arXiv research

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.

We show the existence of strictly almost-Kahler anti-self-dual metrics on certain 4-manifolds by deforming scalar-flat Kahler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of Seiberg-Witten theory. In the process, we provide a simple new proof of the fact t…

If $W_+$ denotes the self dual part of the Weyl tensor of any Kähler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost Kähler 4-manifold with $S \ge 0$, this condition forces the Kähler property. A compact almost Kähler 4-manifold is already Kähler if it satisfie…

We show that any non-Kahler, almost Kahler 4-manifold for which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kahler metric is locally isometric to the (only) proper 3-symmetric 4-dimensional space described by O. Kowalski.

We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gr…

Study proves existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

Study of gauge-theoretic functionals leading to constant scalar curvature almost-Kahler 4-manifolds.

We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Further…

We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.

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

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Conditions for the existence of Kähler-Einstein metrics and central Kähler metrics [MS] along with examples, both old and new, are given on classes of Lorentzian $4$-manifolds with two distinguished vector fields. The results utilize the general construction [AM] of Kähler metrics on such manifolds. The examples includ…

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…

The paper constructs symplectic forms on 4-manifolds using branched coverings and holomorphic line bundles.

In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern cla…

We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible m…

Computes Seiberg-Witten invariants for Kähler families of 4-manifolds.

Proves Kähler-Einstein property for certain Einstein 4-manifolds.

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classificatio…

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observa…

Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.

In this article we show that every closed oriented smooth 4-manifold can be decomposed into two codimension zero submanifolds (one with reversed orientation) so that both pieces are exact Kahler manifolds with strictly pseudoconvex boundaries and that induced contact structures on the common boundary are isotopic. Mean…

We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compa…

Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.

Study on 4D Einstein manifolds with Kähler conformal geometry.

Paper solves a metric-independent problem on almost Kähler 4-manifolds.

Study finds all hyper-Kähler 4-manifolds with specific symmetries and structures.

Classifies scalar-flat toric Kähler instantons in 4D.

We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including Kahler-Einstein metrics and conformally Einstein Kahler metrics. We also extend s…

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.

The paper studies point vortex dynamics on specific Kähler twistor spaces.

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…

Study on 4-manifolds for special Kähler metrics with constant Ricci determinant.

A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We …

Study on smooth moduli spaces of branes in symplectic 4-manifolds.

New proof of instability for certain Einstein metrics.

Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with t…

Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.

Estimates symplectic Calabi-Yau equation using Cheng-Yau method.

The Einstein-Maxwell equations on a smooth compact 4-manifold are reformulated as a purely Riemannian variational problem analogous to Calabi's variational problem for extremal Kahler metrics. Next, Seiberg-Witten theory is used to show that these two problems are in fact intimately related. Extremal Kahler metrics are…

Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.

The paper solves a conjecture on almost complex 4-manifolds using refined Dolbeault cohomology.

New findings on Kähler manifolds restrict orthogonal coordinates existence.

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-Kähler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-Kähler is given.