The paper calculates cohomology for complex 3-folds using new cohomology types.
problem Calculating cohomology for compact complex 3-folds.
method Defined new cohomology types Kp,q and used them to calculate cohomology. result Complete Bott-Chern-Aeppli cohomology for various complex 3-folds.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
Introduces Chern-Dirac bundles on non-Kähler Hermitian manifolds.
problem No specific problem stated; focuses on new mathematical structures.
method Introduces Chern-Dirac bundles and operators, showing isomorphisms with cohomology.
result Spaces of harmonic spinors on V-spinor bundle isomorphic to Dolbeault cohomology.
Left-invariant forms solve Dolbeault cohomology for complex nilmanifolds.
problem Computing Dolbeault cohomology for complex nilmanifolds.
method Inducing isomorphisms in cohomology using left-invariant forms.
result Invariant forms compute Dolbeault cohomology in every bidegree.
The study examines cohomological aspects on complex and symplectic manifolds.
problem Understanding qualitative properties through quantitative cohomological information.
method Analysis of Bott-Chern and Aeppli cohomology groups.
result Established comparisons among cohomology group dimensions and proved the ∂∂-lemma and Hard-Lefschetz condition. Study canonical deformations of complex forms and their cohomology properties.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.
problem Understanding cohomologies and harmonic forms on almost complex manifolds.
method Introducing new cohomologies (Bott-Chern and Aeppli) and studying associated harmonic forms.
result Bott-Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an invariant.
Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.
problem Characterizing and studying cohomologies on complex manifolds.
method Introducing Er-Bott-Chern and Er-Aeppli cohomologies, extending classical cohomologies. result Provides analogues of Serre duality and characterizes page-(r−1)-∂∂ˉ-manifolds. The study explores cohomological invariants and decomposes them into irreducible parts, focusing on zigzags.
problem Finding cohomological invariants and their decomposition into irreducible parts.
method Investigates various cohomological invariants on double complexes, focusing on the multiplicities of zigzags.
result The multiplicities of zigzags in double complexes are not sufficient to distinguish non-isomorphic double complexes.
This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} C manifolds. Our main idea is to exp…