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.
Compute local cohomology of vector fields on manifolds.
problem Understanding cohomology of vector fields on manifolds and complex manifolds.
method Compute local cohomology, use descent for cocycles.
result Explicit representatives for cocycles constructed.
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.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
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.
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 proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.
problem Determining if 4D symplectic manifolds are diffeomorphic based on their equivariant cohomology.
method Proved that equivariant cohomology rings of Hamiltonian circle actions on 4D symplectic manifolds determine their equivariant diffeotypes.
result Isomorphism of equivariant cohomology rings implies equivariant diffeomorphism for 4D symplectic manifolds.
A manifold's co-invariant cohomology can be infinite-dimensional.
problem Infinite-dimensional co-invariant cohomology of manifolds.
method Defined Γ-co-invariant cohomology and cohomology of divergence forms. result Presented a manifold with infinite-dimensional co-invariant cohomology.
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.
New cohomological obstruction found for astheno-Kahler metrics.
problem Existence of astheno-Kahler metrics
method New cohomological obstruction
result Found a new cohomological obstruction
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
This paper extends Dolbeault cohomology and its surrounding theory to arbitrary almost complex manifolds. We define a spectral sequence converging to ordinary cohomology, whose first page is the Dolbeault cohomology, and develop a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic analogues of the t…
Proves Kato manifolds satisfy Hodge decomposition.
problem Proving Hodge decomposition for Kato manifolds.
method Relating cohomology to modification data and studying Bott-Chern and Aeppli cohomology.
result Kato manifolds satisfy Hodge decomposition.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. Cohomology fractals illustrate complex 3-manifold properties.
problem Visualizing complex cohomology classes in hyperbolic 3-manifolds.
method Ray-tracing cohomology fractals and proving their distribution.
result Cohomology fractals converge to a distribution on the sphere at infinity.
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.
Classifies Real line bundles with Real connections on manifolds with involution.
problem Classifying Real line bundles with Real connections on manifolds with involution.
method Defines Real smooth Deligne cohomology to interpolate between equivariant sheaf cohomology and smooth imaginary-valued forms.
result Classifies Real line bundles with Real connections on manifolds with involution.
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. De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. New argument for 3-manifold cohomology with F2 coefficients.
problem Characterization of 3-manifold cohomology rings with F2 coefficients. method New argument based on Postnikov's 1948 characterization using intersection rings.
result A new proof for the characterization of 3-manifold cohomology rings.
Study Hochschild cohomology of dg manifolds linked to integrable distributions.
problem Understanding Hochschild cohomology of dg manifolds associated with integrable distributions.
method Analyzing the Hochschild cohomology of (F[1],dF) and relating it to the algebra of functions on leaf space. result Established a canonical isomorphism between the Hochschild cohomology of (F[1],dF) and the algebra of functions on leaf space. Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.
problem Understanding cohomology groups on foliated manifolds.
method Defined and studied Morse-Novikov cohomology relative to a foliation, proving homotopy invariance and extending to more general forms.
result Proved Hodge theorem and Poincaré duality for reduced leafwise Morse-Novikov cohomology groups on Riemannian foliations.
We relate Lq,p-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of ℓq,p-cohomology, packing cohomology. This implies quasi-isometry invariance of Lq,p-cohomology together with its multiplicative structure. The result partially extends to the Rumin Lq,p-cohomolog…
Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…
We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology. Our result can be considered as a generalization of the identification of the Dolb…
Study primitive cohomology in symplectic manifolds.
problem Understanding primitive cohomology in symplectic geometry.
method Reviewing superbundle-valued forms and proving a transgression formula.
result Introduced primitive characteristic classes and proved a transgression formula.
Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger coh…
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…
In this paper we define coeffective de Rham cohomology for basic forms on a K--contact or Sasakian manifold M and we discuss its relation with usually basic cohomology of M. When M is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Bett…
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.
The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
The study classifies manifolds with specific rational cohomology properties.
problem Classifying smooth manifolds with rational cohomology of a sphere.
method Cohomogeneity one actions on smooth manifolds.
result Such manifolds are diffeomorphic to specific known manifolds.
Paper shows hyperbolic 3-manifolds can sound the same but have different cohomology.
problem Cannot determine rational cohomology ring from sound of hyperbolic 3-manifolds.
method Implemented computer program to find nullity of cup product map.
result Example of strongly isospectral hyperbolic 3-manifolds with nonisomorphic rational cohomology rings.
In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.
Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.
New spectral sequence for K-manifolds, computing cohomology and harmonic forms.
problem Computing cohomology and harmonic forms of K-manifolds. method Introducing a new spectral sequence and using it to generalize theorems from K-contact geometry. result Computed cohomology ring and harmonic forms of S-manifolds. Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
Algorithm to compute cohomology groups of real flag manifolds, proving torsion and Schubert varieties.
problem Computing cohomology groups of real flag manifolds.
method Algorithm based on Schubert cells and incidence coefficients.
result Results on torsion classes and Schubert varieties for real flag manifolds.
Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
problem Quantum cohomology of minuscule flag manifolds.
method Combining Lie-theoretic treatments of tt*-Toda equations and quantum cohomology.
result Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
Introduces θ-almost twisted Poisson structures and their cohomology.
problem Characterizing and understanding θ-almost twisted Poisson structures. method Definition and construction of θ-almost twisted Poisson structures, Lie-Rinehart algebra, cochain complex, and cohomology. result Definition and construction of θ-almost twisted Poisson cohomology. New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
Cohomology fractals are visual representations of cohomology classes on hyperbolic 3-manifolds.
problem Visualizing cohomology classes on hyperbolic 3-manifolds.
method Cohomology fractals are images associated to cohomology classes. They are related to limit sets of Kleinian groups but differ in key aspects. An implementation using ideal triangulations and ray-casting is presented.
result Cohomology fractals allow for real-time zooming in any direction at arbitrary depth.
Study differential operators on specific manifolds and their harmonic forms.
problem Understanding harmonic forms on almost-Hermitian manifolds.
method Analysis of differential operators, Hodge Theory, and cohomologies.
result Comparison of harmonic forms and cohomologies with classical ones.
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
Study cohomology of homeomorphisms and diffeomorphisms of manifolds.
problem Determine bounded and unbounded cohomology of homeomorphism and diffeomorphism groups.
method Analyzing specific manifolds like the circle, 2-disc, and spheres.
result Identify the bounded cohomology of homeomorphisms and diffeomorphisms groups of certain manifolds.