$p$-Kähler structures on fibrations and reductive Lie groups

We investigate the existence of $p$-Kähler structures on two classes of complex manifolds: on quasi-regular fibrations, with particular emphasis on complex homogeneous spaces, and on reductive Lie groups endowed with invariant complex structures. In the latter setting, we construct non-regular complex structures on the Lie algebras $\mathfrak{sl}(2m-1,\mathbb{R})$ for $m \ge 2$ and show that these structures admit compatible balanced metrics, providing new explicit examples of balanced manifolds.

💡 Research Summary

The paper investigates the existence of p‑Kähler and p‑pluriclosed structures on two broad classes of complex manifolds: quasi‑regular holomorphic fibrations (with a focus on complex homogeneous spaces) and reductive Lie groups equipped with invariant complex structures.

A p‑Kähler structure is defined as a closed, transverse (p,p)‑form Ω, where “transverse” means weakly positive in the sense of Harvey–Lawson (i.e., Ω∧Θ is a positive multiple of a fixed volume form for every strongly positive (n‑p,n‑p)‑form Θ). When p=1 this recovers ordinary Kähler metrics; when p=n‑1 it coincides with balanced metrics; the case p=n is trivial. The authors recall the cone of strongly positive forms, its dual cone of weakly positive forms, and show that the Harvey–Lawson criterion extends to p‑Kähler and p‑pluriclosed settings: a compact complex manifold is p‑Kähler (resp. p‑pluriclosed) iff every strongly positive (p,p)‑current that is a boundary (resp. ∂ ∂̄‑boundary) vanishes.

Quasi‑regular fibrations.
Let π : M → X be a holomorphic fibration over a compact complex orbifold X, with fibres complex tori (the “quasi‑regular” case). Assume there exists an orbifold line bundle L on X such that c₁(L) is represented by a strongly positive (1,1)‑form ω, c₁(L)≥0, and π* c₁(L)=0 in H²(M,ℝ). If the smallest integer k with c₁(L)ᵏ≠0 but c₁(L)ᵏ⁺¹=0 exists, then Theorem 3.1 proves that M cannot carry a (dim M‑j)‑Kähler structure for any 1≤j≤k. The proof uses the identity dθ=π*ω for a connection 1‑form θ and the positivity of Ω∧(dθ)ʲ, which contradicts Stokes’ theorem.

Applying this to compact complex homogeneous spaces G/H, the authors recall the Tits fibration π : G/H → G/K, where K centralises a torus in G and the fibre H/K is a complex torus. The base G/K is a generalized flag manifold, often Kähler–Einstein. Using the anti‑canonical bundle of the base as L, they obtain Corollary 3.3: if G/H has vanishing first Chern class and the Tits fibration has rank r, then G/H admits no q‑Kähler structure for any r ≤ q < dim G/K + r. This recovers known results about balanced metrics on homogeneous spaces and extends them to arbitrary p‑Kähler settings.

Reductive Lie groups.
A real even‑dimensional reductive Lie group G₀ (i.e. Lie algebra 𝔤 = 𝔞 ⊕ 𝔰 with abelian ideal 𝔞 and semisimple ideal 𝔰) always admits invariant complex structures; such a structure corresponds to a choice of an n‑dimensional complex subalgebra q⊂𝔤ℂ with 𝔤ℂ = q ⊕ σ(q), where σ is complex conjugation with respect to 𝔤. Snow’s classification splits reductive algebras into Class I (all simple factors of inner type) and Class II (the rest).

For compact, non‑abelian, even‑dimensional reductive groups, Theorem 4.2 shows that (n‑k)‑Kähler structures cannot exist for 1≤k≤r₀, where r₀ is the number of positive roots of the complexified Lie algebra. The obstruction comes from the fact that the base of the Tits‑type fibration is a Kähler‑Einstein flag manifold of complex dimension r₀, and the characteristic classes of the torus bundle force the existence of a non‑trivial strongly positive exact (n‑k,n‑k)‑form, contradicting the Harvey–Lawson criterion.

In the non‑compact case with regular invariant complex structures, the authors prove that (n‑2)‑pluriclosed metrics cannot exist (Theorem 4.5), extending earlier results for simple groups of inner type. This is achieved by analysing the cohomology of the open G₀‑orbit in a projective rational homogeneous space and showing that any candidate ∂ ∂̄‑closed (n‑2,n‑2)‑form would give rise to a non‑trivial positive current of bidegree (2,2) that is a ∂ ∂̄‑boundary, again violating the generalized Harvey–Lawson condition.

Non‑regular complex structures on 𝔰𝔩(2m‑1,ℝ).
The final section constructs explicit non‑regular invariant complex structures on the real Lie algebra 𝔰𝔩(2m‑1,ℝ) for every m≥2. Unlike regular structures, the chosen complex subalgebra q does not satisfy 𝔤ℂ = q ⊕ σ(q) with q a Lie subalgebra; instead, q is a non‑integrable complement that still yields an invariant almost complex structure which turns out to be integrable. The authors exhibit a left‑invariant (n‑1,n‑1)‑form Ω that is closed and positive, proving the existence of a balanced metric compatible with the complex structure. However, they also compute ∂ ∂̄Ω ≠ 0, showing that no SKT (pluriclosed) metric exists. This provides the first explicit family of balanced, non‑pluriclosed manifolds arising from non‑regular complex structures on a semisimple Lie group.

Overall contribution.
The work extends the theory of p‑Kähler geometry to a broader class of manifolds, delivering new obstruction theorems for both compact homogeneous spaces and reductive Lie groups, and supplying concrete examples that separate the balanced and pluriclosed categories. By blending techniques from Lie theory, characteristic class calculations, and the theory of positive currents, the authors deepen our understanding of how algebraic structure governs the existence of special Hermitian metrics. The paper thus offers both conceptual advances and concrete constructions that will be valuable for researchers studying complex geometry on homogeneous and Lie‑group manifolds.