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.
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.
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.
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.
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.
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…
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. 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.
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. 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.
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.
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.
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.
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…
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.
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.
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.
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.
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 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…
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.
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.
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 …
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. 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.
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.
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.
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.
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…
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…
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.
New theory connects non-abelian bundle gerbes to abelian ones.
problem Challenges in extending higher gauge theory beyond fake-flat sector.
method Developed a comprehensive theory of adjusted connections on non-abelian bundle gerbes.
result Established a new coordinate-independent formulation of lifting theorem.
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
Study non-Abelian gauge theories using Poisson bracket structures.
problem Defining a Poisson bracket structure on solution spaces.
method Using coisotropic embedding theorem.
result Defined Poisson bracket structure for non-Abelian gauge theories.
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 …
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
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…
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 describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…
Hodge theory applied to tropical curves.
problem Developing Hodge theory for tropical curves.
method Analytical approach using tropical differential forms and L2−cohomologies. result Construction of Hodge theory analog on tropical curves.
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.
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.
We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for the anomaly cancellation in supersymmetric sigma models, and in a geometric description of so-called non-geometric T-duals.
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.
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.
Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.