The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
The paper studies Killing fields and moment maps for Riemannian manifolds.
problem Understanding Killing fields and moment maps for Riemannian manifolds.
method Analyzes the infinitesimal isometries of connection metrics and generalized moment map equations.
result Proves the relationship between Killing fields and moment maps for Riemannian manifolds.
The paper generalizes the moment map interpretation of scalar curvature in Kähler geometry.
problem Interpreting the variation of the Quillen metric in Kähler geometry.
method Constructing equivariant determinant line bundles and analyzing their curvature forms.
result The moment maps μj coincide with the Z-critical equations introduced by Dervan-Hallam. New approach to scalar curvature using hyperkähler reduction.
problem Finding solutions to scalar curvature equations.
method Explicit construction of hyperkähler metrics and moment map equations.
result Existence of solutions to moment map equations on ruled surfaces.
Paper generalizes equations linking geometry and physics.
problem Finding solutions to complex geometric equations.
method Introducing and studying equations on principal bundles.
result Provides obstructions to solution existence.
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
New equations derived for Kähler metrics, linking stability and curvature.
problem Finding metrics with constant scalar curvature in Kähler geometry.
method Introduced coupled cscK equations and defined K-polystability.
result Proved existence of coupled cscK metrics for small perturbations.
Study global geometry of toric nearly Kähler manifolds using multi-moment maps.
problem Global properties of toric nearly Kähler manifolds.
method Description using multi-moment maps, investigation of polynomial and radial solutions.
result Description of global geometry of toric nearly Kähler manifolds.
New system modifies constant scalar curvature Kähler condition with a 'Higgs field'.
problem Extending constant scalar curvature Kähler condition to higher-dimensional manifolds.
method Explicit construction of hyperkähler metrics, Hitchin's equations for harmonic bundles, and Hermitian Yang-Mills equation.
result Existence of solutions to the modified system on specific cases (Riemann surfaces, ruled surfaces, abelian and toric surfaces).
Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.
problem Existence of solutions to coupled Kähler-Einstein and Hermitian-Yang-Mills equations.
method Moment map interpretation, Futaki invariant, Matsushima-Lichnerowicz theorem, deformation results.
result Nontrivial solutions produced under certain conditions.
Develops moment map theory for twisted scalar curvature in Kähler geometry.
problem Understanding the geometry of holomorphic submersions and foliations.
method Introduces a coupled system of equations on a holomorphic submersion.
result The coupled system appears as a moment map, generalizing to foliations.
Geometric approach to moment maps in complex geometry.
problem Constructing moment maps in complex geometry.
method Introducing universal families and equivariant differential forms.
result New geometric proofs and equations for moment maps.
We revisit the problem of constructing instantons on ADE orbifolds R^4/Γand point out some subtle relations with the complex structure on the orbifold. We consider generalized instanton equations on R^4/Γwhich are BPS equations for the Yang-Mills equations with an external current. The relation between level sets of th…
The paper generalizes a moment map interpretation of scalar curvature in Kähler geometry.
problem Interpreting scalar curvature as a moment map on the space of compatible almost complex structures.
method Constructing equivariant determinant line bundles and analyzing their curvature forms.
result The moment maps μj coincide with the Z-critical equations and generalize Fujiki's fiber integral formula. Study of symplectic Monge-Ampère equations using moment maps and contact structures.
problem Characterizing symplectic Monge-Ampère equations through geometric structures.
method Constructing contact cone structures and using moment maps to relate equations to projective spaces.
result The contact cone structure and the cocharacteristic variety coincide for non-degenerate equations.
Abstract discusses symplectic structure and Maxwell equations on Yang-Mills fields.
problem Formulating Maxwell equations on Yang-Mills fields using symplectic structures.
method Endowed a symplectic structure on the fiber product of tangent and cotangent bundles of connections, derived Hamiltonian equations and moment maps.
result Proved Maxwell equations and derived new conserved quantities from moment maps.
Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.
problem Existence of smooth solutions to complex Hessian equations in unstable cases.
method Parabolic flows and moment-map energy functionals, focusing on J-equation and deformed Hermitian Yang-Mills equation.
result Proves existence of unique canonical solutions with singularities on Kahler surfaces.
Introduces generalized moment maps for almost Hermitian settings.
problem Extending classical moment map theory to almost Hermitian settings.
method Introduces momentumly closed forms and proves a variant of the Darboux-Weinstein theorem.
result Establishes convexity property and constructs reduction space for generalized moment maps.
The paper proves a Kempf-Ness theorem for non-algebraic structures.
problem Non-algebraic symplectic structures and shifted moment maps.
method Proves an affine Kempf-Ness theorem for these structures.
result Describes hyperkahler quotients of T*G.
