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.
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.
This note finds explicit representatives for moduli space of parabolic bundles.
problem Finding explicit representatives in deRham cohomology for the moduli space of parabolic bundles.
method Using results from vector bundles and computing intersection pairing of cohomology.
result Explicit representatives in deRham cohomology for the generators of the moduli space of parabolic bundles.
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.
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.
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 proves a CM line bundle is ample on K-moduli spaces.
problem Proving positivity of the CM line bundle on K-moduli spaces.
method Developed a new invariant for filtrations to test K-stability.
result Proves the CM line bundle is ample on reduced uniformly K-stable Fano varieties.
Defines a metric and form for a bundle moduli space, leading to a zero-curvature formulation.
problem Formulating a metric and form for a bundle moduli space.
method Defines an algebraic metric and closed 3-form on a subspace of the moduli of G-bundles. result Shows a zero-curvature formulation for a σ-model with target the moduli space. The paper proves a theorem about constructing Higgs bundle moduli space.
problem Constructing the moduli space of Higgs bundles on a closed Riemann surface.
method Uses Kuranishi slice method and GIT quotient to prove the moduli space is a complex space locally modeled on a quadratic cone.
result The moduli space of Higgs bundles is a complex space locally modeled on an affine GIT quotient of a quadratic cone.
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.
We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli s…
In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analyt…
Constructs Higgs bundle moduli spaces using Teichmüller space.
problem Holomorphic family of Higgs bundle moduli spaces over a curve.
method Uses a function f on the character variety to define flat Ehresmann connections.
result Reveals various aspects of moduli spaces and their metrics.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
Survey on Higgs bundle moduli spaces and their connected components.
problem Counting connected components of Higgs bundle moduli spaces.
method Analyzes moduli spaces for Higgs bundles associated with real Lie groups and closed Riemann surfaces.
result Explicit descriptions of some moduli space components are possible.
Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.
problem Proving the quasi-projectivity of the moduli space of Higgs bundles.
method Uses analytic methods and the symplectic cut to construct a compactification.
result Proves the quasi-projectivity of the moduli space of Higgs bundles.
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. In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…
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.
Proves Simpson's conjecture about Higgs bundles and moduli spaces.
problem Proving Simpson's conjecture about Higgs bundles and moduli spaces.
method Analyzes Higgs bundles and their associated moduli spaces, proving Simpson's conjecture.
result Proves Simpson's conjecture about holomorphic Lagrangian submanifolds in de Rham moduli space.
Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds
problem Establishing an equivalence between diffeological substacks of Higgs and flat bundles
method Using diffeological moduli stacks
result Shows equivalence of categories between semistable Higgs bundles and flat bundles
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.
problem Characterize the moduli space of stable rank 2 parabolic bundles over an elliptic curve with marked points.
method Explicitly describe the moduli space as a blow-up of an embedded elliptic curve in (CP1)3 and interpret it as the SU(2) character variety of the 3-punctured torus. result The moduli space Ms(X,3) can be described as a blow-up of an embedded elliptic curve in (CP1)3 and interpreted as the SU(2) character variety of the 3-punctured torus. In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern cla…
Existence of symplectic structure shown on Seiberg-Witten moduli space product of Riemann surfaces.
problem Existence of symplectic structure on Seiberg-Witten moduli space.
method Construction of Quillen-type determinant line bundle and showing its curvature proportional to the symplectic form.
result Existence of symplectic structure on the moduli space of Seiberg-Witten equations on product of Riemann surfaces.
Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.
problem Positivity of CM line bundles on moduli space of klt good minimal models with κ=1.
method Construction of a moduli space of numerical equivalence classes, proving projectivity of moduli space of ε-stable quotients, and using K-moduli of quasimaps.
result CM line bundle becomes ample after normalization and moduli space is quasi-projective.
Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.
problem Construction of moduli spaces for Higgs bundles with specific properties.
method Elementary geometric and combinatorial techniques, focusing on orbit stability of automorphism groups.
result Explicit geometric models for moduli spaces of parabolic Higgs bundles over Riemann sphere.
Torsors over moduli spaces of vector bundles with fixed determinant.
problem Understanding connections on vector bundles over curves.
method Algebraic geometry and sheaf theory.
result Moduli space of vector bundles has a natural torsor structure.
The Liouville symplectic form connects various moduli spaces in algebraic geometry.
problem Understanding connections between different moduli spaces in algebraic geometry.
method Using the Liouville symplectic structure on the cotangent bundle of a loop group.
result Induces symplectic structures on moduli stacks and spaces of framed connections.
For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…
Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obt…
We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their…
We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.
Consider the moduli space of parabolic Higgs bundles (E,Φ) of rank two on CP^1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E,Φ)\mapsto (E,-Φ). We study the fixed point locus of this involution. In [GM], this modu…
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern …
Study on the asymptotic geometry of Higgs bundles over projective line.
problem Understanding the asymptotic behavior of Hitchin's metric on moduli spaces of rank two irregular Higgs bundles.
method Analysis of Hitchin's hyperkähler metric and comparison with semiflat and ALG/ALG∗ models. result Hitchin's metric is asymptotic to semiflat and ALG/ALG∗ models at polynomial and exponential rates. The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of S7-bundles over S8 and quotients of Milnor and Shimada spheres. result The moduli space of metrics has infinitely many path components.
We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama m…
Proves existence of flat connection on theta functions for G-bundles.
problem Existence of flat connections on nonabelian theta functions for G-bundles.
method Proves existence of a flat projective connection on nonabelian theta functions on moduli space of parabolic G-bundles.
result Existence of a flat projective connection on nonabelian theta functions for parabolic G-bundles.
The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.
problem Understanding Lagrangian structures in Hitchin moduli space.
method Analyzing semistable and polystable Higgs bundles, focusing on the intersection with Lagrangian sublocus.
result The conformal limit of stable Higgs bundles on a specific Lagrangian sublocus exists under certain conditions.
This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.
The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.
For a Riemann surface X and the moduli of regularly stable G-bundles M, there is a naturally occuring "adjoint" vector bundle over X×M. One can take the determinant of this vector bundle with respect to the projection map onto M. Our aim here is to study the curvature of the determinant bundle as the…
Constructs moduli spaces for complex affine and dilation surfaces.
problem Classifying and understanding moduli spaces of complex surfaces.
method Using Veech's ideas, constructs holomorphic affine bundles and covering spaces.
result Moduli spaces of dilation surfaces are orbifold K(G,1) where G is the framed mapping class group.
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…
The paper studies flow lines on Higgs bundle moduli spaces, classifying them via secant varieties.
problem Classifying flow lines on moduli spaces of Higgs bundles.
method Gradient flow lines for L2 norm of Higgs field, Morse-theoretic compactification, secant varieties. result Flow lines have an algebro-geometric classification via secant 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.
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.