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Paul Gauduchon

Publications and source records attributed to Paul Gauduchon.

At least 19 recordsLinked to original sources
arXiv2023

Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry

In a previous paper, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this article, we consider general Ricci-flat deformations of such spaces, assuming only suitable fall-off conditions. Quite generally, we are able to show that such a deformation must be Hermitian, and must carry a non-trivial Killing vector field with fixed asymptotics. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must in fact belong to the family of previously classified metrics.

math.DG↗
arXiv2023

About a Family of ALF Instantons with Conical Singularities

We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities.

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arXiv2022

On toric Hermitian ALF gravitational instantons

We give a classification of toric, Hermitian, Ricci flat, ALF Riemannian metrics in dimension 4, including metrics with conical singularities. The only smooth examples are on one hand the hyperKaehler ALF metrics, on the other hand, the Kerr, Taub-NUT and Chen-Teo metrics. There are examples with conical singularities with infinitely many distinct topologies. We provide explicit formulas.

math.DG↗
arXiv2019

Non-existence of orthogonal coordinates on the complex and quaternionic projective spaces

DeTurck and Yang have shown that in the neighbourhood of every point of a $3$-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex projective spaces $\mathbb{CP}^m$ and the quaternionic projective spaces $\mathbb{HP}^q$, endowed with their canonical metrics, do not have local systems of orthogonal coordinates for $m,q\ge 2$.

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arXiv2017

Toric contact geometry in arbitrary codimension

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

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arXiv2017

Levi-Kahler reduction of CR structures, products of spheres, and toric geometry

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.

math.DG↗
arXiv2015

Killing 2-forms in dimension 4

A Killing $p$-form on a Riemannian manifold is a $p$-form whose covariant derivative is totally anti-symmetric. In this paper we give the complete (local) description of 4-dimensional Riemannian manifolds (M,g) carrying non-parallel Killing 2-forms $φ$. If $M$ is connected and oriented, we show that there exists a dense open subset of $M$ on which one of the three exclusive situations holds: either $ϕ$ is everywhere degenerate and $g$ is conformal to a product metric, or $g$ is conformal to an ambikähler metric obtained via the Calabi construction from a polarized Riemannian surface, or $g$ is conformal to an ambitoric structure of hyperbolic type, and depends locally on two functions of one variable. We also give compact examples, by constructing infinite-dimensional families of Riemannian metrics carrying Killing 2-forms of each of the above types on $S^4$ and on Hirzebruch surfaces.

math.DG↗
arXiv2013

Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds

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 one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to either orientation. This can be considered as a riemannian analogue of a result in General Relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. We discuss toric compactifications of these metrics on orbifolds and provide infinite discrete families of compact toric extremal Kaehler orbifolds. Our examples include Bach-flat Kaehler orbifolds which are conformal to complete smooth Einstein metrics on an open subset. We illustrate how these examples fit with recent conjectures relating the existence of extremal toric metrics to various notions of stability.

math.DG↗
arXiv2013

Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures

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 and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.

math.DG↗
arXiv2013

Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds

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 sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.

math.DG↗
arXiv2011

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 $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.

math.DG↗
arXiv2010

Extremal Kähler metrics on projective bundles over a curve

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to Σ$ over a compact complex curve $Σ$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of $E$ as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in sufficiently small Kähler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.

math.DG↗
arXiv2005

Hamiltonian 2-forms in Kahler geometry, III Compact examples

This paper has been withdrawn in order to replace it by two separate submissions: 1. Hamiltonian 2-forms in Kahler geometry III: Extremal metrics and stability, math.DG/0511118; 2. Hamiltonian 2-forms in Kahler geometry IV: Weakly Bochner-flat Kahler manifolds, math.DG/0511119. As the titles indicate, the first paper covers the parts of this withdrawn submission concerning extremal Kahler metrics, while the second one deals the weakly Bochner-flat Kahler metrics. However, the material for the first paper has been substantially revised and extended with several new results. 1. The exposition has been expanded and clarified, and some technical errors and missing arguments have been corrected. 2. A new computation of the modified K-energy is used to obtain a characterization of the admissible Kahler classes which contain an extremal Kahler metric. In particular, these results complete the classification of extremal Kahler metrics on ruled surfaces. 3. The existence of extremal Kahler metrics is related to the notion of relative K-stability leading in particular to some examples of projective varieties which are destabilized by a non-algebraic degeneration. We believe that these results add considerable interest to our work, and go far beyond the original paper, which is why we have chosen to withdraw this paper and post the replacements as new submissions.

math.DG↗
arXiv2005

Hamiltonian 2-forms in Kahler geometry, IV Weakly Bochner-flat Kahler manifolds

We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series. It follows that WBF Kahler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kahler 6-manifolds, extending work by the first three authors on weakly selfdual Kahler 4-manifolds. The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kahler manifolds with a hamiltonian 2-form in math.DG/0401320 as well as some of the results in math.DG/0511118.

math.DG↗
arXiv2005

Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability

This paper concerns the explicit construction of extremal Kaehler 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 those `admissible' Kaehler classes (i.e., the ones compatible with the bundle structure in a way we make precise) which contain an extremal Kaehler metric. In many cases, such as on geometrically ruled surfaces, every Kaehler class is admissible. In particular, our results complete the classification of extremal Kaehler 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 Kaehler 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.

math.DG↗
arXiv2004

Hamiltonian 2-forms in Kahler geometry, II Global Classification

We present a classification of compact Kaehler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest. The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kaehler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kaehler manifolds with a rigid hamiltonian torus action are bundles of toric Kaehler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kaehler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of Kaehler--Einstein 4-orbifolds. Combining these two themes, we prove that compact Kaehler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kaehler products, and we describe explicitly how the Kaehler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kaehler metrics - in particular a subclass of such metrics which we call weakly Bochner-flat. We also provide a self-contained treatment of the theory of compact toric Kaehler manifolds, since we need it and find the existing literature incomplete.

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