arXiv research

Extends moment map concept to locally conformally Kähler manifolds.

The paper generalizes the moment map interpretation of scalar curvature in Kähler geometry.

Deform quantization recovers scalar curvature in complex structures.

The paper generalizes a moment map interpretation of scalar curvature in Kähler geometry.

Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.

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.

Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ 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^{-…

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM eq…

Geometric approach to moment maps in complex geometry.

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In…