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 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.
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
New cohomology ring for complex manifolds with group action.
problem Constructing cohomology for complex manifolds with group action.
method Introduced a new cohomology ring HG,cs∗(X) for equivariant almost complex pairs. result New cohomology ring provides insights into stringy orbifold theory.
We prove properties of the Schweitzer complex and its cohomologies.
problem Properties of the Schweitzer complex and its cohomologies.
method Elliptic complex and cohomological functors.
result Finite dimensionality and equivalence to Hirzebruch-Riemann-Roch relations.
Study cohomologies of complex bundles and sections.
problem Understanding cohomologies of Landau-Ginzburg models.
method Investigate cohomologies of Koszul complex induced by a section.
result Cohomologies are isomorphic and self-dual when the manifold is complete.
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.
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 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.
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 complexes derived from any filtered cochain complex compute the same cohomology.
problem Constructing cohomologically equivalent subcomplexes from filtered cochain complexes.
method Presenting a general construction that produces subcomplexes from any filtered cochain complex of finite depth.
result The construction of subcomplexes depends only on the filtration up to isomorphism.
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.
The paper studies cohomology of sheaf complexes and proves a relative de Rham theorem.
problem Cohomology of sheaf complexes and their representations.
method Intersection of Cech theory and derived categories, proving a relative de Rham theorem.
result Cohomology of sheaf complexes is canonically isomorphic to the relative cohomology of the sheaf.
Study anti-invariant cohomology on almost complex manifolds, showing infinite and finite dimensions.
problem Understanding the cohomology of anti-invariant forms on almost complex manifolds.
method Construction of specific almost complex structures and analysis of cohomology groups.
result Found manifolds with anti-invariant cohomology of infinite, 0-1, and 2 dimensions.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
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.
We find a splitting in a special cohomology theory for complex manifolds.
problem Finding a splitting in a specific cohomology theory.
method Construct Hodge filtered function spaces and show they satisfy an unstable splitting.
result Obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson 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.
Complex group cohomology surprisingly simple.
problem Computing the cohomology of a complex group's centralizer.
method Explicit computation of rational cohomology.
result Rational cohomology of universal centralizer coincides with that of a point.
We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.
Study cohomology of Bigolin complex on complex manifolds.
problem Characterize cohomology of Bigolin complex on compact complex manifolds.
method Analyze the decomposition of the double complex into squares and zigzags, focusing on the zigzags contributing to cohomology.
result In complex dimension 3, multiplicities of zigzags are characterized by Betti, Hodge, Aeppli numbers plus Bigolin numbers.
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. In this note we study a new cohomology attached to a function along the leaves of complex foliations. We also explain how this cohomology depends on the function and we study a relative cohomology and a Mayer-Vietoris sequence related to this cohomology.
The cohomology of complex irreducible polynomials stabilizes with degree or variables.
problem Understanding the cohomology of complex irreducible polynomials.
method Proving homological stability in cohomology as degree or variables increase.
result Cohomology stabilizes with respect to both degree and number of variables.
Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are c…
Formula derived for cohomology of local systems on complex manifolds.
problem Cohomology of local systems on compact complex manifolds.
method Derive a blow-up formula for de Rham cohomology of local systems.
result Blow-up invariance of E1-degeneracy of Hodge-de Rham spectral sequence. While small deformations of Kähler manifolds are Kähler too, we prove that the cohomological property to be C∞-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symp…
We show that the complex cohomologies of Bott, Chern, and Aeppli and the symplectic cohomologies of Tseng and Yau arise in the context of type II string theory. Specifically, they can be used to count a subset of scalar moduli fields in Minkowski compactification with RR fluxes in the presence of either O5/D5 or O6/D6 …
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.
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.
Abstract reviews cohomologies on complex and hypercomplex manifolds.
problem Comparing cohomologies on complex and hypercomplex manifolds.
method Review and comparison of cohomological aspects.
result Similarities found between compact complex surfaces and compact hypercomplex manifolds.
New proof of blow-up formula for Morse-Novikov cohomology.
problem Blow-up formula for Morse-Novikov cohomology.
method Introducing relative Morse-Novikov cohomology and using sheaf cohomology.
result Explicit isomorphism in relative Morse-Novikov cohomology.
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.
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.
Researchers compute the cohomology of an elliptic tangent bundle.
problem Computing the cohomology of a specific Lie algebroid.
method Direct computation of cohomology.
result The cohomology of the elliptic tangent bundle is computed.
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.
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. Classifies foliated complex manifolds using marked fans.
problem Understanding foliated complex manifolds similar to toric varieties.
method Classifies by marked fans and describes cohomology algebras.
result Basic cohomology and Dolbeault cohomology algebras described in terms of marked fans.
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
The paper studies twisted Morse homology and cohomology on manifolds.
problem Computing homology and cohomology with local coefficients on manifolds.
method Morse theory, CW-complexes, de Rham cohomology, Lichnerowicz cohomology.
result Isomorphisms between different cohomology theories.
Study on symplectic structures and their deformations.
problem Preservation of complex symplectic structures under deformations.
method Analyzes various cohomologies and conditions for deformations.
result Obtains topological obstructions for compact complex symplectic manifolds.
This paper generalizes L2 cohomology theory for complex manifolds.
problem Developing a L2 cohomology theory for Hodge modules on infinite covering spaces.
method Formulating a conjectural generalization of L2-Mixed Hodge structures using Saito's Mixed Hodge Modules.
result Partial results in the conjectural generalization of L2-Mixed Hodge structures.
A Morse complex for Axiom A flows on smooth manifolds.
problem Constructing a finite-dimensional cohomological complex for Axiom A flows.
method Defining anisotropic Sobolev spaces and spectral projectors.
result The cohomology of the constructed complex is isomorphic to De Rham cohomology.
We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precise…
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.
The paper explores cohomologies on almost complex manifolds and their applications.
problem Distinguishing and understanding cohomologies on non-integrable almost complex structures.
method Defined and studied cohomologies HN∙(M) and HJ∙(M) using Nijenhuis-Lie derivations. result The J-cohomology encodes whether an almost complex structure satisfies the dLJ-lemma.