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.
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show th…
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…
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex 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.
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.
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.
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.
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.
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
We use Hodge theory and a construction of Merkulov to construct A∞ structures on de Rham cohomology and Dolbeault cohomology.
Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Hodge theory applied to tropical curves.
problem Developing Hodge theory for tropical curves.
method Analytical approach using tropical differential forms and L2−cohomologies. result Construction of Hodge theory analog on tropical curves.
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
Study on Lp cohomology and Hodge decomposition for ALE manifolds.
problem Understanding Lp cohomology dimensions and harmonic forms in ALE manifolds. method Relating dimensions of Lp cohomology spaces to decaying harmonic forms, proving independence and jumps in dimensions, and providing Hodge decompositions. result Dimension of Lp reduced cohomology spaces in degree k is independent of p for k not equal to 1 or n-1, and jumps by a factor N-1 for k equal to 1 or n-1. A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.
Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the L2 de Rham and Hodge cohomology and the intersection cohomology of X associated to some perversities. More precisely, to a kind of metric whi…
Embeds cohomology of hyperkahler manifolds into torus cohomology.
problem Embedding cohomology of hyperkahler manifolds into torus cohomology.
method Kuga-Satake construction and embedding of graded cohomology spaces.
result Compatibility of embeddings with Hodge structures and Lie algebra actions.
We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class C of Fujiki, first proven by D.…
Proves cohomology theorems for tropical varieties.
problem Cohomology of smooth projective tropical varieties.
method Introduces and proves new results in tropical geometry.
result Establishes tropical analogs of three fundamental theorems.
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.
Let V be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the L2 harmonic theory and construct a pure Hodge structure on the L2-cohomology of V. If the dimension of V is two, we put a cohomological Hodge stru…
Discretizes Hodge-Dirac operators on a torus.
problem Capturing geometric aspects of continuum Hodge theory in discrete settings.
method Discrete exterior calculus framework, Hodge-Dirac and Laplace operators.
result Proves discrete Hodge decomposition theorem on combinatorial torus.
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.
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian
Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.
problem Investigating conditions for spectral sequence degeneration in holomorphic Poisson cohomology.
method Analyzing spectral sequences associated with bi-complexes on nilmanifolds.
result A Hodge-type decomposition of holomorphic Poisson cohomology is established for a specific class of structures.
Proves a tropical version of Clemens-Schmid sequence for tropical varieties.
problem Existence of tropical cycles with specific cohomology classes.
method Tropical Hodge theory and Clemens-Schmid sequence analogy.
result Proves tropical Hodge conjecture for rationally triangulable varieties.
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.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.
The paper shows how to represent cohomology classes on Kähler manifolds using differential forms.
problem Representing cohomology classes on compact Kähler manifolds as differential forms.
method Representing Chern characters as Čech cocycles and analyzing their behavior under Hodge structure.
result Every rational cohomology class of type (p,p) on a compact Kähler manifold can be represented by a differential (p,p)-form.
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.
Paper introduces signal processing on cell complexes.
problem Processing signals on non-Euclidean domains.
method Signal processing on abstract regular cell complexes.
result Hodge Laplacians for cell complexes enable convolutional filters.
This paper decomposes Hodge cohomology of manifold cylinders over graphs.
problem Decomposing Hodge cohomology of manifolds with embedded cylinders.
method Develops adiabatic decomposition using combinatorial and geometric structures.
result Generalizes Cappell-Lee-Miller splicing map to finite number of edges.
The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.
problem Generalizing Hodge theory to semisimple local systems.
method Establishing a canonical isomorphism and proving a global invariant cycle theorem.
result A new geometric proof of the Decomposition theorem for semisimple local systems.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
Study on complex variation of Hodge structures for non-Kähler manifolds.
problem Understanding complex variation of Hodge structures for non-Kähler manifolds.
method Analyzes holomorphic families of compact complex manifolds with specific cohomology properties.
result Period map is holomorphic and transversal under given conditions.
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
Notes describe geometric interpretations of cohomology in trisected 4-manifolds.
problem Understanding geometric interpretations of cohomology classes in trisected 4-manifolds.
method Analogy with Hodge theory and sheaf cohomology in algebraic geometry.
result Classes in H2(X) can be interpreted as (1,1)-classes. The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
The paper proves estimates for Hodge Laplacians on Lie groups.
problem Estimating Hodge Laplacians on semisimple Lie groups.
method Proves Schwartz estimates for Hodge Laplacian and Dirac operators.
result Generalizes results on symmetric spaces to Lie groups.
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.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.
problem Proving log-concavity of characteristic polynomials of matroids.
method Combinatorial approach, conditional proof of Kähler package.
result Conditional proof of Kähler package for tropical cohomology.
The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, th…
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 cohomologies on manifolds with locally conformally symplectic structures.
problem Understanding cohomologies on manifolds with locally conformally symplectic structures.
method Introduced lcs cohomologies, studied elliptic Hodge theory, dualities, and Hard Lefschetz Condition.
result Oeljeklaus-Toma manifolds with precisely one complex place and under an arithmetic condition satisfy the Mostow property.