Introduces Lax-Kirchhoff moduli spaces for quivers and Lie groups.
problem Constructing moduli spaces for quivers and Lie groups.
method Introduces Lax equations and Kirchhoff conditions, constructs slices, and uses Marsden-Weinstein reduction.
result Proves M(Γ) is a finite-dimensional smooth symplectic manifold with a Hamiltonian action of G∂Γ. We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
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.
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.
Study special Lagrangian moduli spaces with boundary.
problem Understanding geometric structures on moduli spaces of special Lagrangians.
method Investigates geometric structures and constructs special affine structures and a Hessian metric.
result Constructs a pair of special affine structures and a Hessian metric on the moduli space.
Constructs projective moduli spaces for Calabi-Yau pairs.
problem Creating projective moduli spaces for Calabi-Yau surface pairs.
method Constructs projective asymptotically good moduli spaces.
result Provides a wall crossing between KSBA and K-moduli spaces.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. Study of Hitchin moduli spaces over Teichmüller space.
problem Metric aspects of Hitchin moduli spaces over varying complex structures.
method Gauge theoretical approach, Kähler fibrations, moment map interpretation, symplectic reduction.
result Establishes natural complex and pseudo-Kähler structures on universal Hitchin moduli spaces.
The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.
problem Comparing constrained and decoupled moduli spaces of manifolds with embedded particles and discs.
method Generalized Bödigheimer--Tillmann's work to higher dimensions and different tangential structures.
result New results for surfaces with different tangential structures and higher dimensional manifolds.
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…
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.
Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.
problem Constructing moduli spaces over Riemann surfaces.
method Finite-dimensional construction using holomorphic symplectic reduction.
result Moduli spaces over Riemann surfaces as stratified holomorphic symplectic spaces.
We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under B-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decompo…
Researchers compute the heterotic moduli-space metric up to α′2.
problem Computing the heterotic moduli-space metric up to a specific order.
method Using α′2 for backgrounds with a smooth α′o0 limit, focusing on the Kaehler potential and metric corrections. result Metric corrections from the deformation of the Hull connection are identified.
Homological stability fails for 4-manifold moduli spaces, detected by new class.
problem Homological instability for moduli spaces of 4-manifolds.
method Used Seiberg-Witten equations to construct a new characteristic class.
result Homological discrepancy between smooth and topological 4-manifold moduli spaces.
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
We shall construct a moduli space of pairs of Kähler-Einstein structures and special lagrangians and obtain smoothness of the moduli space of these pairs. Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group.
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. 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 study finds non-trivial elements in moduli spaces of curvature metrics.
problem Identifying non-trivial elements in moduli spaces of curvature metrics.
method Constructing elements in homotopy groups of moduli spaces of positive curvature metrics.
result Non-trivial elements found in moduli spaces of positive curvature metrics.
Study moduli spaces of solutions to Bogomolny equations on surfaces.
problem Understanding solutions to Bogomolny equations on surfaces with specific boundary conditions.
method Refined Kobayashi-Hitchin correspondence and identification with Higgs bundles.
result Identified moduli spaces as holomorphic lagrangian sub-manifolds.
Study of tautological forms on curve moduli spaces.
problem Understanding tautological forms on moduli spaces of curves.
method Defined and studied a system of tautological rings on moduli spaces of marked curves, showing certain 2-forms are tautological and rings are finite dimensional.
result Characterized the Kawazumi-Zhang invariant as a tautological form.
We construct proper good moduli spaces parametrizing K-polystable Q-Gorenstein smoothable log Fano pairs (X,cD), where X is a Fano variety and D is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as c varies. The main applicatio…
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.
We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the A∞ structure of the both …
We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.
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.
Universal geometry organizes heterotic moduli spaces.
problem Understanding the structure of heterotic compactifications.
method Fibering compactification data over moduli space and analyzing universal curvatures.
result Deformations are components of universal curvatures, incorporating α′2 corrections. Same cohomology for curves with levels, proving stable range.
problem Stability of cohomology in moduli spaces with level structures.
method Proving cohomology equality through stable range analysis.
result Rational cohomology of moduli space with levels matches ordinary space.
We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
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.
This paper shows moduli spaces of RCD(0,2) structures are contractible.
problem Understanding moduli spaces of RCD(0,2) structures.
method Established a list of compact topological spaces admitting RCD(0,2) structures and described their associated moduli spaces.
result All moduli spaces of RCD(0,2) structures are contractible.
Mirzakhani volumes of moduli spaces are polylogarithmic.
problem Understanding the volume of moduli spaces of hyperbolic surfaces.
method Expressed as a sum of polylogarithms evaluated at specific points.
result Mirzakhani volumes are polylogarithmic.
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.
Proves properness of K-moduli spaces for Fano varieties.
problem Proving properness of moduli spaces of K-polystable Fano varieties.
method Algebraic approach, studying test configurations, constructing stratification.
result Proves properness under specific divisorial valuation condition.
Constructs a space for stable holomorphic submersions over a fixed base.
problem Stability of holomorphic submersions over a compact Kaehler base.
method Geometric invariant theory combined with geometric PDEs.
result Moduli space is a Hausdorff complex space with a Weil-Petersson type Kaehler metric.
We show that the moduli space M of holomorphic vector bundles on CP3 that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on R4, called the space of twistor sections. We then use this c…
In this paper we continue our study on the moduli spaces of flat G-bundles, for any semi-simple Lie group G, over a Riemann surface by using heat kernel and Reidemeister torsion. Formulas for intersection numbers on the moduli spaces over a Riemann surface with several boundary components, over non-orientable Riemann s…
Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.
problem Understanding K-moduli spaces of curves on quadrics and K3 surfaces.
method Using log Fano pairs and VGIT quotients, the study compares K-moduli spaces of curves on P1imesP1 and quartic hyperelliptic K3 surfaces. result K-moduli spaces of curves on quadrics and K3 surfaces form a natural interpolation.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Holomorphic maps between moduli spaces are shown to be forgetful for large g.
problem Characterizing holomorphic maps between moduli spaces.
method Proving that only forgetful maps are non-constant for large g.
result Forgetful maps are the only non-constant holomorphic maps between moduli spaces for g≥4. We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.
problem Calculating volumes of moduli spaces of flat surfaces with prescribed conical singularities.
method Induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.
result Explicit computation of volumes is possible.
In generalization of the classical Atiyah-Bott Poisson brackets on the moduli spaces of surfaces we define quasi-Poisson brackets on the moduli spaces of quasi-surfaces.
We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.