Holomorphic Deformations of Balanced Calabi-Yau ∂∂-Manifolds

2013/04/01 by Dan Popovici, Popovici, Dan · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AG #math.CV #math.DG

paper · pdf · doi:10.48550/arxiv.1304.0331

To appear in the Annales de l'Institut Fourier

openalex publication_date 2013/04/01 · arxiv created 2018/03/14 · arxiv updated 2018/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a compact complex n-fold X satisfying the ∂∂-lemma and supposed to have a trivial canonical bundle K_X and to admit a balanced (=semi-Kähler) Hermitian metric ω, we introduce the concept of deformations of X that are \bf co-polarised by the balanced class [ω^n-1]∈ H^{n-1, n-1}(X, \C)⊂ H^2n-2(X, \C) and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi-Yau or even holomorphic symplectic compact complex manifolds. The concept of Weil-Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.

Cited by

• Several special complex structures and their deformation properties
• The Monge-Ampère equation for (𝑛-1)-plurisubharmonic functions on a compact Kähler manifold

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