Study on GL(1∣1) Higgs bundles over Riemann surfaces.
problem Investigate the moduli space of GL(1∣1) 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.
Authors compute monodromy groups for SL(n) and GL(n) Hitchin fibrations.
problem Computing monodromy groups for Higgs bundles on Riemann surfaces.
method Using spectral curves and Picard-Lefschetz transformations, they construct and classify vanishing lattices.
result They determine the structure of monodromy groups for SL(n) and GL(n) Hitchin fibrations.
Classifies very stable Higgs bundles for complex groups.
problem Classifying Higgs bundles for arbitrary complex groups.
method Classification based on stability and Higgs field properties.
result Extends previous classification for GLn to arbitrary groups.
Study of Hitchin map on specific Higgs bundles.
problem Understanding the Hitchin map on even very stable Higgs bundles.
method Defined even very stable Higgs bundles and studied the Hitchin map on their upward flows.
result Classification of type (1,...,1) examples and their relation to root systems.
Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.
problem Determine the topology and polynomials of moduli spaces of Higgs bundles over Abelian varieties.
method Analyzing the Poincaré polynomials and mixed Hodge polynomials of moduli spaces MAH(G) for various groups G and dimensions d. result Explicit formulas for Poincaré polynomials and mixed Hodge polynomials in specific cases, including rank 2 and 3 Higgs bundles.
This paper generalizes a topological invariant to cyclic Higgs bundles.
problem Defining and studying a topological invariant for cyclic Higgs bundles.
method Using a complex semisimple Lie group and its Lie algebra, the authors construct special cyclic Higgs bundles and define a topological invariant.
result The authors generalize the definition and properties of the Toledo invariant to arbitrary (G0,g1⊕g1−m)-Higgs pairs. The paper constructs a symplectic structure on moduli spaces of framed G-Higgs bundles.
problem Understanding symplectic structures on moduli spaces of framed G-Higgs bundles.
method Construction of a holomorphic symplectic structure on the moduli space of stable framed G-Higgs bundles.
result A natural morphism from the moduli space of framed G-Higgs bundles to the moduli space of twisted G-Higgs bundles is Poisson.
We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G=GL(2,C), SL(2,C) and PSL(2,C). We also determine the twisted Chern class of the regula…
Computes intersection cohomology of moduli space of Higgs bundles on a genus 2 curve.
problem Computing the intersection cohomology of the moduli space of Higgs bundles.
method Constructs a semismall desingularization and uses the decomposition theorem to compute the cohomology.
result Proves the mixed Hodge structure on the intersection cohomology is pure.
In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality…
Study stability of GL and YMH functionals on spheres and CP spaces.
problem Stability and critical points of Ginzburg-Landau and Yang-Mills-Higgs functionals.
method Analysis of critical points using Lawson-Simons methods.
result Lower bounds on Morse index and no stable critical points for YMH on Sn for n≥4. We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.
Energy functional on Teichmüller space is plurisubharmonic but not strictly so.
problem Characterizing points where energy functional fails to be strictly plurisubharmonic.
method Analyzing the kernel of the Levi form and relating it to Higgs bundles and Hitchin fibration.
result For generic choices, energy functional is strictly plurisubharmonic.
The paper establishes a correspondence between Higgs torsors and connections on curves.
problem Establishing a correspondence between Higgs torsors and connections on curves.
method Introduced a stability condition on filtered Stokes local systems and used it to prove a one-to-one correspondence.
result One-to-one correspondence between stable meromorphic parahoric Higgs torsors and stable meromorphic parahoric connections.
Classifies Higgs and co-Higgs bundles on symmetric spaces.
problem Classifying Higgs and co-Higgs bundles over Hermitian symmetric spaces.
method Defined homogeneous principal Higgs and co-Higgs bundles, provided a classification up to isomorphism.
result Defined and classified moduli spaces for each type of bundle.
Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.
problem Analyzing harmonic metrics and deformations on Higgs and flat bundles on compact Kähler manifolds.
method Relative analytic theory, Sobolev completions, elliptic regularity, normalized gluing, plotwise smoothness, heat flow, harmonic filtrations, obstruction theory.
result Global smooth harmonic metrics exist for smooth stable Higgs families under certain conditions, and this theory extends to reduced singular parameter spaces.
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
Higgs bundles used in new applications.
problem None explicitly stated in the abstract.
method Overview of recent applications.
result Applications of Higgs bundles.
Researchers extend Higgs bundle theory to parabolic bundles, counting components.
problem Counting components in moduli spaces of Higgs bundles with parabolic structures.
method Generalized Beauville-Narasimhan-Ramanan correspondence, Bott-Morse theory.
result Exact component count for maximal parabolic Sp(2n,ℝ)-Higgs bundles.
Identifies images of determinant morphism for specific co-Higgs bundles.
problem Determining images of determinant morphism for co-Higgs bundles.
method Identifying images of the determinant morphism of trace-free co-Higgs bundles modeled on rank 2 Schwarzenberger bundles.
result Identified images of the determinant morphism for specific co-Higgs bundles.
The paper examines critical points in Higgs bundle moduli spaces.
problem Understanding critical points in Higgs bundle moduli spaces.
method Analyzes the Higgs field vanishing on divisors and related integrable systems.
result Topological and differential-geometric properties of critical loci are addressed.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
The paper extends orthogonal decomposition results to hermitian Higgs bundles.
problem Classical propositions on holomorphic vector bundles do not always extend to Higgs bundles.
method The approach involves extending propositions on orthogonal decompositions and the second fundamental form to hermitian Higgs bundles.
result Extended propositions concerning orthogonal decompositions and the second fundamental form have applications in Higgs bundles.
Analyzes harmonic metrics for Higgs bundles over punctured surfaces.
problem Analyzing the moduli space of parabolic Higgs bundles with varying weights.
method Investigates the analytic dependence of harmonic metrics on weights and stable Higgs bundles.
result Shows the harmonic metric depends analytically on weights and stable Higgs bundles.
The paper solves equations for Higgs bundles on non-Kähler manifolds.
problem Analytically stable Higgs bundles on non-Kähler manifolds.
method Solving the Hermitian-Einstein equation on analytically stable Higgs bundles under specific conditions.
result Solutions to the Hermitian-Einstein equation for analytically stable Higgs bundles on non-Kähler manifolds.
Study proves Higgs fields non-existent on Calabi-Yau manifolds.
problem Proving non-existence of Higgs fields on Calabi-Yau manifolds.
method Used Yang-Mills-Higgs flow to prove Higgs field triviality.
result Higgs field is trivial for semistable Higgs bundles with vanishing Chern classes.
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.
problem Finding compatible harmonic metrics for rank 3 Higgs bundles in the Hitchin section.
method Defined a symmetric pairing and studied spectral curves as 2-sheeted branched coverings.
result Gave a condition for Higgs bundles on C or C∗ to have compatible harmonic metrics. Equivalence between flat and Higgs bundles on compact Sasakian manifolds.
problem Establishing an equivalence between flat and Higgs bundles on compact Sasakian manifolds.
method Extending the equivalence from Kähler manifolds to Sasakian manifolds, proving the equivalence for semi-simple flat bundles and polystable Higgs bundles with trivial Chern classes.
result Equivalence between semi-simple flat bundles and polystable Higgs bundles on compact Sasakian manifolds with trivial Chern classes.
Study uses Vinberg pairs for Higgs bundles, revealing their role.
problem Understanding the role of Vinberg pairs in Higgs bundle theory.
method Exploring Vinberg pairs defined by cyclic gradings of a Lie algebra in Higgs bundle theory.
result Vinberg pairs play a significant role in Higgs bundle theory.
