Samelson Locally Homogeneous BHE Geometries

• Samelson locally homogeneous BHE geometries are compact complex 3-folds with invariant pluriclosed metrics and Bismut–Ricci flat conditions derived from left-invariant structures on semisimple Lie groups.
• Their construction integrates Hermitian geometry and analytical techniques, centering on a canonical 6th-order PDE that facilitates reduction over Kähler bases.
• Classification results reveal rigidity, linking explicit models like SU(2)×SU(2) with extended non-Samelson cases via orthotoric constructions.

Samelson locally homogeneous Bismut–Hermitian–Einstein (BHE) geometries form a distinguished class of compact complex 3-folds characterized by a confluence of Hermitian and topological properties, notably being pluriclosed and Bismut–Ricci flat. These geometries are intimately related to left-invariant complex structures on compact semisimple Lie groups and exhibit rigid classification features in complex dimension three. Their study combines Hermitian geometry, complex Lie groups, and analytical techniques centered on a canonical 6th-order partial differential equation arising from dimensional reduction, with significant implications for the structure of non-Kähler manifolds.

1. Fundamental Definitions and Preliminaries

Let $(N^{2n},J,g)$ be a Hermitian manifold with fundamental form $\omega = g(J\cdot, \cdot)$. The Bismut connection $\nabla^B$ is the unique Hermitian connection with totally skew-symmetric torsion: $$\nabla^B_X Y = \nabla^{\mathrm{LC}}_X Y - \tfrac{1}{2}(d^c\omega)(X,Y,-).$$ Its curvature $R^B$ defines the Bismut Ricci form by

$$\rho^B(X,Y) = \sum_{i=1}^n \langle R^B_{X,Y} e_i, J e_i \rangle \in 2\pi c_1(N),$$

with $\{e_i\}$ an orthonormal basis.

A Hermitian manifold is pluriclosed (also called SKT) if $dd^c\omega = 0$. It is Bismut–Ricci flat if $\rho^B \equiv 0$. A structure satisfying both is termed a Bismut–Hermitian–Einstein (BHE) manifold (Apostolov et al., 8 Jan 2026). This pluriclosed condition is a non-Kähler analogue of Kählerness relevant for the study of Hermitian and generalized Kähler geometries.

2. Construction of Samelson Locally Homogeneous BHE Geometries

Samelson’s construction starts with a compact semisimple Lie group $G$ of even real dimension. Taking a decomposition of its Lie algebra $$\mathfrak{g} = \mathfrak{t} \oplus \bigoplus_{\alpha>0} \mathfrak{g}_\alpha$$ and an invariant inner product, a positive system of roots yields a left-invariant complex structure $J$. The bi-invariant Riemannian metric $g$ is pluriclosed, and the corresponding Bismut connection is flat ($\rho^B = 0$), forming a BHE structure.

In complex dimension three, this yields explicit homogeneous BHE 3-folds: $N = \SU(2) \times \SU(2) \quad \text{or} \quad N = \SU(2) \times S^1 \times S^1,$ with the product Samelson structure. These and their finite quotients are referred to as Samelson BHE geometries (Apostolov et al., 8 Jan 2026).

The explicit metric on $\SU(2) \times \SU(2)$ is

$$g = \tfrac{1}{4} \sum_{a=1}^6 (\sigma^a \otimes \sigma^a),$$

with $\{\sigma^a\}$ denoting the Maurer–Cartan forms. The associated complex structure identifies each $\mathfrak{su}(2)$ summand with the imaginary quaternions.

3. Classification and Rigidity in Complex Dimension Three

A non-Kähler compact BHE 3-fold $(N^3,J,g,\omega)$ satisfies $\dim_\mathbb{R} H_{BC}^{1,1}(N) \ge 2$, has two commuting $\nabla^B$-parallel Killing fields of constant length, and admits a reduction to a Kähler quotient $(S,J_S,\omega_S)$ with a closed primitive $(1,1)$-form $\theta$. The transverse geometry satisfies a 6th-order nonlinear PDE: $\tfrac{1}{2} dd^c(\Scal_{\omega_S}) \wedge \omega_S = \theta \wedge \theta + \rho(\omega_S) \wedge \rho(\omega_S), \quad \Scal_{\omega_S} > 0.$ Conversely, any compact toric $T^2$-bundle over a Kähler surface $(S, \omega_S, \theta)$ satisfying this PDE yields a BHE structure.

Classification theorem: $N$ is (covered by) a Samelson geometry if and only if $\dim H_{BC}^{1,1}(N) = 2$ and $\Scal_{\omega_S}$ is constant on the quotient (Apostolov et al., 8 Jan 2026). Corollary: In the regular case, such compact non-Kähler BHE 3-folds are isometric to $\SU(2)\times \SU(2)$ or $\SU(2)\times T^2$.

