Vestislav Apostolov

Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability

This paper concerns the existence and explicit construction of extremal Kähler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of Hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory.We obtain a characterization, on a large family of projective bundles, of the ‘admissible’ Kähler classes (i.e., those compatible with the bundle structure in a way we make precise) which contain an extremal Kähler metric. In many cases every Kähler class is admissible. In particular, our results complete the classification of extremal Kähler metrics on geometrically ruled surfaces, answering several long-standing questions.We also find that our characterization agrees with a notion of K-stability for admissible Kähler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.

A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kahler–Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Without the compactness assumption, irreducible Kahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kahler metrics of any even real dimension greater than 4.

The Geometry of Weakly Self-dual Kähler Surfaces

We study Kähler surfaces with harmonic anti-self-dual 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 Kähler metrics, including Kähler–Einstein metrics and conformally Einstein Kähler metrics. We also extend some of our results to almost Kähler 4-manifolds, providing new examples of Ricci-flat almost Kähler metrics which are not Kähler.

Kähler reduction of metrics with holonomy G2

A torsion-free G2 structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G2.

A REMARK ON KÄHLER METRICS OF CONSTANT SCALAR CURVATURE ON RULED COMPLEX SURFACES

In this paper we point out how some recent developments in the theory of constant scalar curvature Kahler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces.

Local Rigidity of Certain Classes of Almost Kähler 4-Manifolds

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

Symplectic 4-manifolds with Hermitian Weyl tensor

It is proved that any compact almost Kihler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kihler.

Generalized Goldberg-Sachs theorems for pseudo-Riemannian four-manifolds

Abstract It has been recently observed that the generalized Goldberg-Sachs theorem in general relativity as well as some of its corollaries admit appropriate Riemannian versions. In this paper we use the formalism of spinors to give alternative proofs of these results clarifying the analogy between positive Hermitian structures of oriented Riemannian four-manifolds and shear-free congruences of oriented Lorentzian four-manifolds. We also prove similar results for oriented pseudo-Riemannian four-manifolds when the metric is of zero signature. This allows us to describe compact oriented four-manifolds possibly admitting a pseudo-Riemannian Einstein metric of zero signature whose positive Weyl tensor has two distinct eigenvalues corresponding to non-isotropic eigenspaces.

An integrability theorem for almost Kähler 4-manifolds†

Abstract We prove that every compact almost Kahler 4-manifold which satisfies the second curvature condition of Gray [4] is necessarily Kahler.

Compact self-dual Hermitian surfaces

In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kahler.