Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds

Abstract

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an analog of Dolbeault-cohomology for Lie-algebras with complex structure.

19 pages; updated first part of arXiv:0709.0467v1 containing the results on small deformations. The results on deformations in the large will appear in an independent paper

References in corpus (5)

• The Kuranishi-space of complex parallelisable nilmanifolds
• Differential Gerstenhaber algebras associated to nilpotent algebras
• Complex product structures on 6-dimensional nilpotent Lie algebras
• Extended Deformation of Kodaira Surfaces
• Hermitian structures on six dimensional nilmanifolds

Cited by in corpus (9)

• Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
• Deformations of Dolbeault cohomology classes for Lie algebra with complex structures
• The Kuranishi-space of complex parallelisable nilmanifolds
• Abelian Complex Structures and Generalizations
• Maximal nilpotent complex structures
• Holomorphic Poisson Cohomology
• Frobenius Structures and Generalized Deformation of Kodaira Manifolds
• Invariant and non-invariant almost complex structures on compact quotients of Lie groups
• On the Bott-Chern cohomology and balanced Hermitian nilmanifolds