In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle (E,H0) over a compact Kähler manifold (M,ω). We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorph…
Paper proves a theorem for Higgs bundles on certain non-compact manifolds.
problem Proving a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
method Generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over non-compact Gauduchon manifolds.
result Proved a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
New Poisson structures found on Higgs bundle moduli spaces.
problem Constructing Poisson structures on moduli spaces of Higgs bundles.
method Via Lie algebroids on stacky curves, focusing on parabolic Higgs bundles.
result Provides new examples of Poisson structures on moduli spaces.
Survey on 2k-Hitchin equations and Higgs bundles from geometric perspective.
problem Understanding 2k-Hitchin equations through Higgs bundles and complex geometry. method Review of Higgs bundles, holomorphic vector bundles, and Hermite-Yang-Mills equations; geometric tools applied to simplify equations.
result Simplified 2k-Hitchin equations to a set of two equations for Higgs bundles. We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle (E,H0) over a Riemann surface X. It is already known the gradient flow with initial data (A0,φ0) converges to a critical point (A∞,φ∞) of this functional. Using a modified Chern-Wei…
Study Higgs bundles and their reductions to prove stability and cohomology properties.
problem Analyzing Higgs bundles and their stability conditions.
method Introduced H-nflatness, proved stability conditions, and used cohomology rings.
result H-nflat Higgs bundles are either stable or reducible to a parabolic subgroup.
Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.
problem Existence of diagonal pluriharmonic metrics in G-Higgs bundles. method Analyzes Higgs bundles over compact Kähler manifolds, decomposes vector bundles, and uses torus action to relate stability and conditions.
result Necessary and sufficient conditions for the existence of diagonal metrics solving Hermitian-Einstein equations.
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, C∗-actions, Hecke transformations, Fourier-Mukai transforms. result Precise formula for multiplicity of very stable components of global nilpotent cone.
Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
problem Analytic and geometric properties of harmonic maps.
method Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
result Proves Dai-Li's conjecture on the monotonicity of the energy density and negative curvature conjecture for Coxeter cyclic G-Higgs bundles.
Defines and classifies toric co-Higgs bundles on projective toric varieties.
problem Classifying co-Higgs bundles on toric varieties.
method Using Klyachko's fan filtration and studying the co-Higgs bundle fiber at a closed point.
result Provides a Lie-theoretic classification of toric co-Higgs bundles.
In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.
Construct model Higgs bundles in exceptional components of Sp(4, R) character variety.
problem Constructing model Higgs bundles in exceptional components of the Sp(4, R) character variety.
method Using the gluing construction for Higgs bundles over connected sums of Riemann surfaces and solutions to the Sp(4, R)-Hitchin equations.
result Provide model Higgs bundles in all 2g-3 exceptional components of the maximal Sp(4, R)-Higgs bundle moduli space.
Defines a new functional for Higgs bundles, linking it to known concepts.
problem Understanding the geometry of Higgs bundles.
method Introducing a new functional J(h) and studying its properties. result The functional J(h) has absolute minima corresponding to Hermite-Yang-Mills metrics. We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…
Paper uses flow to prove theorem on Higgs bundles.
problem Proving generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles.
method Affine Hermitian-Yang-Mills flow
result Generalized Donaldson-Uhlenbeck-Yau theorem proved.
We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…
Symplectic structure found on moduli space of framed Higgs bundles.
problem Understanding the symplectic structure of moduli spaces of Higgs bundles.
method Trivializing vector bundles over divisors to construct framed Higgs bundles, proving symplectic structure.
result Moduli space of framed Higgs bundles admits a natural holomorphic symplectic structure.
The paper studies the moduli space of Higgs pairs and their geometric properties.
problem The moduli space of Higgs pairs and its geometric properties.
method Introduced τ-stability of Higgs pairs and established the Kobayashi-Hitchin correspondence. result Proved that the moduli space is a non-singular complex manifold for a suitable choice of τ.