4. The 6th-Order PDE and Momentum Map Formalism

The reduced 6th-order PDE is central to the theory. For a compact Kähler surface $(S^2, \omega)$ and closed, primitive $(1,1)$-form $\theta$ with $\int_S \theta \wedge \omega = 0$, the equation is: $\frac{1}{2} dd^c(\Scal_\omega) \wedge \omega = \theta \wedge \theta + \rho(\omega) \wedge \rho(\omega), \qquad \Scal_\omega > 0.$ This is equivalent (via Hodge theory) to

$\Delta\Scal_\omega + \frac{1}{2} \Scal_\omega^2 - \|{\rm Rc}_\omega\|^2 - \|\theta\|^2 = 0.$

The generalization to higher dimension involves real $(1,1)$-classes $\alpha, \beta$ and a constant $c_{\alpha,\beta}$, with solutions interpreted as zeros of an infinite-dimensional moment map on the space of complex structures compatible with a fixed symplectic form.

This formalism enables the definition of a Mabuchi functional (whose critical points are solutions to the PDE), a Futaki invariant (obstruction to existence), and a Calabi functional (whose critical points are “extremal” for the PDE). The Mabuchi functional is convex along geodesics of positive scalar curvature, and vanishing of the Futaki invariant is necessary for existence, paralleling the structure of constant scalar curvature Kähler (cscK) theory.

5. Explicit Models: Samelson Structures on Homogeneous 3-Folds

The bi-invariant Samelson structure on $G = \SU(2)\times\SU(2)$ or $\SU(2)\times T^2$ yields a free torus action; the Kähler quotient is $S^2\times S^2$ (product of round metrics) or an elliptic ruled surface $$\mathbb{P}(\mathcal{O}_E \oplus \mathcal{O}_E) \to E$$ with locally symmetric cscK metric. The total space metric is explicitly given: $J_N(V) = W, \quad J_N(W) = -V, \quad g_N = \pi^*\Big(\tfrac{\Scal_S}{2} g_S\Big) + \eta_V^2 + \eta_W^2,$ with $V, W$ generating the torus action and $\eta_V, \eta_W$ dual connection forms.

These explicit constructions realize the Samelson BHE metrics as total spaces over Kähler bases with constant scalar curvature and trivial transverse data $\theta=0$.

6. Non-Samelson BHE Examples: Orthotoric Construction

A significant development is the construction of infinitely many non-Samelson BHE structures on $S^3\times S^3$ and $S^1 \times S^2 \times S^3$ via the orthotoric Kähler ansatz on toric surfaces of quadrilateral type (Apostolov et al., 8 Jan 2026). For a generic Delzant quadrilateral $\Delta$ with suitable affine normals, coordinates $(x, y)$ and functions $A(x), B(y)$ yield orthotoric metrics. For quadratic choices of $A,B$ satisfying

$4(a_0 + b_0)(a_2 + b_2) \ge (a_1 + b_1)^2,$

and suitable constants $\lambda$, the reduced BHE PDE becomes an ODE in $x, y$ and is satisfied.

Applying the Delzant–Lerman–Tolman construction, one obtains a compact 4-orbifold Kähler base $M$, over which a principal $T^2$-bundle (with Chern data $(\theta, \rho(\omega))$) yields total spaces diffeomorphic to $S^3\times S^3$ or $S^1\times S^2\times S^3$, carrying pluriclosed, Bismut–Ricci flat Hermitian metrics. For generic $A,B$, the scalar curvature is non-constant, so these are genuinely non-Samelson structures. This generates the first known examples of such BHE 3-folds beyond the Samelson case.

7. Variational and Geometric Invariants

The analytic framework for the BHE reduction PDE extends classical functional-analytic tools. On the space of Kähler potentials $\varphi$ (with $\omega_\varphi = \omega_0 + dd^c \varphi > 0$), the 1-form

$\Theta_\varphi(\dot\varphi) = -\int_M \dot\varphi \left[ -\frac12\, dd^c \Scal_\varphi \wedge \omega_\varphi^{n-1} + \rho_\varphi^2 \wedge \omega_\varphi^{n-2} + \theta_\varphi^2 \wedge \omega_\varphi^{n-2} - \tfrac{c_{\alpha,\beta}2\,\omega_\varphi^n \right]$

is closed, admitting a primitive Mabuchi-type functional $\mathcal{E}$. The derivative of $\mathcal{E}$ encodes the PDE. The Futaki invariant for holomorphic vector fields and the Calabi functional characterizing extremal solutions also extend to this setting and satisfy analogues of classical theorems: e.g., vanishing Futaki invariant is a necessary condition for existence of solutions; $\mathcal{E}$ is convex along geodesics.

Standard GIT arguments enforce reductivity of the automorphism group if solutions exist and provide uniqueness up to automorphisms.

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References: Core results, explicit formulae, and classification theorems are detailed in (Apostolov et al., 8 Jan 2026). Key foundational contributions are due to Samelson (complex structures on Lie groups), Gauduchon–Ivanov (Hermitian–Einstein surfaces), and Apostolov–Barbaro–Lee–Streets ([ABLS], (Apostolov et al., 2024). The orthotoric construction generalizes work by Couzens–Gauntlett–Martelli–Sparks ([CGMS], in the context of AdS$_3\times Y_7$ geometries). For further historical and classification context, see the relevant series by Apostolov–Calderbank–Gauduchon ([ACG], [ACGTF]) and standard references on Kähler geometry and Hermitian surfaces.