Hodge and Teichmüllermath.GT
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.
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48 results for “Hodge norm”
Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.
problem Finding a sharp lower bound for the spectrum of the Hodge Laplacian.
method Explicitly expressed in terms of the supremum norm of the 1-form.
result Explicit spectral lower bounds for bounded symmetric domains.
The more important difference between Riemann and pseudo-Riemann manifolds is the metric signature and its theoretical consequences. The practical application for Physics Theories becomes often impossible due to the signature consequences. Eg., some of the rich results in Riemann Geometry and Topology become invalid fo…
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend t…
In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold isometrically immersed into another Riemannian manifold for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of bounded from below, and obtain an extrinsic…
Study on -structures using Laplacian coflow and solitons.
problem Characterizing and understanding -structures and their solitons.
method Using the irreducible -decomposition of the Hodge Laplacian and Lie derivative, characterizing infinitesimal symmetries and soliton conditions.
result Proof of the absence of compact shrinking solitons for the Laplacian coflow.
Optimizes metrics for the first curl eigenvalue on 3-manifolds.
problem Finding optimal metrics for minimizing the first curl eigenvalue.
method Analyzes metrics that minimize the first curl eigenvalue among metrics of the same volume in the same conformal class.
result Proves that and are local minimizers for the first curl eigenvalue.
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.
We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .
Almost complex Hodge theorymath.DG
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.
We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…
We refine the Morgan's work on mixed Hodge structures on Sullivan's --minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent variations of mixed Hodge structures over compact K"ahler manifolds.
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 -bundles.
result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
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.
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 -inner product, codifferential, and Laplacian.
result Established de Rham-Hodge theory for closed and boundary manifolds.
Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.
problem Understanding non-abelian Hodge loci for quasi-projective varieties.
method Analyzes -local systems and polarized variations of Hodge structures.
result Proves algebraicity of non-abelian Hodge loci for -anisotropic monodromy.
Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology classmath.AG
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 give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan's -minimal models of certain differential graded algebras associated with real variations of Hodge structures.
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.
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.
For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an n-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of n-dimensional smooth complex projective va…
The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.
problem Formulating Lichnerowicz type formulas and Kastler-Kalau-Walze theorems for statistical de Rham Hodge operators.
method Developed Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
result Proved Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
Hodge theory applied to tropical curves.
problem Developing Hodge theory for tropical curves.
method Analytical approach using tropical differential forms and cohomologies.
result Construction of Hodge theory analog on tropical curves.
We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized -Hodge structures. Basing on this consideration, we introduce the class of abelian Cousin groups and prove an analogue of Poincar…
The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.
problem Analyzing perturbations of de Rham Hodge operators on manifolds with boundaries.
method Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
result Proves Kastler-Kalau-Walze type theorems for perturbations of de Rham Hodge operators on 4D and 6D manifolds with or without boundaries.
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 shows equality in Hodge Laplacian bound occurs only on spheres.
problem Understanding when equality holds in Hodge Laplacian bounds for submanifolds.
method Analyzes closed submanifolds in space forms, proving equality on spheres.
result Equality in Hodge Laplacian bound occurs only on topological spheres.
For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; if k=2m is even, then we have to impose that h^{m,m} is bigger than some quadratic bound in m. Combining these results for different weights, we so…
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).
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.
The paper broadens a mathematical correspondence to include more balanced metrics.
problem Extending a mathematical correspondence to a broader class of metrics.
method Using key observations and known theorems to apply results to a new class of metrics.
result The known results can be applied to a larger class of metrics, including those arising from multipolarizations.
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.
On the Interplay Between Hodge Projections and Bounded Harmonic Functions on Manifolds with Endsmath.AP
Study examines -boundedness of Hodge projection on manifolds with ends.
problem Understanding -boundedness of Hodge projection on manifolds with ends.
method Investigates the relationship between Hodge projection, Riesz transform, and bounded harmonic functions.
result Connects -boundedness of Hodge projection to the structure of harmonic one-forms and bounded harmonic functions.
New proof classifies orbit closures in Hodge bundle.
problem Classifying -orbit closures in Hodge bundle.
method Using deformations of flat pairs of pants.
result Short proof of absolute period foliation classification.
Lower bounds for Hodge-Laplacian spectrum on orbifolds.
problem Finding bounds for the essential spectrum of Hodge-Laplacian.
method Deriving lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.
result Lower bounds for the essential spectrum of the Hodge-Laplacian.
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 inequalities generalize Li's theorem on mixed Hodge structures.
problem Generalizing Li's theorem on mixed Hodge structures.
method Develop new Hodge-Riemann bilinear relations in mixed settings.
result New Khovanskii-Teissier type inequalities and log-concavity results.
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.
This note presents an analytic construction of the optimal unit-norm direction hat(x) = x/|x| that maximizes or minimizes the objective linear expression, B . hat(x), subject to a system of linear constraints of the form [A] . x = 0, where x is an unknown n-dimensional real vector to be determined up to an overall norm…
In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric f…
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 -action.
result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over .
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
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.
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.
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.
Filtration and splitting of the Hodge bundle on the non-varying strata of quadratic differentialsmath.AG
Study describes splitting and filtration of Hodge bundle on quadratic differentials.
problem Understanding the structure of Hodge bundles on quadratic differentials.
method Harder-Narasimhan filtration and splitting as direct sum of line bundles.
result Determine all Lyapunov exponents of algebraically primitive Teichmüller curves.
The Hodge spectra help distinguish orbifolds from manifolds with singularities.
problem Distinguishing orbifolds from manifolds based on their singular sets.
method Computing heat invariants of Hodge Laplacians on -forms.
result The Hodge spectra of - and -forms distinguish orbifolds from manifolds with singularities.