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Study of hyperkähler reduction on abelian varieties and toric manifolds.
problem Understanding hyperkähler reduction on specific manifolds.
method Lifts canonical Kähler reduction to hyperkähler, studies on abelian varieties and toric manifolds.
result Obtains decoupling result, variational characterisation, relation to K-stability, and proves existence and uniqueness under suitable assumptions. We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of …
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
We study generalized moment maps for a Hamiltonian action on a connected compact H-twisted generalized complex manifold introduced by Lin and Tolman and prove the convexity and connectedness properties of the generalized moment maps for a Hamiltonian torus action.
We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this form…
In this paper, we consider generalized moment maps for Hamiltonian actions on H-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact H…
We study equations on a principal bundle over a compact complex manifold coupling connections on the bundle with Kähler structures in the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of th…
The paper applies Fisher-Rao geometry to beta distributions for moment analysis.
problem Comparing and analyzing moments of probability distributions.
method Derived geodesic equations and sectional curvature on beta distributions' parameter space. Used Fisher-Rao geometry to map canonical moments to beta distributions.
result Uniqueness of Riemannian centroid in beta distributions' parameter space.
Study of dHYM connections on ruled surfaces with variable background metrics.
problem Finding new dHYM connections on ruled surfaces with variable metrics.
method Using momentum construction and moment map partial differential equations, coupled to scalar curvature of the background.
result Provide many new examples of dHYM connections coupled to a variable background Kähler metric.
Introduces gauge theory for string algebroids, solving Calabi system.
problem Solving coupled equations for Calabi problem and Hull-Strominger system.
method Moment map picture with Hamiltonian gauge action, inner automorphisms of Courant algebroids.
result Moduli space carries pseudo-Kähler metric with Kähler potential given by dilaton functional.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…
moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originat…
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
The paper trivializes moment maps for various geometric structures.
problem Trivializing moment maps for different geometric structures.
method General framework of a reductive group G acting on a smooth affine variety, using Kempf-Ness theory, Morse theory, and ideas from Nakajima and Kronheimer. result Locally trivial fibration of moment maps over a regular locus of the center of the Lie algebra of a maximal compact subgroup.
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
Study of hyperkähler reduction on Riemann surfaces, finding more solutions.
problem Finding solutions to the constant scalar curvature equation on Riemann surfaces.
method Infinite-dimensional hyperkähler reduction associated with the constant scalar curvature equation on a Riemann surface.
result Obtained a more general existence result, leading to a larger hyperkähler moduli space.
Given a multisymplectic manifold (M,ω) and a Lie algebra g acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an L∞-algebra-homomorphism from g to the observable algebra L(M,ω) associated to (M,ω), in analogy with and generalizing the notio…
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
We describe the multi-moment map associated to an almost Hermitian manifold which admits an action of a torus by holomorphic isometries. We investigate in particular the case of a T3 action on the homogeneous nearly Kähler S3×S3. We find that the multi-moment map in this case acts more-or-less sim…
This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow.…
Study moment maps coupled with convex functions to find critical points.
problem Understanding critical points of moment maps coupled with convex functions.
method Develop a theory of moment maps coupled with an Ad_K-invariant convex function f on k*.
result Interpret Kähler-Ricci solitons as a special case of generalized extremal metrics.
Deformation quantization yields a new moment map on symplectic diffeomorphisms.
problem Formalizing moment maps on diffeomorphism groups of symplectic manifolds.
method Deformation quantization framework applied to extrmDiff0(M). result Obtained a deformation of the Donaldson moment map.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
Let ρ:(D2)m→Im be the orbit map for the diagonal action of the torus Tm on the unit poly-disk (D2)m, Im=[0,1]m is the unit cube. Let C be a cubical subcomplex in Im. The moment-angle complex $\ma(C)$ is a Tm-invariant bigraded cellular decomposition of the subset ρ−1(C)⊂(D2)m wit…
Study of multi-moment maps on specific six-manifolds.
problem Understanding multi-moment maps on nearly Kähler six-manifolds.
method Explicit derivation of multi-moment maps and analysis of fixed-points and orbits.
result Explicit expression and configuration of fixed-points and orbits derived.
Extends moment map concept to locally conformally Kähler manifolds.
problem No specific problem stated; extends existing concept.
method Extends classical moment map interpretation to locally conformally Kähler geometry.
result Scalar curvature as moment map in locally conformally Kähler geometry.
Investigates properties of moment maps and stratifications on Lie groups.
problem Understanding moment maps and stratifications on real reductive Lie groups.
method Functorial, algebraic approach to moment map and Kirwan-Ness stratification.
result Properties and properties of moment maps and stratifications established.