Bismut Ricci Flat Pluriclosed Hermitian Manifolds

• BHE manifolds are Hermitian spaces with pluriclosed metrics and vanishing Bismut Ricci form that extend Calabi–Yau theory to non-Kähler settings.
• They feature a unique Hermitian connection with totally skew torsion, providing insights into heterotic string theory and generalized Einstein equations.
• Recent studies classify explicit low-dimensional examples and reveal rigidity linked to cohomological constraints and the dynamics of the pluriclosed flow.

A Bismut Ricci flat pluriclosed Hermitian manifold (commonly abbreviated BHE or “Calabi–Yau with torsion”) is a Hermitian manifold $(M, J, g)$ admitting a pluriclosed metric ($dd^c\omega=0$), with vanishing Bismut Ricci form ($\rho^B=0$) for the unique Hermitian connection with totally skew torsion. BHE geometry arises naturally in mathematical physics (notably in the Strominger system for heterotic string theory), as the non-Kähler extension of Calabi–Yau theory and as critical points of generalized Einstein–Hilbert functionals in generalized geometry (Lee, 2024, Garcia-Fernandez et al., 2021).

1. Pluriclosed Metrics and the Bismut Connection

A Hermitian metric $g$ on a complex manifold $(M, J)$ is pluriclosed if the associated (1,1)-form $\omega(X,Y) = g(JX,Y)$ satisfies

$dd^c\omega = 0,$

where $d^c = i(\bar\partial - \partial)$, ensuring the torsion 3-form $H = -d^c\omega$ is real, skew, and closed ($dH=0$). The unique metric– and $J$–compatible connection with totally skew torsion $T^B=H$ is the Bismut connection: $$\nabla^B_XY = \nabla^g_XY + \tfrac12 g^{-1} H(X,Y,\cdot),$$ where $\nabla^g$ is the Levi–Civita connection. The curvature tensor $R^B$, and in particular the Ricci form

$$\rho^B(X,Y) = \tfrac12 \sum_{i=1}^{2n} g\left(R^B(X,Y) J e_i, e_i \right),$$

can be written locally in terms of $\omega$ (Lee, 2024, Wang et al., 2016, Brienza et al., 9 May 2025).

2. Definition and Characterization of BHE Metrics

A BHE metric is pluriclosed ($dd^c\omega = 0$) and solves

$\rho^B = 0.$

This equivalent to the generalized Einstein condition

$\mathrm{Rc} - \tfrac14 H^2 = 0, \quad d^*H = 0,$

where $\mathrm{Rc}$ is the Riemannian Ricci tensor and $H^2(X,Y) = \langle i_XH, i_YH \rangle$. On BHE manifolds, the first Chern class necessarily vanishes ($c_1(M) = 0$). The static BHE condition is a special case of the steady pluriclosed Ricci soliton system for the generalized Einstein–Hilbert functional (Lee, 2024, Garcia-Fernandez et al., 2021).

Notably, in the pluriclosed case the Bismut–Ricci form becomes

$$\rho^B(\omega) = -i \partial \bar\partial \log \det g,$$

so Ricci–flatness is equivalent to $\log \det g$ being pluriharmonic (Barbaro, 2021).

3. Geometric Structure, Holonomy, and Rigidity

On any compact non-Kähler BHE manifold, there exists a nonzero $\nabla^B$–parallel holomorphic and Killing vector field $$V = \tfrac12 \left(\theta^\sharp - \nabla f \right)$$, forcing holonomy reduction

$\mathrm{Hol}^0(\nabla^B) \subset \mathrm{SU}(n-1),$

i.e., unless the metric is Kähler, $\nabla^B$ never has full SU(n) holonomy (Brienza et al., 9 May 2025). Invariant BHE metrics on solvmanifolds necessarily reduce to Kähler metrics.

For SKT+CYT manifolds with parallel torsion ($\nabla^B H = 0$), the Riemannian–holomorphic universal cover splits holomorphically and isometrically

$$(\widetilde{M},\widetilde{J},\widetilde{g}) \cong (N,\text{Ricci–flat K\"ahler}) \times (G',J_L,\text{bi-invariant}),$$

where $G'$ is a Samelson space (Brienza et al., 2024, Wang et al., 2016). The only BHE manifolds admitting full holonomy must be Kähler (Brienza et al., 9 May 2025).

4. Classification and Explicit Examples

In complex dimension 2, the only compact non-Kähler BHE manifolds are diagonal Hopf surfaces $(\mathbb{C}^2 \setminus \{0\}) / \langle f \rangle$ with Boothby metrics. In dimension 3, classification yields:

• Calabi–Eckmann threefolds $S^3 \times S^3$ with bi-invariant complex structure,
• Products $S^3 \times T^2$ up to finite quotients, both admitting left-invariant pluriclosed metrics with parallel torsion and flat Bismut connection (Wang et al., 2016, Apostolov et al., 2024, Apostolov et al., 8 Jan 2026, Barbaro, 2023).

