The abstract proves a Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology.
problem Cohomology on complex manifolds, especially non-compact ones.
method Proves a Leray-Hirsch theorem and an explicit blow-up formula.
result Explicit blow-up formula for Dolbeault cohomology on complex manifolds.
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.
Defines relative Dolbeault homology and proves its equivalence with Čech-Dolbeault cohomology.
problem Comparing relative Dolbeault cohomology groups of complex manifolds.
method Uses Čech approach to define relative Dolbeault homology and proves equivalence with Čech-Dolbeault cohomology.
result Relative Dolbeault homology and Čech-Dolbeault cohomology are equivalent.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
Study on complex manifolds and their cohomology properties.
problem Stability of Dolbeault cohomology under modifications of compact complex manifolds.
method Using Čech cohomology theory to study Dolbeault cohomology of blow-ups.
result Construction of a compact complex non-Kähler manifold satisfying the ∂∂ˉ-Lemma. The paper solves a conjecture on almost complex 4-manifolds using refined Dolbeault cohomology.
problem Proving a condition for almost complex 4-manifolds to be symplectic or almost Kähler.
method Defining refined Dolbeault cohomology and proving conditions equivalent to known results.
result The condition ildeh1,0=ildeh0,1 is equivalent to a generalized ∂∂ˉ-lemma on almost complex 4-manifolds. We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particula…
Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.
problem Compute cohomologies of complex solvmanifolds.
method Build finite-dimensional double subcomplexes and decompose them into indecomposable ones.
result Characterize the ∂∂ˉ-Lemma property and compute triple ABC-Massey products. Study on properties of Oeljeklaus-Toma manifolds, including cohomology and metrics.
problem Characterizing and understanding the metric and cohomological properties of Oeljeklaus-Toma manifolds.
method Analysis of double complex of differential forms, Bott-Chern cohomology, and explicit formulas for Dolbeault cohomology.
result Proved that Oeljeklaus-Toma manifolds do not admit certain types of metrics and provided explicit formulas for their Dolbeault cohomology.
The paper proves a property of complex geometry under transformations.
problem The ∂∂-lemma property in complex geometry. method Simple proof using a blow-up formula and morphism.
result Heredity and bimeromorphic invariance of the ∂∂-lemma property. Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.
problem Analyzing complex structures with perturbed differential operators.
method Perturbing the standard differential operator to a first-order operator Dη and computing Bochner-Kodaira-Nakano-type formulae. result Obtained vanishing results for certain harmonic spaces and Dolbeault cohomology.
Updates and rewrites a 1974 AMS Memoir on Lie groups.
problem Real reductive Lie groups and their representations.
method Rewriting and updating a 1974 AMS Memoir.
result Ties with recent approaches to geometric realization of unitary representations.
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
Study on twisted Dolbeault cohomology in Kähler foliations.
problem Exploring cohomology in transverse Kähler foliations.
method Analysis of twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem.
result Proved Kodaira-Serre type duality for twisted basic Dolbeault cohomology.
The paper extends Dolbeault cohomology to almost complex manifolds and provides new tools for studying their properties.
problem Extending Dolbeault cohomology to almost complex manifolds.
method Developed a spectral sequence and harmonic theory for Dolbeault cohomology.
result Dolbeault cohomology can be used to prohibit the existence of nearly Kähler metrics.
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
The paper develops L2 theory for foliations on manifolds with boundary.
problem Analyzing the cohomology of foliated manifolds with boundary.
method Develops L2 theory, establishes decomposition and vanishing theorems, and proves duality and extension theorems. result Establishes Dolbeault decomposition of basic forms and proves global regularity for ∂ˉB-equations. We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality à la Frölicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equa…
The paper introduces a new class of manifolds based on the Hodge decomposition and spectral sequences.
problem Characterizing and understanding new classes of compact complex manifolds.
method Introducing a new class of page-r-∂∂ˉ-manifolds and using spectral sequences. result Characterized and provided examples of the new class of manifolds.
Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
New cohomology η for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.
problem Developing a new cohomology structure for deformed Sasaki-Einstein manifolds.
method Introducing η-cohomology defined by a CR structure and a holomorphic function f with non-vanishing η≡df. result Established a direct relation between the cyclic homologies of the Calabi-Yau algebra and the η-cohomology groups. Study on Dolbeault cohomology and Kähler foliations with new formulas and restrictions.
problem Investigating Dolbeault cohomology on transversely Kähler foliations.
method Developed new Weitzenböck formulas and conditions on curvature.
result Proved absence of nonzero basic-harmonic forms on foliations with positive Ricci curvature.
Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.
problem Dolbeault and Bott-Chern cohomology of Oeljeklaus-Toma manifolds.
method Explicit harmonic representatives and geometric analysis.
result Showed geometric Dolbeault formality and studied Angella-Tomassini inequality.
Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.
problem Investigating holomorphic Koszul-Brylinski homology on Poisson manifolds.
method Using Dolbeault cohomology to derive properties of holomorphic Koszul-Brylinski homology.
result Obtained the Leray-Hirsch theorem, Mayer-Vietoris sequence, and Künneth theorem for holomorphic Koszul-Brylinski homology.
Study Dolbeault cohomology on complex manifolds with torus action.
problem Describe Dolbeault cohomology of complex manifolds with torus action.
method Describe Dolbeault cohomology algebra of canonical foliation, provide dga model, prove Hodge decomposition.
result Hodge decomposition for basic Dolbeault cohomology proved.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
The paper develops a deformation theory for Dolbeault cohomology classes.
problem Understanding the variations of Dolbeault cohomology classes.
method Established a deformation theory using the power series method and proved the extension equation.
result Proved the existence and unobstructedness of deformations under certain conditions.
We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.
Formula for Dolbeault cohomologies of complex manifolds via relative cohomology.
problem Calculating Dolbeault cohomologies of complex manifolds after blowing up.
method Introducing relative Dolbeault cohomology, proving blow-up formula, and using bimeromorphic invariance.
result Uniform proof of bimeromorphic invariance of Dolbeault cohomology numbers.
Proves Dolbeault cohomology conjecture for complex nilmanifolds foliated by tori.
problem Computing Dolbeault cohomology of complex nilmanifolds.
method Generalizes previous methods by assuming foliation in toroidal groups.
result Conjecture holds in real dimension up to six.
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.
Kirwan surjectivity proven for equivariant Dolbeault cohomology.
problem Understanding cohomology of Kähler quotients for Hamiltonian actions.
method Using Cartan-Chern-Weil theory to define and prove a Kirwan map.
result A natural surjective Kirwan map established between equivariant Dolbeault cohomologies.
Constructs a Dolbeault-Hilbert complex for spaces with isolated singular points.
problem Cohomology of spaces with isolated singular points.
method Constructs a Dolbeault-type Hilbert complex.
result Cohomology of the complex is isomorphic to the structure sheaf cohomology.
Study cohomologies of complex manifolds with symplectic forms and their stability.
problem Analyzing cohomologies of complex manifolds with symplectic forms.
method Investigate the Hard Lefschetz Condition on Dolbeault cohomology groups using a double complex.
result Stability of the ∂∂Λ-Lemma under small deformations of ω but not under complex structure. Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.
problem Determining when Dolbeault DGAs on complex nilmanifolds are formal.
method Using quasi-isomorphisms and cohomology connections, proving formality conditions.
result Formality of Dolbeault DGAs on complex nilmanifolds is restricted to tori.
Formula for Dolbeault cohomology groups on complex manifolds with vector bundles.
problem Calculating Dolbeault cohomology groups on complex manifolds with vector bundles.
method Sheaf-theoretic approach to derive a blow-up formula.
result Uniform presentation of vanishing theorems and blow-up invariance of holomorphic invariants.
We consider semi-direct products $\C^{n}\ltimes_φN$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct …
We prove a relation between the ∂ˉM cohomology of a minimal orbit M of a real form G0 of a complex semisimple Lie group G in a flag manifold G/Q and the Dolbeault cohomology of the Matsuki dual open orbit X of the complexification K of a maximal compact subgroup K0 of G0, under the assum…
Study Hodge decomposition for special geometric manifolds.
problem Compute Dolbeault cohomology of specific geometric domains.
method Compute Dolbeault cohomology of geodesically convex domains in Cousin groups with strong dispersiveness condition.
result Hodge decomposition holds for Oeljeklaus-Toma manifolds.
Nilpotent Lie algebra cohomology vanishes for certain local systems.
problem Cohomology of nilpotent Lie algebras with specific local systems.
method Proved Dolbeault cohomology vanishes for nilpotent Lie algebras.
result Dolbeault cohomology of nilpotent Lie algebras with non-trivial local systems vanishes.
This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.
problem Calculating characteristic class relations in Dolbeault cohomology.
method Representation theory of structure group, without metric or connection.
result First results on Chern-Simons invariants of Cartan geometries.
Study of split Nakamura manifolds and their automorphisms.
problem Understanding the structure and automorphisms of Nakamura manifolds.
method Analyzing cohomology, spectral sequences, and automorphism groups.
result Detailed description of the automorphism group of split Nakamura manifolds.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Paper connects cohomologies on almost complex manifolds.
problem Relating J-invariant cohomology to Dolbeault cohomology. method Relating cohomologies through necessary and sufficient conditions.
result Conditions for isomorphism between cohomologies found.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Complex foliation cohomology is shown to be simple.
problem Analyzing cohomology of a specific complex foliation.
method Explicitly computed the Dolbeault cohomology in degree 1.
result Foliated Dolbeault cohomology in degree 1 is isomorphic to C.