arXiv research

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…

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…

Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler structures of symplectic type, without requiring them to be anti-diagonal. Such a structu…

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 …

We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a ce…

Study cohomology of quaternionic foliations and orbifolds.

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…

We prove a bubble tree convergence theorem for a sequence of closed Hamiltonian Stationary Lagrangian surfaces with bounded areas and Willmore energies in a complete K{ä}hler surface. We also prove two strong compactness theorems on the space of Hamiltonian stationary Lagrangian tori in $\mathbb C^2$ and $\mathbb{CP}^2…

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 $…

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…

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

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

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…

We revisit generalized K$\ddot{a}$hler reduction introduced by Lin and Tolman in \cite{LT} from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary K$\ddot{a}$hler reduction can be generalized without much ef…

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

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

The paper studies Ricci curvature on Kähler-Ricci flow.

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…

The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.

In a recent paper (math.DG/0701278) we constructed a series of new Moishezon twistor spaces which is a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on nCP^2 for arbitrary n>2, which can be regarded as a generalizati…

In this paper we investigate a family of Moishezon twistor spaces on the connected sum of 4 complex projective planes, which can be regarded as a direct generalization of the twistor spaces on 3CP^2 of double solid type studied by Poon and Kreussler-Kurke. These twistor spaces have a natural structure of double coverin…

Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.

The paper examines curvature properties of twistor spaces.

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…

In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied by Poon and Kreussler-Kurke to the case of arbitrary signature. In particular, the branch divisor o…

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space-time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the 6-dimensional case in which twisto…

Study geometric structures on twistor and reflector spaces of paraquaternionic contact manifolds.

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their ho…

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

Study of complex structures on product twistor spaces for 4D manifolds.

We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three comp…

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

Study of twistor spaces and minitwistor spaces for ALE gravitational instantons.

The paper connects complex spacetimes to twistor spaces and finds an almost contact structure.

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

In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called…

Survey of twistor lifts of surfaces in 4-dimensional spaces.

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this s…

Develops twistor theory for foliated manifolds, proving orbifold results.

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian spaces as generalizations of conformal transforms. We introduce the nearly autoparal…

In this paper we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalizes the twistor space of quaternion-Hermitian manifolds and weak-Spin(9) structures. We also construct almost complex structures on the twistor space for parallel even Clifford structures and check…

In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the a…

New examples of structures found using twistor space.

Twistor space of hypercomplex manifolds is never Moishezon.

We show that the algebraic dimension of a twistor space over n#CP^2 cannot be two if n>4 and the fundamental system (i.e. the linear system associated to the half-anti-canonical bundle, which is available on any twistor space) is a pencil. This means that if the algebraic dimension of a twistor space on n#CP^2, n>4, is…

We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map equation …