Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

2024/07/05 by Liviu Ornea, Misha Verbitsky, Ornea, Liviu +1 · 2 citations
Mathematics · #32H04 #53C55 #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.2407.04623

openalex publication_date 2024/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A complex Hermitian n-manifold (M,I, ω) is called locally conformally Kahler (LCK) if dω=θ\wedgeω, where θ is a closed 1-form, balanced if ω^n-1 is closed, and SKT if dIdω=0. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form -d(Iθ) is Bott--Chern homologous to a positive (1,1)-current. This conjecture implies that (M,I) does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.

Cited by

• Singular Gauduchon Conjecture
• Calabi-Yau locally conformally Kähler manifolds

Related

• Compatibility between non-Kähler structures on complex (nil)manifolds
• Calabi-Yau locally conformally Kähler manifolds
• Currents on locally conformally Kähler manifolds
• On Hermitian manifolds with vanishing curvature
• Balanced Manifolds and SKT Metrics