Overview of Homogeneous SKT Structures

• Homogeneous SKT structures are Hermitian metrics on homogeneous complex manifolds with invariant complex structures and closed torsion forms, reflecting precise Lie algebra constraints.
• They are characterized by algebraic equations on the Lie algebra that simplify the SKT condition, enabling concrete classifications on nilmanifolds, solvmanifolds, and compact Lie groups.
• Applications include the analysis of rigidity results, classification of invariant structures, and insights into pluriclosed flows and geometric obstructions in non-Kähler settings.

A homogeneous SKT (strong Kähler with torsion) structure is a Hermitian metric $g$ on a homogeneous complex manifold $(M, J)$ whose fundamental (1,1)-form $\omega$ satisfies $\partial\bar\partial\omega = 0$, subject to the additional symmetry condition that both the complex structure and the metric are invariant under a transitive Lie group action. This property distinguishes homogeneous SKT structures from the general SKT class and connects them deeply with algebraic and representation-theoretic data of the underlying Lie group or homogeneous space.

1. Definitions and Background

Given a complex manifold $(M, J)$ and a $J$-Hermitian Riemannian metric $g$, the fundamental form is defined by $\omega(X, Y) = g(JX, Y)$. The SKT condition requires this form to be $\partial\bar\partial$-closed: $\partial\bar\partial\,\omega = 0$ Alternatively, in real notation, $dJd\omega = 0$, or in terms of the Bismut connection $\nabla^B$, one asks that the totally skew-symmetric torsion 3-form $c = Jd\omega$ is closed: $dc = 0$ A homogeneous SKT structure is one in which both $(J, g)$ are invariant under a transitive (typically Lie) group action, so $(M, J, g)$ is a homogeneous Hermitian space with an SKT metric. Invariant complex structures on Lie groups (Samelson, Wang) and their quotients under discrete subgroups provide central sources of such examples, especially on nilmanifolds, solvmanifolds, and compact Lie groups (Enrietti et al., 2010, Poddar et al., 2016, Enrietti et al., 2011, Fino et al., 2022, Pham, 2024).

2. SKT Condition in the Homogeneous and Lie-Algebraic Context

For a homogeneous space $M = G/H$ or a Lie group $G$ with left-invariant complex structure, the SKT condition reduces to algebraic equations on the Lie algebra $\mathfrak{g}$, due to the invariance of both $J$ and $g$.

For a left-invariant Hermitian structure on a real Lie algebra $\mathfrak{g}$, with basis $\{e^i\}$ of 1-forms and $J$ the complex structure, the fundamental form is

$\omega = \sum_{p<q} \omega_{pq} e^p \wedge e^q$

and its differential

$$d\omega(X, Y, Z) = -\omega([X, Y], Z) - \omega([Y, Z], X) - \omega([Z, X], Y)$$

The condition $d(J d\omega) = 0$ (so $dc=0$) translates to a system of quadratic equations in the structure constants and metric coefficients. This gives a finite algebraic system to be solved for possible SKT structures on a given (homogeneous) complex manifold (Enrietti et al., 2010, Fino et al., 2022, Enrietti et al., 2011, Alekseevsky et al., 2014).

3. Rigidity Results and Classification: Nilmanifolds and Solvmanifolds

For compact homogeneous nilmanifolds $M = \Gamma \backslash G$ (with $G$ nilpotent, $\Gamma$ cocompact) carrying an invariant complex structure $J$, the existence of a homogeneous SKT metric imposes strong rigidity:

• If $(M, J)$ admits an invariant SKT metric, then $\mathfrak{g}$ is at most 2-step nilpotent. In particular, every such $(M, J)$ is a holomorphic principal torus bundle over an abelian complex torus, and no higher-step nilmanifold admits a homogeneous SKT metric (Enrietti et al., 2010).
• In dimension eight, all possible nilpotent SKT structures are classified into two explicit families via their structure equations and SKT constraints; every homogeneous SKT nilmanifold in dimension eight fits into these families. Minimal examples include $h_3 \oplus \mathbb{R}^5$ and Heisenberg-like extensions (Enrietti et al., 2010).
• For solvmanifolds (compact quotients of solvable Lie groups), the situation is even more restrictive. Homogeneous SKT metrics can exist only in extremely constrained settings:
  • For almost-abelian or abelian complex structures, or if $J$ is invariant under a nilpotent complement, the only SKT solvmanifolds are tori, i.e., Kähler (Fino et al., 2013, Beaufort et al., 2022).
  • For codimension-two nilradical solvable Lie algebras, explicit classification (e.g., in dimension six) shows only two types admit SKT: $\mathfrak{t}_{3,0}\times\mathfrak{t}_{3,0}$ and $\mathfrak{g}_{5.35}\oplus\mathbb{R}$, with explicit coframes and SKT metrics (Brienza et al., 2024).
  • General SKT extensions exist via semidirect products of nilpotent SKT Lie algebras with abelian Hermitian ones, under precise compatibility conditions on derivations (Brienza et al., 2024).

