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.
Study invariant structures on flag manifolds using transformations and pure spinors.
problem Understanding invariant generalized complex and Kähler structures on flag manifolds.
method Description of moduli spaces using invariant structures, Weyl group action, and pure spinors.
result Alternative description and cell decomposition of moduli spaces.
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.
Twistors help show complex moduli spaces can be infinite-dimensional.
problem Moduli spaces of complex structures can be infinite-dimensional globally.
method Using twistor theory of hyper-Kaehler manifolds.
result Examples demonstrate moduli spaces can contain regions with infinite local dimension.
The paper constructs a moduli space for opers and proves its symplectic properties.
problem Understanding the moduli space of opers and its symplectic structure.
method Constructing the moduli space of marked oper structures and proving properties of the holonomy map.
result The symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space of marked opers.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
Study of tropical moduli spaces using symmetric Delta-complexes.
problem Understanding the fundamental groups and homology of tropical moduli spaces.
method Develop techniques for symmetric Delta-complexes, apply to moduli spaces of tropical curves.
result Delta_g and Delta_{g,n} are simply connected for positive g.
Study of heterotic superpotential and moduli in compactified gravity.
problem Investigate properties of four-dimensional Minkowski vacua in heterotic supergravity.
method Corrected heterotic superpotential, F-terms, D-terms, and constraints from complex structure and bundle deformations.
result Identify and calculate the infinitesimal massless moduli space of the theory.
Study constructs local moduli space for scalar-flat Kähler ALE surfaces.
problem Understanding the local moduli space of scalar-flat Kähler ALE surfaces.
method Construction of local moduli space and explicit formula derivation.
result Explicit formula for the dimension of moduli space.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
The paper provides complex coordinates on moduli space of vector bundles.
problem Constructing coordinates on the moduli space of vector bundles.
method Two ways of constructing complex coordinates, computing transformation between them.
result Complex coordinates agree to second order but not to third order at the center.
Study automorphisms of complex spaces linked to Riemann surfaces.
problem Computing automorphism groups of moduli spaces.
method Analysis of Dolbeault, de Rham, and Betti moduli spaces.
result Computed automorphism groups for C∗ moduli spaces. It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.
No nontrivial automorphisms for cubic surfaces moduli space.
problem Understanding automorphisms of cubic surfaces moduli space.
method Analyzing the fundamental group of the moduli space.
result No nontrivial biholomorphic automorphisms for cubic surfaces moduli space.
We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space.
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 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.
Researchers create a new moduli space for Fano manifolds with special geometric properties.
problem Constructing a new moduli space for Fano manifolds with Kähler-Ricci solitons.
method Developed a moment map picture and used complex analytic charts to construct the moduli space.
result Created a larger moduli space that includes Fano manifolds with Kähler-Einstein metrics.
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.
We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli space of holomorphic structures on a complex vector bundle over a compact complex …
Decouples moduli groups in heterotic string theory cohomology.
problem Decomposing moduli groups in heterotic string theory cohomology.
method Computing cohomology groups at the standard embedding and showing their direct sum decomposition.
result Decomposes moduli groups into a direct sum of cohomologies at the standard embedding.
We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors.…
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.
Study of metrics on a specific Lie group with complex structure.
problem Determine moduli space of left-invariant metrics on a six-dimensional Lie group.
method Investigate left-invariant metrics on a Lie group with complex structure and first Betti number 4.
result Classify metrics and complex structures on the Lie group.
The harmonic volume is a complex analytic invariant that captures detailed complex structure information.
problem Capturing detailed complex structure information of compact Riemann surfaces.
method Defined using Chen's iterated integrals, the harmonic volume extends the period.
result Enabled a quantitative study of the local structure of the moduli space.
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.
Study on real algebraic curves' moduli spaces using a new complex.
problem Understanding the topology of moduli spaces of real algebraic curves.
method Defined a new complex, the ABC-complex, to encode intersection patterns. result Showed that mapping class groups are virtual duality groups and deduced orbifold homotopy group results.
We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pair…
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.
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.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
Study presents a twistor correspondence for specific geometric structures.
problem Twistor theory for almost-Grassmannian manifolds.
method Utilizes moduli of curves-with-boundary for global correspondence.
result Foundational results in complex setting, global correspondence for real Grassmannian.
Study on null curves in Klein's quartic moduli space.
problem Existence of minimal surfaces with specific planar ends.
method Methods developed by Robert Bryant.
result Minimal surfaces with 9 embedded planar ends do not exist.
Researchers smooth out moduli space of Spin(7)-instantons.
problem Regularizing the moduli space of Spin(7)-instantons.
method Constructing and perturbing the moduli space to achieve smoothness and expected dimension.
result Smooth and of expected dimension over the irreducible locus.
Study on moduli spaces of branched projective structures on surfaces.
problem Characterizing and understanding moduli spaces of branched projective structures.
method Analytic and geometric methods to study the moduli spaces of branched projective structures.
result The moduli space of marked branched projective structures is a complex analytic space with specific dimensions and singular points.
Study instanton moduli spaces on class VII surfaces, proving complex structure extension.
problem Extend complex structure on Donaldson-Uhlenbeck compactification of instanton moduli spaces.
method Analyzing moduli spaces of SU(2)-instantons on class VII surfaces.
result Prove complex structure extension for moduli spaces of instantons with c2=1.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
An \emph{ω-admissible almost complex structure} on a 2n-dimensional symplectic manifold (M,ω) is a ω-calibrated almost complex structure J admitting a nowhere vanishing ∂ˉJ-closed (n,0)-form ψ. After giving some examples we consider the moduli space of admissible almost complex structures a…
Extends Masur's divergence theorem to complex tori and Kummer surfaces.
problem Establishing uniquely ergodic horizontal foliations for geodesic flows on moduli spaces.
method Defined and calculated horizontal foliations and geodesic flows on moduli spaces of Kähler metrics.
result Proved that horizontal foliations are uniquely ergodic if geodesic flows are recurrent.
Constructs the moduli space of super J-holomorphic curves.
problem Defines and constructs the moduli space of super J-holomorphic curves.
method Uses component fields of a super differential equation and a transversality argument.
result Constructs the moduli space of super J-holomorphic curves as a smooth subsupermanifold.
Study complex deformations of the circle using group cohomology and Virasoro algebra.
problem Complexification of circle diffeomorphism group and its geometric properties.
method Real-analytic maps, group cohomology, Witt algebra, Frölicher structures.
result Virasoro uniformization theorem for moduli spaces of Riemann surfaces.
Extends Higgs bundle theory to cell complexes and finitely presented groups.
problem Extending Higgs bundle theory to cell complexes and finitely presented groups.
method Defined Higgs bundles over cell complexes, described moduli spaces, and used harmonic maps.
result Character variety of a finitely presented group is homeomorphic to the moduli space of Higgs bundles.
The study proves deformations of scalar-flat Kähler ALE surfaces and constructs a moduli space.
problem Deformation theory of scalar-flat Kähler ALE surfaces.
method Kuranishi-type theorem for complex structures, scalar-flat metrics, local moduli space construction.
result All small deformations of complex structures on scalar-flat Kähler ALE surfaces also admit scalar-flat Kähler ALE 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.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
Researchers mapped the moduli space of a specific group in 3D complex hyperbolic geometry.
problem Mapping the moduli space of a discrete, faithful representation of the modular group in PU(3,1). method Constructed the entire moduli space M by parameterizing it with a square, relating it to PU(2,1) representations. result The moduli space M is divided into subspaces parameterized by a square, each corresponding to different geometries. The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classificatio…