Paul Gauduchon

Generalized cylinders in semi-Riemannian and spin geometry

Abstract.We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

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.

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.

Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4

Let (M , g, J ) be a Hermitian manifold of (real) dimension n = 2m > 2, where g is a Riemannian metric and J a g-orthogonal, integrable, almostcomplex structure on M . Let E be a holomorphic vector bundle of rank r over M . Then, a (fibred) Hermitian metric h on E is said to be Einstein (with respect to g) if the curvature R∇ of the associated Chern connection ∇ satisfies the following condition, cf. [7]:

Compact homogeneous lcK manifolds are Vaisman

We prove that any compact homogeneous locally conformally Kähler manifold has parallel Lee form.

Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr2(ℂn+2). We also prove that irreducible inner symmetric spaces M4n of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.

Hirzebruch Surfaces and Weighted Projective Planes

For any positive integer, we show that the standard self-dual orbifold Kahler structure of the weighted projective surface ℙ1,1,k can be realized as a limit of the Hirzebruch surface F k , equipped with a sequence of Calabi extremal Kahler metrics whose Kahler classes tend to the boundary of the Kahler cone, and that this collapsing process is compatible with the natural toric structures of ℙ1,1,k and F k .

Killing 2-Forms in Dimension 4

A Killing p-form on a Riemannian manifold (M, g) is a p-form whose covariant derivative is totally antisymmetric. If M is a connected, oriented, 4-dimensional manifold admitting a non-parallel Killing 2-form ψ, we show that there exists a dense open subset of M on which one of the following three exclusive situations holds: either ψ is everywhere degenerate and g is locally conformal to a product metric, or g gives rise to an ambikahler structure of Calabi type, or, generically, g gives rise to an ambitoric structure of hyperbolic type, in particular depends locally on two functions of one variable. Compact examples of either types are provided.

Weyl–Einstein structures on K-contact manifolds

We show that a compact K-contact manifold