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 .
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48 results for “Hodge correspondence”
Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.
problem None explicitly stated, focuses on introduction.
method Pedagogical introduction of concepts linking algebraic structures to geometry.
result Explains the non-abelian Hodge correspondence and its connections.
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.
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We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized -Hodge structures.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…
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…
Paper extends Hodge correspondence to singular Kähler spaces.
problem Establishing Hodge correspondence over Kähler spaces with singularities.
method Using equivalence of polystable Higgs bundles and semi-simple flat bundles over regular loci, and descent theorem for semistable Higgs bundles.
result Non-abelian Hodge correspondence established over compact Kähler klt spaces and their regular loci.
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.
Establishes correspondence between Calabi-Yau and Landau-Ginzburg structures.
problem Preserving real structures in the Calabi-Yau/Landau-Ginzburg correspondence.
method Detailed analysis of period integrals and modification of real structures.
result Full CY/LG correspondence for structures established.
The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This…
New representation theory for surface groups to SO0(2,3).
problem Understanding representations of surface groups into special orthogonal groups.
method Non-maximal Anosov representations via Higgs bundles and non-Abelian Hodge correspondence.
result Generalization of Filip's result on weight 3 variation of Hodge structures.
Study describes moduli spaces of flat bundles on Sasakian manifolds.
problem Understanding moduli spaces of flat bundles on Sasakian manifolds.
method Shows moduli space of simple flat bundles is a union of spaces with fixed basic structures.
result Detailed description of non-abelian Hodge correspondence on compact Sasakian manifolds.
Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundlesmath.DG
Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.
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 Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
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.
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open --orbits in flag varieties . We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, w…
An interesting theme in complex differential geometry is to find a correspondence between algebraic objects and differential geometric objects. One of the most attractive is the non-abelian Hodge theory of Simpson. In this paper, pursuing an analogue of the non-abelian Hodge theory in the context of -difference modu…
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms f…
A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is e…
Two projective structures on Riemann surfaces are described and shown not to be identical.
problem Characterizing and distinguishing projective structures on Riemann surfaces.
method Construction and analysis of projective structures and using the period map and moduli space.
result The Hodge projective structure is not the same as the uniformization projective structure .
Constructs a connection for Hodge theoretic projective structures on Riemann surfaces.
problem Describes connections between projective structures and Hodge theory on Riemann surfaces.
method Uses complex connections on the dual of the determinant of the Hodge line bundle, described in three ways.
result Constructs a connection on the dual of the Hodge line bundle for Hodge theoretic projective structures.
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.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
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 .
We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of , Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…
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.
T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irr…
We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-struc…
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.
In this paper we prove the following results: We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Ho…
We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stron…
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.
Let be a compact and irreducible Hermitian complex space of complex dimension . In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…
The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classific…
On Higgs bundlesmath.AG
Study on Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of Higgs bundles.
method Explicit description of moduli space, study of Narasimhan-Seshadri theorem, nonabelian Hodge correspondence, Hitchin equations.
result Derive an explicit description of the moduli space and study its properties.
New natural presentation of supergravity c-map using Hodge structures.
problem Presenting a new natural presentation of the supergravity c-map.
method Explicit description of correspondence between projective special Kähler manifolds and variations of Hodge structure, and twist construction.
result General isomorphisms can be naturally lifted along the deformed c-map.
The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.
problem Distinguishing orbifolds from manifolds using Hodge spectra.
method Computing heat invariants of Hodge Laplacians on -forms.
result The Hodge spectra, particularly the - and -spectra, can distinguish orbifolds from manifolds in low dimensions.
Let and be Riemannian metrics on a noncompact manifold , which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the Hodge-Laplacians and acting on differential forms exist and are c…
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.
This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.
Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition…
The paper classifies Lie algebroids and their connections, modulating principal objects.
problem Classifying and studying Lie algebroids and their connections.
method Classifying integrable transitive Lie algebroids, introducing Higgs bundles, and using Tannakian categories.
result Moduli spaces of principal \(G\)-bundles and Higgs bundles are semiprojective varieties.
Holomorphic curves found in compact quotients of SL(2,C).
problem Proving the existence of holomorphic curves in compact quotients of SL(2,C).
method Non-Abelian Hodge correspondence, WKB analysis, and Morgan-Shalen compactification.
result Every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two.
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.
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.
We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…
The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.
problem Relating Kapustin-Witten equations to nonabelian Hodge theory.
method Utilizing nonabelian Hodge theory correspondence to describe moduli spaces.
result Computing expected dimension of moduli spaces for Kapustin-Witten equations.