4. Compact Lie Groups, Flag Manifolds, and Principal Bundle Constructions

A central source of homogeneous SKT structures is the theory of compact Lie groups equipped with Samelson (left-invariant) complex structures:

• Every even-dimensional compact classical group admits left-invariant SKT metrics—specifically, bi-invariant (Killing-form) Hermitian metrics are SKT with respect to Samelson $J$ (Poddar et al., 2016, Fino et al., 2022, Fino et al., 2016).
• On each such $K$, the SKT condition for left-invariant Hermitian metrics can be recast as an explicit linear constraint on root weights and Killing-form eigenvalues, e.g., as a system for parameters $\lambda_\alpha$ indexed by roots (Fino et al., 2022, Pham, 2024).
• The exceptional Lie group $G_2$ admits a 3-parameter family of left-invariant SKT metrics with respect to its Samelson complex structure, which includes all bi-invariant metrics. These are characterized explicitly via seven parameters $\mu_j$ subject to linear constraints, and any $T$-invariant SKT metric must belong to this family (Pham, 2024).
• Homogeneous SKT structures also arise as explicit pullbacks on total spaces of principal torus bundles over Kähler bases, provided certain characteristic-class vanishing conditions (e.g., a quadratic Chern or Pontryagin class) are met. This unifies compact group and torus-bundle examples (Poddar et al., 2016).

The following table summarizes principal cases for homogeneous SKT structures:

Homogeneous Space	Existence of SKT	SKT Structure Classification
Nilmanifolds (not torus)	iff 2-step nilpotent	8D: two explicit families (Enrietti et al., 2010)
Solvmanifolds	very rare	Only specific low-dim. types (Brienza et al., 2024)
Compact Lie groups	always (even dim)	Bi-invariant, extended families (Fino et al., 2022, Pham, 2024)
Principal torus bundles	Characteristic-class constraint	Explicit metric formulas (Poddar et al., 2016)

5. Homogeneous SKT on Six-dimensional Manifolds with Semi-simple Isotropy

Classification of left-invariant SKT Hermitian structures with semi-simple isotropy is complete in dimension six:

• Only three non-Kähler SKT families exist, all realized on solvable Lie groups as symmetric extensions of modules for $\mathrm{SU}(2)$ or $\mathrm{SU}(1,1)$, with explicit structure equations and Gray-Hervella type $\mathcal{W}_3$ or $\mathcal{W}_3\oplus\mathcal{W}_4$.
• All further structures are invariant on solvable Lie groups, with no “exceptional” cases arising (Alekseevsky et al., 2014).

6. Cohomological and Geometric Obstructions

Homogeneous SKT existence is obstructed in several ways:

• For compact quotients of solvmanifolds, SKT implies Kähler under strong conditions, e.g., when the complex structure is abelian or compatible with a nilpotent complement (Fino et al., 2013).
• Balanced versus SKT compatibility: on two-step solvable unimodular Lie algebras, the existence of both a balanced and an SKT metric implies the existence of a Kähler metric; outside the unimodular case, explicit counterexamples exist (Freibert et al., 2022).
• For almost abelian Lie algebras, the spectrum of the adjoint must satisfy strict algebraic constraints (real parts $0$ or $-a$), and only certain block diagonalizations are possible (Beaufort et al., 2022).
• In the context of Wang's C-spaces (compact complex homogeneous spaces with finite $\pi_1$), non-Kähler SKT structures exist only if the space is a product $G_1 \times F$ of a compact group and a flag manifold; no strictly non-product C-space admits an SKT metric (Fino et al., 2016).

7. Further Directions and Open Problems

Current research avenues and open questions in homogeneous SKT geometry include:

• Complete classification of root-coefficient SKT systems on arbitrary compact semisimple Lie groups, especially in the right $T$-invariant case, and study of the structure of the solution locus (Fino et al., 2022, Pham, 2024).
• Analysis of pluriclosed and generalized Ricci flows for homogeneous SKT metrics on non-Kähler Lie groups, with focus on their dynamical stability and Gromov-Hausdorff limits (Fino et al., 2022).
• Extension of SKT theory to non-compact, non-simply connected homogeneous spaces or to homogeneous structures beyond classical Samelson complex structures (Poddar et al., 2016).
• Construction and classification of examples with nonflat Bismut curvature, including connections to generalized Kähler geometry and hypercomplex structures, and detailed study of their cohomological invariants (e.g., Hodge, Dolbeault, and Bismut-Ricci spectra) (Brienza et al., 2024, Enrietti et al., 2010).

Homogeneous SKT geometry thereby connects complex Lie groups, representation theory, characteristic class theory, and the precise algebraic structure of Lie algebras, providing a unified framework for the exploration of special Hermitian metrics beyond the Kähler regime.