arXiv research

The paper explores Hamiltonian stationary Lagrangian fibrations in various geometries.

Holomorphic Euler number vanishes for certain Kähler manifolds.

The paper proves conditions for Kähler manifolds with negative curvature.

Develops Kähler geometry on new varieties for canonical metrics.

Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …

In this note we prove the following result: There is a positive constant $ε(n,Λ)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $Λ$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -ε(n,Λ)$, then $M^n$ is dif…

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

The paper studies Ricci curvature on Kähler-Ricci flow.

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizin…

The paper explores metric reduction in generalized geometry and constructs holomorphic bundles.

Positive scalar curvature implies small 2-systoles in Kähler manifolds

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …

Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.

In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the …

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

New constraints found for Lagrangian embeddings in symplectic fillings.

New Sasaki-Einstein structures found on odd connected sums of spheres.

The paper describes Calabi-Yau metrics on complex flag manifolds using Lie theory.

We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjectur…

New structures on symplectic manifolds derived from convex functions and matrices.

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk…

Book teaches how Lagrangian torus fibration base geometry can be read off.

Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.

The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.

Minimal hypersurfaces in nearly G2 manifolds have special properties.

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable po…

Study cohomology of quaternionic foliations and orbifolds.

Abstract revisits Kähler reduction using GIT, generalizing results.

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

Abstract: Characterizes special Kähler manifolds with specific properties.

The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galick…

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $…

Stability proven for complex equations on Kähler manifolds.

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kahler toric ALE spaces. A corollary of t…

Develops theory for Kähler-Ricci flow on singular varieties.

The paper proves stability for Einstein metrics with special twisted spinors.

Study of toric generalized Kähler structures with strong Hamiltonian torus actions.

The study proves compactness of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces.

We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifo…

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ and $\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ in dimension 11 and 15 respectively. The metric cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ is a generalization of the Kronheimer hyperkähle…

In 2D, a conjecture about the Ricci tensor of Kähler-Einstein metrics for convex bodies is verified.

A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyp…

The study shows that positive scalar curvature 4-manifolds can be desingularized.

The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.