and Variational Structures: Reduction to Scalar PDE and Moment Map Picture

On Bismut Hermite Einstein 3-folds, the geometry reduces via symmetry to a scalar sixth-order nonlinear PDE on the transversal Kähler base. This PDE has infinite-dimensional moment map and variational interpretations, parallel to the theory of Kähler metrics with constant scalar curvature: \begin{align*} \tfrac12 dd^c(\mathrm{Scal}(\omega_K))\wedge\omega_K &= \theta\wedge\theta + \rho(\omega_K)\wedge \rho(\omega_K),\ \mathcal{S}(\omega) := \tfrac12 \Delta \mathrm{Scal} + \tfrac14 \mathrm{Scal}^2 - \tfrac12 |\mathrm{Rc}|^2 - \tfrac12|\theta|^2 &= 0. \end{align*} Functionals such as the Mabuchi and Calabi functionals, and obstruction invariants (Futaki type), yield criteria for existence, uniqueness, and openness under deformations (Apostolov et al., 8 Jan 2026).

Explicit solutions—particularly orthotoric solutions on quadrilateral Delzant polytopes—give rise to non-homogeneous compact BHE examples on $S^3\times S^3$ and $S^1\times S^2\times S^3$ (Apostolov et al., 8 Jan 2026). These constructions also have applications to the differential geometry underlying Type IIB AdS$_3\times Y_7$ supergravity backgrounds.

7. Open Questions and Outlook

Existing work suggests strong rigidity: in compact dimension 6, non-Kähler Bismut Hermite Einstein geometries with constant soliton potential and minimal Bott–Chern number are exhausted by Calabi–Eckmann and Vaisman models (Apostolov et al., 2024). There now exist non-homogeneous compact BHE examples outside this classification, showing the possible diversity of non-Kähler Calabi–Yau structures in higher dimensions (Apostolov et al., 8 Jan 2026). Analytical methods point toward a Yau–Tian–Donaldson type correspondence for the existence of BHE metrics, with invariants and PDE structure closely paralleling the cscK and general extremal Kähler theory.

Future research avenues include the general existence theory for the associated sixth-order PDE, relations with algebraic (K-)stability, and further explicit constructions—potentially relevant to mathematical physics, particularly compactifications with fluxes. The full generalization of Calabi–Yau theory to non-Kähler pluriclosed geometry, including the classification, flow dynamics, and moduli of Bismut Hermite Einstein spaces, remains an active area of research (Apostolov et al., 8 Jan 2026, Apostolov et al., 2024, Garcia-Fernandez et al., 2021).