Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.
problem Understanding non-abelian Hodge loci for quasi-projective varieties.
method Analyzes Z-local systems and polarized variations of Hodge structures. result Proves algebraicity of non-abelian Hodge loci for Q-anisotropic monodromy. 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.
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 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 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 describes how Hodge loci are typically equidistributed in complex varieties.
problem Understanding the distribution of Hodge loci in complex varieties.
method Analyzing polarized variations of Hodge structures over smooth complex quasi-projective varieties.
result Hodge loci are either empty or equidistributed with respect to a pull-push form.
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 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.
We refine the Morgan's work on mixed Hodge structures on Sullivan's 1--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.
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.
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 q-difference modu…
Notes on harmonic maps between manifolds, existence and regularity covered.
problem Existence and regularity of harmonic maps between Riemannian manifolds.
method Lecture-based approach covering harmonic maps, pluriharmonic maps, and related theorems.
result Coverage of existence and regularity of harmonic maps, including Siu-Sampson formula and Donaldson-Corlette theorem.
Develops a new non-abelian framework for Riemann surfaces and differential equations.
problem Analyzing second-order differential equations on Riemann surfaces.
method Gauge-theoretic framework and non-abelian approach.
result Extends Dedekind's Schwarzian approach to generic one-parameter families of curves of genus g.
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. 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 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 study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs …
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…
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized Z-variation of Hodge structure V on a smooth complex quasi-projective variety S, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-…
Holomorphic Higgs bundles on Teichmüller space for surface groups.
problem Characterizing representations of surface groups admitting holomorphic Higgs data.
method Non-abelian Hodge correspondence, unitarity conditions, and existence proofs for higher ranks.
result Holomorphic dependency of Higgs data is equivalent to unitarity for SL(2,C) representations but fails for higher ranks. We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…
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.
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.
Compact character varieties of punctured spheres are proven.
problem Compact relative SO0(2,q)-character varieties of punctured spheres. method Non-abelian Hodge correspondence and Geometric Invariant Theory (GIT).
result Proves the existence of compact, totally non-hyperbolic character varieties.
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.
In this paper we explain how non-abelian Hodge theory allows one to compute the L2 cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new algebraic description for the fiberwise L2 cohomology of a tame harmonic bundle o…
In this paper we prove the following results: 1) 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 the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…
In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside Ag. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic…
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.
The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.
problem Refining Alexander polynomials and bounds BNSR Σ-invariants for 3-manifolds and Kähler manifolds.
method Introduces twisted homology jump loci and uses tropical geometry to obtain bounds.
result Sharp bounds for BNSR Σ-invariants and obstructions to geometric realizability.
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.
Study finite group actions on Higgs bundle moduli spaces.
problem Describe fixed points of finite group actions on Higgs bundle moduli spaces.
method Use twisted Γ-equivariant bundles and Prym-Narasimhan-Ramanan construction.
result Provide description of fixed-point subvarieties of certain finite group actions on G-character varieties.
The paper proves a conjecture about the positivity of the Euler characteristic for complex projective manifolds.
problem Proving the positivity of the Euler characteristic for complex projective manifolds with large fundamental groups.
method Introducing a vanishing cycle functor of multivalued one-forms and applying non-abelian Hodge theory techniques.
result The paper proves a stronger statement about the positivity of the Euler characteristic for complex projective manifolds with large fundamental groups and almost faithful linear representations.
Study shows no hyperkähler fourfolds in specified conditions.
problem Identifying hyperkähler fourfolds in specific geometric settings.
method Classification and computation of Hodge numbers for fourfolds over rational homogeneous varieties.
result No hyperkähler fourfolds found in the specified conditions.
We use superconnections to define and study some natural differential forms on period domains D that parametrize polarized Hodge structures of given type on a rational quadratic vector space V. These forms depend on a choice of vectors v1,…,vr∈V and have a Gaussian shape that peaks on the locu…
We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…
Maps asymptotically embed conic transforms from circle bundles.
problem Embedding conic transforms from circle bundles.
method Asymptotic embeddings using equivariant Szegő projectors.
result Maps embed conic transforms from circle bundles.
Study actions of mapping class groups on surface representations, proving finite image for certain representations.
problem Finite image of representations of mapping class groups on surfaces.
method Hodge-theoretic and arithmetic techniques, including non-abelian Hodge theory and isomonodromic deformations.
result Proves finite image for representations with specific properties.
The paper constructs solutions for Higgs fields on a 4-punctured sphere.
problem Constructing self-duality solutions for Higgs fields on a 4-punctured sphere.
method Complex analytic methods, twistor approach, λ-connections interpretation.
result Identifies the rescaled limit hyper-Kähler moduli space at t=0.
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
A new complex space resolves projective structures on surfaces.
problem Understanding projective structures on compact surfaces.
method Proposed a complex analytic space Pg and analyzed it for g=1. result The space Pg naturally resolves the orbifold locus of Ag=1. This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined wi…
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character variet…
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.
As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irredu…
We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups SU(p,q) admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an …