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 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 construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology fo…
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.
We use Hodge theory and a construction of Merkulov to construct A∞ structures on de Rham cohomology and Dolbeault cohomology.
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.
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.
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. 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…
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…
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
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.
We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomol…
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 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.
We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bo…
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…
We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of inv…
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
Develops Hodge theory on ALG∗ manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG∗ manifolds. method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG∗ manifolds. result Non-existence of ALG∗ manifolds with non-negative Ricci curvature at infinity. Explains quantum cohomology of Grassmannians using tt* equations.
problem Relates quantum cohomology of complex Grassmannians to projective space.
method Uses tt* equations and Lie-theoretic connections.
result Illustrates relations between tt* equations and quantum cohomology.
This paper constructs a Hodge theory of noncompact topologically tame manifolds M. The main result is an isomorphism between the de Rham cohomology with compact supports of M and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure dμ which has sufficiently rapid growth at infinity o…
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…
Study Betti and Hodge numbers of solvmanifolds from integer polynomials.
problem Computing Betti and Hodge numbers of solvmanifolds constructed from integer polynomials.
method Analyzing de Rham and Dolbeault cohomology of solvmanifolds under algebraic conditions.
result Explicit generating polynomials for Hodge numbers in quasi full rank case.
We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomol…
Study intersection cohomology and Lagrangian fibrations in symplectic varieties.
problem Understanding the intersection cohomology and perverse filtration of Lagrangian fibrations in symplectic varieties.
method Analyzes the deformation equivalence class, computes the border of the perverse diamond, and identifies perverse and Hodge numbers.
result Complete description of intersection cohomology and invariant cohomology classes of fibers.
Derives integral formula for Hodge and Teichmüller norms.
problem Relationship between Hodge and Teichmüller norms.
method Integrals and cohomology classes.
result Comparison between Hodge and Teichmüller norms.