Related papers: Bochner-Kahler metrics

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…

The property of admitting an astheno-K\"ahler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence…

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the…

Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $\beta \in (0,1)$. We show that in any positive co-homology class on $X$ there is a K\"ahler metric with cone angle $2\pi\beta$ along $D$ which has bounded Ricci…

We continue the study of the geometry and topology of compact submanifolds of arbitrary codimension in space forms that satisfy a pinching condition involving the length of the second fundamental form and the mean curvature. Our primary…

Let $X$ be a complex manifold and let $g$ be a polyhedral metric on it inducing its topology. We say that $g$ is a polyhedral K\"ahler (PK) metric on $X$ if it is K\"ahler outside its singular set. The local geometry of PK metrics is…

In general relativity, astrophysical black holes are uniquely described by the Kerr metric. Observational tests of the Kerr nature of these compact objects and, hence, of general relativity, require a metric that encompasses a broader class…

Let $f\colon M^{2n}\to\mathbb{R}^{2n+\ell}$, $n \geq 5$, denote a conformal immersion into Euclidean space with codimension $\ell$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $\ell=1,2$ we show…

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c--projective geometry, which is the almost complex analogue of projective differential geometry.…

In this note we present, for every $n \geq 4$, a non-K\"ahler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-K\"ahler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…

We study how the existence of a negatively pinched K\"ahler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete K\"ahler metric, with pinched…

In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact K\"ahler manifold $X$: this notion include constant scalar curvature K\"ahler metrics, weighted solitons, Calabi's…

We consider a compact K\"ahler manifold admitting a constant scalar curvature K\"ahler metric and with no nontrivial holomorphic vector fields. After blowing up the manifold at finitely many points, we prove the existence of constant scalar…

It is proved that if an almost Hermitian manifold of dimension greater than 4 has vanishing (classical) Bochner curvature tensor and is not Kaehlerian at a point, then it is flat in a neighbourhood of this point.

Given a semi-Riemannian $4$-manifold $(M,g)$ with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of K\"ahler metrics $g_K$ is constructed, defined on an…

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article…

A locally conformally K\"ahler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a K\"ahler metric in the neighbourhood of each point. In this paper we obtain three classification…

In this note we study the Bochner formula on smooth metric measure spaces. We introduce weighted curvature conditions that imply vanishing of all Betti numbers.

The Bochner technique is a classical tool in global differential geometry for proving vanishing and rigidity results by exploiting curvature conditions. Building on recent extensions of this method to complete non-compact settings by…