The universal covers of compact Bismut–flat manifolds are local Samelson spaces $G \times \mathbb{R}^k$, where $G$ is a compact semisimple Lie group with bi-invariant metric and left-invariant complex structure. Non–Bismut–flat examples such as K3 mapping tori and their generalizations arise via twisted fibered constructions and may carry genuinely new BHE structures (Brienza et al., 2024, Apostolov et al., 8 Jan 2026).

In higher dimensions, principal $T^2$-bundles over rational homogeneous varieties with Picard number $\varrho(X)>1$ yield infinite families of non–Kähler BHE metrics (Strominger–Bismut Ricci–flat), as in (Correa, 2023).

5. Cohomological Constraints and Stability

The Bott–Chern (1,1)-cohomology satisfies $h^{1,1}_{BC} \geq 2$ on any steady BHE 3-fold; equality holds if and only if the metric is Bismut–flat (Apostolov et al., 2024, Apostolov et al., 8 Jan 2026). For any compact simply-connected Lie group (Samelson space), the (1,1)-Aeppli cohomology groups are generated by the bi-invariant metrics on each simple factor; every pluriclosed metric lies in the same Aeppli class as a Bismut–flat metric, and the pluriclosed flow converges globally to a Bismut–flat limit (Barbaro, 2023).

Obstructions to existence appear via Mumford–Takemoto slope stability: for the holomorphic Courant algebroid associated to a pluriclosed metric, the polystability of subsheaves is necessary for a Hermitian–Einstein metric and hence a BHE solution. Infinitely many topological types with $c_1=0$ admit no BHE metric (Garcia-Fernandez et al., 2021).

6. Analytical Theory and Pluriclosed Flow

The pluriclosed flow

$$\partial_t \omega(t) = - (\mathrm{Ric}^B(\omega(t)))^{1,1}, \qquad \omega(0)=\omega_0,$$

formally corresponds to a Hermitian–Yang–Mills flow on the holomorphic Courant algebroid. On Bismut–flat backgrounds, the flow preserves torsion class and converges globally to a Bismut–flat metric (Barbaro, 2023, Garcia-Fernandez et al., 2021).

On Kodaira surfaces (principal $T^2$-bundles over curves), long-time existence for the flow is guaranteed from any starting pluriclosed metric. On Bismut–flat manifolds, every pluriclosed metric flows to the unique Bismut–flat limit, providing canonical representatives in the Aeppli class (Garcia-Fernandez et al., 2021, Barbaro, 2023).

7. Recent Developments, Classification in Dimension Three, and Open Problems

Recent results on pluriclosed Hermitian 3–folds with $\rho^B=0$ reveal a reduction mechanism to a transverse Kähler geometry, with the BHE condition corresponding to a 6th-order nonlinear PDE for the scalar curvature and Ricci forms on the quotient. The momentum map framework and associated Mabuchi–Calabi functionals yield obstructions analogous to Kähler cscK metrics (Futaki invariant, Calabi–Lichnerowicz–Matsushima rigidity), yielding rigidity for the Samelson–homogeneous BHE cases (Apostolov et al., 8 Jan 2026).

New infinite families of quasi-regular, non–locally-homogeneous BHE structures have been constructed on $S^3 \times S^3$ and $S^1 \times S^2 \times S^3$ via toric orbifold reduction methods, suggesting richer moduli in dimension three than previously observed (Apostolov et al., 8 Jan 2026).

All known compact BHE examples outside the Kähler or hyperKähler field have parallel Bismut torsion ($\nabla^B H = 0$), and any fully holonomic BHE metric must be Kähler. It remains open whether one can construct compact BHE manifolds with nonparallel torsion or with more intricate Ricci–flat structures (Brienza et al., 9 May 2025, Brienza et al., 2024, Apostolov et al., 8 Jan 2026).

---

Summary Table: Compact Non-Kähler BHE Structures

Type/Fiber Construction	Canonical BHE Example	Torsion Properties
Diagonal Hopf surface	$(\mathbb{C}^2 \setminus \{0\})/\langle f\rangle$	Parallel torsion
Calabi–Eckmann threefold $S^3 \times S^3$	Bi-invariant metric, Samelson complex structure	Parallel torsion
Samelson space Lie group quotients	$S^3 \times T^2$, $SU(3)$, etc.	Parallel torsion
Mapping tori and torus bundles	K3 mapping torus, principal $T^2$-bundle over Fano threefold	May admit nonparallel torsion, see open questions
Toric orbifold reductions	$S^3 \times S^3$, $S^1 \times S^2 \times S^3$ (quasi-regular)	New non-homogeneous examples
Solvmanifold quotient	No non-Kähler BHE unless Kähler	-

---

Bismut Ricci flat pluriclosed Hermitian manifolds constitute a rigid and rich class of geometries, unifying generalized Calabi–Yau theory with pluriclosed flow, Courant algebroid Hermitian–Einstein equations, and advanced symmetry reduction techniques. The interplay with cohomological and GIT obstructions, as well as new families of explicit examples in dimension three, positions BHE geometry as a central topic for further investigation in both differential and algebraic geometry and mathematical physics (Lee, 2024, Brienza et al., 2024, Apostolov et al., 8 Jan 2026, Barbaro, 2023).