Cohomological properties and Hermitian metrics of complex non-Kähler manifolds

We give a partial account of some problems concerning cohomological invariants and metric properties of complex non-Kähler manifolds.

💡 Research Summary

The manuscript “Cohomological properties and Hermitian metrics of complex non‑Kähler manifolds” is a comprehensive set of lecture notes that surveys recent developments at the interface of complex cohomology, special Hermitian metrics, and geometric flows on non‑Kähler manifolds. The authors begin with a concise review of basic complex‑geometric notions—holomorphic manifolds, holomorphic vector bundles, almost‑complex structures, and the Newlander‑Nirenberg integrability theorem—setting the stage for a systematic discussion of cohomological tools.

Section 2 introduces the double complex of (p,q)‑forms equipped with the ∂ and ∂̄ operators. The associated Frölicher spectral sequence is described, and the authors emphasize that its degeneration at the first page is equivalent to the ∂∂‑Lemma. They then present a “numerical characterization” of the ∂∂‑Lemma via Bott‑Chern and Aeppli cohomology: the lemma holds precisely when the natural maps H^{p,q}{BC} → H^{p,q}{\bar∂} and H^{p,q}{\bar∂} → H^{p,q}{A} are isomorphisms for all (p,q). This viewpoint allows one to detect the lemma by comparing dimensions of these cohomology groups.

The authors study the stability of the ∂∂‑Lemma under various geometric transformations, such as blow‑ups, modifications, and deformations, showing that it is preserved under many standard operations. Concrete computations are carried out for nilmanifolds and solvmanifolds, where the Bott‑Chern and Aeppli groups can be expressed in terms of the underlying Lie algebra structure. These examples illustrate how non‑Kähler phenomena are reflected in cohomological invariants.

Section 3 shifts to Hermitian geometry. After recalling the definition of a Hermitian metric and its associated (1,1)‑form ω, the authors discuss several distinguished metrics: Gauduchon, balanced, pluriclosed, and locally conformally Kähler (LCK). They review known existence results, such as the universal existence of Gauduchon metrics on any compact complex manifold, and the more restrictive conditions required for balanced or pluriclosed metrics. A notable contribution is the partial classification of three‑dimensional complex manifolds admitting LCK metrics, which extends the classical Kähler‑Einstein classification into the non‑Kähler realm.

Section 4 is devoted to the Chern‑Yamabe problem, i.e., the search for Hermitian metrics with constant Chern scalar curvature. By expressing the Chern scalar curvature in terms of the metric potential, the problem reduces to a Liouville‑type nonlinear PDE. The authors distinguish three sign regimes (zero, negative, positive) and summarize existence results: zero curvature metrics exist under topological constraints, negative curvature solutions follow from a maximum principle argument, while positive curvature requires delicate variational techniques and Moser–Trudinger type inequalities. The “moment map picture” is also sketched, linking the problem to symplectic reduction.

Section 5 introduces geometric flows on Hermitian manifolds, focusing on the Chern‑Ricci flow. After defining the flow and recalling short‑time existence, the authors compare it with the Kähler‑Ricci flow, highlighting that the Chern‑Ricci flow preserves the complex structure but not necessarily the Hermitian condition. They analyze its behavior on non‑Kähler Calabi–Yau manifolds and on Inoue surfaces. In the latter case, the flow exhibits a periodic “cycling” phenomenon and does not converge to a canonical metric, illustrating the richness of dynamics in the non‑Kähler setting.

The final Section 6 collects a curated list of open problems, grouped into six themes: (1) geometric representatives of cohomology classes, (2) topological obstructions to complex structures, (3) almost‑complex cohomologies, (4) topological obstructions to Hermitian metrics, (5) existence and classification of canonical Hermitian metrics, and (6) broader classification problems for complex manifolds and surfaces. Each problem is accompanied by references to known partial results, thereby providing a roadmap for future research.

Overall, the paper succeeds in weaving together three major strands—cohomology, special Hermitian metrics, and geometric flows—into a coherent narrative. By coupling abstract theory with explicit calculations on nilmanifolds, solvmanifolds, and Inoue surfaces, the authors illuminate how cohomological invariants control metric properties and flow behavior. The extensive list of open questions makes the manuscript a valuable reference point for researchers aiming to advance the understanding of complex non‑Kähler geometry.