Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
Refines mixed Hodge structures using non-abelian Hodge theory.
problem Mixed Hodge structures on Sullivan's minimal models.
method Non-abelian Hodge theory.
result Explicit representatives of real unipotent variations.
Develops Hodge theory for foliations using perturbed Laplacians.
problem Creating a Hodge theory for foliations.
method Mimicking Witten's approach to Morse theory with perturbations of the Laplacian.
result Establishes a Hodge theory for tangential cohomology of foliations.
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.
Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.
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.
The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
The paper solves complex structure changes and Beltrami equations using Hodge theory.
problem Changes of complex structures on Kähler manifolds and solving Beltrami equations.
method Global geometric methods using Hodge theory and L2-Hodge theory. result Closed explicit extension formulas for holomorphic canonical forms and pluricanonical forms.
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. 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.
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.
Anabelian geometry reformulated using Hodge theory for hyperbolic curves.
problem Determining varieties over number fields using their étale fundamental groups.
method Formulating a Hodge-theoretic version of anabelian conjecture, replacing Galois action with Cimes-action. result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over C. The paper connects Hodge theory and modular forms to prove inequalities.
problem Proving a conjectural inequality on weights of modular forms.
method Using nonabelian Hodge theory and vector valued modular forms.
result New instances of the three-term inequality for nonunitary representations.
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 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.…
Paper shows Goldman bracket's Hodge structure properties.
problem Hodge structure of Goldman bracket on loops.
method Analyzes loops on algebraic curves, proving Hodge structure properties.
result Constructs torsors of isomorphisms with graded Lie algebra.
The study explores the implications of curvature positivity in vector bundles and Hodge theory.
problem Understanding the existence of sections in semi-positive but not strictly positive vector bundles and the nature of Hodge metrics.
method Analyzes various positivity measures and their implications in algebraic geometry and Hodge theory.
result Provides insights into the existence of sections in semi-positive vector bundles and the nature of Hodge metrics.
We use Hodge theory and a construction of Merkulov to construct A∞ structures on de Rham cohomology and Dolbeault cohomology.
Study Hodge theory on non-compact Riemannian manifolds with Lr estimates.
problem Solving Hodge Laplace equation on p forms in non-compact Riemannian manifolds. method Generalization of Raising Steps Method for non-compact manifolds, spectral gap hypothesis.
result Non-classical Lr Hodge decomposition theorems without bounded Riesz transforms. 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.
New connection found between Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
problem Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.
method Established a compact analog of the P = W conjecture for holomorphic symplectic varieties with Lagrangian fibrations.
result Perverse numbers match Hodge numbers of the total space.
Study Hodge theory for Landau-Ginzburg models on Calabi-Yau manifolds.
problem Hodge theory for non-compact Calabi-Yau manifolds with holomorphic functions.
method Introduce f-twisted Sobolev spaces and prove Hodge-to-de Rham degeneration via L2-Hodge theory.
result Construct Frobenius manifolds for Landau-Ginzburg models and orbifolds.
New class of singular complex manifolds studied with degenerate theory.
problem Understanding singular complex manifolds.
method Developed degenerate Kodaira-Hodge theory for new class.
result New degenerate theory for singular 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 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 paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
The paper reformulates Hodge index theorem on Kähler manifolds using L2-index theory.
problem Proving the Hodge index theorem on compact Kähler manifolds.
method Using Atiyah's L2-index theory and Galois coverings. result Established the relationship between signature and L2-Hodge numbers. Here we survey questions and results on the Hodge theory of hyperkaehler quotients, motivated by certain S-duality considerations in string theory. The problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic mo…
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.
New method solves Beltrami equation using Hodge star.
problem Solving the Beltrami equation.
method Using Hodge star operator and elliptic PDE theory.
result Essentially unique homeomorphic solution found.
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
Paper connects Turaev cobracket to Kashiwara--Vergne problem via Hodge theory.
problem Solving the Kashiwara--Vergne problem in algebraic geometry.
method Combining Turaev cobracket and Goldman bracket results, constructing torsors of solutions.
result Found a torsor of solutions to the Kashiwara--Vergne problem that depends only on topology.
Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.
problem Representation theory of Hopf link complements with n twists.
method Combinatorial problem and equivariant Hodge theory.
result Close formulas for E-polynomials of representation and character varieties for ranks 2 and 3.
Hodge theory proven for elliptic complexes over compact operator C∗-algebras.
problem Proving Hodge theory for elliptic complexes over compact operator C∗-algebras. method Analyzing pseudodifferential operators and finitely generated projective Hilbert bundles.
result Topological isomorphism between cohomology groups and harmonic elements.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
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.
Generalizes nonabelian Hodge theory to klt singularities.
problem Applying nonabelian Hodge theory to spaces with klt singularities.
method Uses descent theorems for numerically flat vector bundles and a new restriction theorem for semistable Higgs sheaves.
result Establishes a new restriction theorem for semistable Higgs sheaves.
Survey of Hodge theory on stratified spaces and related compactifications.
problem Analyzing Hodge theory on complex stratified spaces.
method Resolution of Thom-Mather stratified spaces to manifolds with corners, introduction of mezzoperversity, definition of Cheeger spaces.
result Novikov conjecture verified for Cheeger spaces.
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.
The paper studies heat equations and Hodge theory on incomplete cusp edge spaces.
problem Analyzing the spectral and Hodge theory of incomplete cusp edge spaces.
method Constructing a fundamental solution to the heat equation and proving essential self-adjointness of the Hodge-Laplacian.
result The space of L2-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop …
Defines loop Hodge structures and connects them to harmonic bundles.
problem Understanding harmonic bundles using Hodge theory.
method Introduces loop Hodge structures and proves their equivalence to harmonic bundles.
result Establishes equivalence between loop Hodge structures and harmonic bundles, enabling classical Hodge theory tools.
Abstract: Almost Kähler Hodge numbers vary with metric choices.
problem Variation of almost Kähler Hodge numbers with metrics.
method Analysis of almost complex Hodge numbers under different almost Kähler metrics.
result The almost Kähler Hodge number h0,1 varies with metric choices. The abstract explores analogues of Hodge theory in Lie algebroids.
problem Exploring analogues of Hodge theory in Lie algebroids.
method Establishing equivalence of conditions and applying algebraic theory to geometric setting.
result Equivalence of analogues of Hodge theory conditions in Lie algebroids.
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.