arXiv research

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…

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…

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

Study characterizes toric LCK manifolds, proving conjecture and showing differences from symplectic case.

Study on compact toric locally conformally Kähler manifolds, finding specific properties.

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 …

New metrics found on toric LCS manifolds.

Study of finite energy quasiplurisubharmonic functions on toric Kähler manifolds.

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…

Classification of toric self-dual Einstein gravitational instantons with negative cosmological constant

We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sens…

Develops Kähler geometry on new varieties for canonical metrics.

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

We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of o…

The paper studies Ricci curvature on Kähler-Ricci flow.

We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki…

Study of special Kato manifolds derived from toric geometry.

The abstract investigates convexity in locally conformally symplectic geometry.

This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.

Uniform K-stability ensures existence of special metrics on toric manifolds.

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech …

We prove local well-posedness of the Schrödinger flow from $R^n$ into a compact K\{"a}hler manifold $N$ with initial data in $H^{s+1}(R^n, N)$ for $s\geq n/2+4$.

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal str…

Proves Witten genus vanishes for certain string intersections.

The paper develops quaternionic toric geometry and classifies local actions.

Formula proves invariant matches for smooth and orbifold test configurations.

Local equivalence found between certain solitons and generalized Kähler-Ricci solitons.

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…

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…

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

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…

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on $\mathbb{CP}^{2}\sharp2\overline{\mathbb{CP}}^{2}$, which is conformal to an Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various g…

The paper examines the topology of quaternionic toric actions on manifolds.

Classifies all toric Kahler surfaces with twistor 2-forms.

Study cohomology of quaternionic foliations and orbifolds.

Abstract revisits Kähler reduction using GIT, generalizing results.

We give a direct geometric proof of a Danilov-type formula for toric origami manifolds by using the localization of Riemann-Roch number.

Study toric gravitational instantons using rod structures and inequalities.

Holomorphic Euler number vanishes for certain Kähler manifolds.

Existence proved for specific types of gravitational instantons.

Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.

A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" action-angle coordinat…

We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A a…

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

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…