arXiv research

We analyze degenerate homogeneous structures of linear type in the pseudo-Kähler and para-Kähler cases. The local form and the holonomy of pseudo-Kähler or para-Kähler manifolds admitting such structure are obtained. In addition the associated homogeneous models are studied exhibiting their relation with the incomplete…

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…

Research explores Kähler and semi-para-Kähler structures on specific Lie groups.

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

The paper explores $p$-Kähler structures on Lie group quotients.

The paper explores $p$-Kähler structures on complex manifolds and their properties.

Constructs special Kähler structures on Lie groups.

Study on existence of $p$-Kähler structures on nilmanifolds with nilpotent complex structures.

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähle…

We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.

Kähler manifold loop space inherits Kähler structure and is complete.

Study describes conformal product structures on compact Kähler manifolds.

We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…

Two Kähler structures are PCR equivalent in the Siegel domain.

Derives criteria for Kähler structures on holomorphic submersions.

We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.

Defines pre-Kähler structures and their properties.

The paper analyzes Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.

Spin-structures on real Bott manifolds with Kähler structure are characterized.

Study semi-Kähler structures on specific Lie groups without symplectic structures.

Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case.

We develop a theory of reduction for generalized Kahler and hyper-Kahler structures which uses the generalized Riemannian metric in an essential way, and which is not described with reference solely to a single generalized complex structure. We show that our construction specializes to the usual theory of Kahler and hy…

Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.

The paper explores Kähler and anti-Kähler structures on quasi-statistical manifolds.

New symplectic structures found on complex manifolds without Kähler structures.

Solves generalized Kähler Calabi-Yau problem on compact manifolds.

In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler …

Study para-complex structures on specific Lie groups, finding explicit forms and properties.

This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.

The paper explores families of almost complex structures and transverse (p,p)-forms.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.

We investigate the linear stability of Kähler-Ricci solitons for perturbations induced by varying the complex structure within a fixed Kähler class. We calculate stability for the known examples of Kähler-Ricci solitons.

New extremal Kähler metrics found on 4-manifolds with U(2) symmetry.

We study the class $\mathcal{K}_{2}+\mathcal{K}_{4}$ of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure is obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is …

Study the structure of Kähler foliations with negative Ricci curvature.

The paper describes a new type of Kähler surfaces and their properties.

It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we ob…

Study of Hermitian structures on Lie groups with 2D commutator subgroups.

Study on astheno-Kähler structures on product manifolds.

Complete classification of para-Kähler structures on Lie groups.

In this article, we consider the almost Hermitian structure on $TM$ induced by a pair of a metric and an affine connection on $M$. We find the conditions under which $TM$ admits almost Kähler structures, Kähler structures and Einstein metrics, respectively. Moreover, we give two examples of Kähler-Einstein structures o…

New structure found on Lie group tangent bundle.

Let $\,Ξ\,$ be the crown domain associated with a non-compact irreducible hermitian symmetric space $\,G/K$. We give an explicit description of the unique $\,G$-invariant adapted hyper-Kähler structure on $\,Ξ$,$\$i.$\,$e.$\$compatible with the adapted complex structure $\,J_{ad}\,$ and with the $\,G$-invariant Kähle…

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first example…

The paper studies deformations of astheno-Kähler metrics on complex manifolds.

We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting Kähler structures all have identical Chern classes. We construct Bott di…

Study of complex and Hermitian structures on specific Lie groups.