Deformations of Dolbeault cohomology classes

Abstract

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology.

arXiv comment46 pages, published version (https://doi.org/10.1007/s00209-021-02900-w)

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Cited by in corpus (5)

• Deformations of Dolbeault cohomology classes for Lie algebra with complex structures
• On the deformed Bott-Chern cohomology
• Deformed Aeppli cohomology: canonical deformations and jumping formulas
• On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-Kähler structures
• On the obstructions of deforming vector forms