p-Kähler structures on compact complex manifolds

2025/06/16 by Ettore Lo Giudice, Giudice, Ettore Lo
Mathematics · #32G05 #32J27 #53C15 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2506.13546

openalex publication_date 2025/06/16 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

Let (M,J) be a complex manifold of complex dimension n. A p-Kähler structure on (M,J) is a real, closed (p,p)-transverse form. In this paper, we address the conjecture of L. Alessandrini and G. Bassanelli on (n-2)-Kähler nilmanifolds equipped with nilpotent complex structures and holomorphically parallelizable nilmanifolds. We also derive necessary conditions for the existence of smooth curves of p-Kähler structures, starting from a fixed p-Kähler structure, along a differentiable family of compact complex manifolds. In addition, we study the cohomology classes of p-Kähler (resp. p-symplectic, p-pluriclosed) structures on compact complex manifolds. We provide several examples of families of compact complex manifolds admitting p-Kähler or p-symplectic structures.

Citations

• Some criteria for positive forms and applications
• p-symplectic and p-pluriclosed structures on solvmanifolds
• Deformations of astheno-Kähler metrics
• On cohomological and formal properties of Strong Kähler with torsion and astheno-Kähler metrics
• p-Kähler and balanced structures on nilmanifolds with nilpotent complex structures
• Families of almost complex structures and transverse (p,p)-forms
• Deformations of balanced metrics
• Deformations of Strong Kähler with torsion metrics
• Aeppli cohomology and Gauduchon metrics
• Several special complex structures and their deformation properties
• Proper modifications of generalized p-Kähler manifolds
• Squares of positive (p,p)-forms
• A parabolic flow of pluriclosed metrics
• Blow-ups and resolutions of strong Kähler with torsion metrics
• Complex structures on nilpotent Lie algebras
• Complex structures on nilpotent Lie algebras
• Moishezon Manifolds
• Complex parallisable manifolds

Related

• p-symplectic and p-pluriclosed structures on solvmanifolds
• Families of almost complex structures and transverse (p,p)-forms
• p-Kähler and balanced structures on nilmanifolds with nilpotent complex structures
• ∂∂-Lemma and p-Kähler structures on families of solvmanifolds
• Transverse Kähler structures on central foliations of complex manifolds