Deformations of (p,q)-forms and degenerations of the Frölicher spectral sequence

2023/12/21 by Xueyuan Wan, Wei Xia, Wan, Xueyuan +1 · 1 citation
Mathematics · #32A05 #32G05 #55T05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2312.13692

openalex publication_date 2023/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well-known that Hodge numbers are invariant under deformations of complex structures if the Frölicher spectral sequence of the central fiber degenerates at the first page (i.e. E₁=E_∞). As a result, the deformations of (p,q)-forms are unobstructed for all (p,q) if E₁=E_∞. We refine this classical result by showing that for any fixed (p,q) the deformations of (p,q)-forms are unobstructed if the differentials d_r^p,q in the Frölicher spectral sequence satisfy \bigoplus_{r≥ 1}d_r^p,q=0\quadand \bigoplus _{r≥ i≥ 1} d_r^p-i,q+i=0. Moreover, the deformation stability of the degeneration property for Frölicher spectral sequences in the first page and higher pages is also studied. In particular, we have found suitable conditions to ensure the deformation stability of E_r^p,q=E_∞^p,q (r≥1) for fixed (p,q).

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