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.
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. 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 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. Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.
problem Eigenvalue estimates for the Hodge Laplacian on Fano manifolds.
method Application of Bochner-Kodaira formula to study the geometry of Fano manifolds.
result The original Kähler form remains Kähler for deformations, and an explicit Ricci potential is given.
Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed i…
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. Let (M,g) be a compact Kähler manifold and f a positive smooth function such that its Hamiltonian vector field K=Jgradgf for the Kähler form ωg is a holomorphic Killing vector field. We say that the pair (g,f) is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-…
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.
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.
On a given compact complex manifold or orbifold (M,J), we study the existence of Hermitian metrics g~ in the conformal classes of Kähler metrics on (M,J), such that the Ricci tensor of g~ is of type (1,1) with respect to the complex structure, and the scalar curvature of g~ is constant. In…