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.
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.
New proof classifies orbit closures in Hodge bundle.
problem Classifying mGL+(2,R)-orbit closures in Hodge bundle. method Using deformations of flat pairs of pants.
result Short proof of absolute period foliation classification.
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.
We define the notion of a loop Hodge structure -- an infinite dimensional generalization of a Hodge structure -- and prove that a suitable variation of this object over a complex manifold is equivalent to the datum of a harmonic bundle. Hence one can study harmonic bundles using classical tools of Hodge theory, especia…
Study locates divisors in Hodge bundle with specific properties.
problem Locating effective divisors in the projectivized Hodge bundle.
method Computing the class of closures of loci of canonical divisors with specific conditions.
result Strata of canonical and bicanonical divisors with double zeros span extremal rays of pseudoeffective cones.
Study singularities of metrics on Hodge bundles and their topological invariants.
problem Understanding the singularities of metrics on Hodge bundles.
method Analyzing degenerations of Calabi-Yau varieties and studying metrics on Hodge and determinant bundles.
result Dominant and subdominant terms in the expansions of metrics are related to topological invariants of singularities.
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 Higgs bundle structure on complexified Kähler cone found.
problem Constructing a Higgs bundle on complexified Kähler cone.
method Using commutator identity and generalized Hodge metric.
result Complexified Kähler cone has negative holomorphic sectional curvature.
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.
Survey on Higgs bundles via harmonic maps and their role in non-abelian Hodge correspondence.
problem Understanding the non-abelian Hodge correspondence and equivariant harmonic maps.
method Explains the non-abelian Hodge correspondence and reviews progress on equivariant harmonic maps.
result Review of current progress on open problems in equivariant harmonic maps.
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.
Proves GAGA-style result for toric vector bundles.
problem None explicitly stated in the abstract.
method Algebraic construction of Frölicher approximating vector bundle.
result Proves GAGA-style result for toric vector bundles.
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.
Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.
problem Understand nilpotent Higgs bundles and their metrics.
method Algebraic inequality for nilpotent matrices, geometric applications.
result Sharp upper bound of holomorphic sectional curvatures on Calabi-Yau moduli.
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space…
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.
This work connects Higgs bundles to Calabi-Yau manifolds via mirror symmetry.
problem Analyzing Higgs bundles associated with Calabi-Yau manifolds.
method Applying mirror symmetry and techniques from Hodge theory.
result Flat connections are gauge equivalent to opers, linking special functions.
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.
We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is Kähler. Several applications in cases of geometric interest are given, such as for normal functions and…
The paper extends Hodge theory results to Hermitian vector bundles on Kähler manifolds.
problem Studying L2-Hodge theory on Hermitian vector bundles over Kähler manifolds. method Extending Gromov's results to Hermitian vector bundles, proving a gap result for Yang-Mills connections.
result Proves a gap result for Yang-Mills connections on the bundle E over X. We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interp…
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.
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.
We study rank 1 flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which c…
A gauge-invariant form of the nonlinear Hodge equations is studied.
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.
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.
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 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.
Determinant of twisted Hodge filtration is positive if the line bundle is positive.
problem Analyzing positivity of determinant of twisted Hodge filtration.
method Explicit formula for curvature tensor of higher direct images.
result Determinant of twisted Hodge filtration is positive if the line bundle is positive.
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…
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.
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.
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.
Polystable bundles on Kaehler manifolds are stable if their classes are stable.
problem Stability of holomorphic vector bundles on Kaehler manifolds.
method Quasi-linear Hodge theory and geometric invariant theory.
result Stability of vector bundles is equivalent to stability of their classes in geometric invariant theory.
Computes algebraic hull of Kontsevich-Zorich cocycle.
problem Finiteness of GL^+_2(R) invariant subvarieties of the Hodge bundle.
method Computes algebraic hull over GL^+_2(R) invariant subvarieties.
result Finiteness results on invariant subvarieties.
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.
We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.
The paper provides a new algebraic method to compute L2 cohomology of harmonic bundles.
problem Computing L2 cohomology of harmonic bundles and twistor D-modules. method Non-abelian Hodge theory and algebraic computation of L2 cohomology. result A new algebraic formula for the Dolbeault version of L2 pushforward. Develops SGH bundles and theories for GC manifolds.
problem No specific problem stated; focuses on new bundle theory.
method Introduces SGH bundles, develops cohomology, and establishes theories.
result Established a Chern-Weil theory and Hodge theory for SGH bundles.
Introduces a new geometric product for differential forms.
problem Constructing a Clifford algebra from differential forms.
method Introduces and revisits the Graf product with a new framework.
result Provides a structure for constructing a Clifford algebra.
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…
Develops Weil bundles over \( p \)-adic manifolds for arithmetic geometry.
problem Connecting differential calculus and arithmetic geometry over \( p \)-adic fields.
method Systematic theory of Weil bundles, developing analytic structures.
result Establishes canonical analytic structures on Weil bundles and their cohomological comparison.
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…
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.