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.
Study of multi-moment map for nearly Kähler S³ × S³.
problem Investigating the multi-moment map for nearly Kähler S³ × S³.
method Analyzing the multi-moment map associated with an almost Hermitian manifold with a torus action.
result The multi-moment map behaves similarly to the moment map of a toric manifold in the nearly Kähler S³ × S³ case.
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.
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…
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…
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.
The scalar curvature is redefined in generalized Kahler geometry as a moment map.
problem Defining scalar curvature in generalized Kahler geometry.
method Introducing a moment map in generalized Kahler geometry to define a generalized scalar curvature.
result Infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite-dimensional.
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.
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.
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.
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. 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 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.
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.
New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.
problem Constructing real algebraic maps with specific properties and compositions.
method Explicit construction of real algebraic hypersurfaces and maps with prescribed images and compositions.
result Explicit families of functions represented as compositions of constructed maps with canonical projections.
The study connects moment maps, star products, and automorphism groups on Kaehler manifolds.
problem Analyzing the structure of automorphism groups on Kaehler manifolds with specific curvature properties.
method Using star products, moment maps, and Hessian formulas to study holomorphic vector fields.
result Proves a reductive Lie algebra structure for holomorphic vector fields on Kaehler manifolds.
Conditions for pre-quantizability of G-invariant forms are derived using moment maps.
problem Conditions for equivariant prequantizability of G-invariant forms.
method Conditions derived using moment maps and obstructions computed.
result Necessary and sufficient conditions for equivariant pre-quantizability are computed.
New method for moment maps in multisymplectic geometry using Lie 2-algebras.
problem Existence and construction of moment maps in multisymplectic geometry.
method Introducing homotopy moment maps defined on a Lie 2-algebra.
result Existence criteria and construction of homotopy moment maps.
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
We study quantum moment maps of G-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a G-invariant star product is differentiable. This property gives us a new method for the class…
New constructions and examples from moduli spaces.
problem Understanding quasi-Poisson G-spaces and their moment maps. method Lifting Theorem establishing a bijective correspondence.
result Simple constructions of fusion and conjugation.
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…
In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally…
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.
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.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
In [GMPS] we proved that the moment map image of a b-symplectic toric manifold is a convex b-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on b-symplectic manifolds. The modular weights of the action on the connected components of the exceptio…
Given a Lie group acting on a manifold M preserving a closed n+1-form ω, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This descr…
We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-u…
A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-h…
We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra…
We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of …
We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…
New quaternionic toric manifolds introduced, studied with properties.
problem Developing a new type of toric manifold in quaternionic geometry.
method Construction from Delzant polytopes, 4-plectic structure, generalized moment map. result Quaternionic toric manifolds are a large class for testing new quaternionic geometry results.
We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension 4n with a tri-Hamiltonian action of a torus of dimension n, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyp…
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
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.
The paper constructs examples of Lagrangian flows using moment maps.
problem Constructing Lagrangian mean curvature flows in Calabi-Yau manifolds.
method Using moment maps for abelian Lie group actions.
result Examples of Lagrangian self-shrinkers and translating solitons.
Study Hessian geometry of multi-Taub-NUT metrics and their phase changes.
problem Understanding phase changes in multi-Taub-NUT metrics.
method Analysis via moment maps of Hessian geometry.
result Generalization of earlier work on toric Gibbons-Hawking metrics.
Constructs a moment map for maps to balanced manifolds.
problem Understanding maps from complex manifolds to balanced manifolds.
method Constructs a moment map for a specific action of biholomorphisms.
result Lays groundwork for balanced quotients.
We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.