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David M. J. Calderbank

Publications and source records attributed to David M. J. Calderbank.

At least 37 records · Page 2Linked to original sources
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.

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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.

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arXiv2005

Continued fractions and Einstein manifolds of infinite topological type

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient singularities (governed by continued fraction expansions of rational numbers) on which we constructed complete self-dual Einstein metrics in previous work.

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arXiv2005

Toric selfdual Einstein metrics on compact orbifolds

We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.

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arXiv2004

Ricci-corrected derivatives and invariant differential operators

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time extend these formulae from the context of AHS structures (which include conformal and projective structures) to the more general class of all parabolic structures (including CR structures).

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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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arXiv2004

Hamiltonian 2-forms in Kahler geometry, I General Theory

We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective) line bundles. They also arise on conformally-Einstein Kaehler manifolds. We explore these connections and ramifications.

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arXiv2003

Toric self-dual Einstein metrics as quotients

We use the quaternion Kahler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kahler quotients of (semi-)quaternion Kahler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov-Gauduchon and Bryant.

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arXiv2002

Einstein metrics and complex singularities

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 instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kaehler metric (which is hyperkaehler if and only if c_1(X)=0), and that if c_1(X)<0 then X also admits a complete (non-Kaehler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

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arXiv2001

Selfdual Einstein metrics with torus symmetry

It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain `multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kahler quotients of HP(m-1) by an (m-2)-torus first studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kahler quotients of HP(2) (and noncompact analogues), and the more recently studied selfdual Einstein hermitian metrics in a unified framework, and give new complete examples.

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arXiv2001

The geometry of weakly selfdual Kahler surfaces

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 some of our results to almost Kahler 4-manifolds, providing new examples of Ricci-flat almost Kahler metrics which are not Kahler.

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arXiv2000

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A-infinity algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

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arXiv2000

Two dimensional Einstein-Weyl structures

All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples which I obtained in "Moebius structures and two dimensional Einstein-Weyl geometry" J. reine angew. Math. 504 (1998).

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arXiv2000

The geometry of the Toda equation

I show that solutions of the SU(infinity) Toda field equation generating a fixed Einstein-Weyl space are governed by a linear equation on the Einstein-Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein-Weyl space are found. I also give a classification of Einstein-Weyl spaces arising from the Toda equation in more than one way. This classification coincides with a class of spaces found by Ward and hence clarifies some of their properties. I end by discussing the simplest examples.

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arXiv2000

Selfdual Einstein metrics and conformal submersions

Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compatible Einstein metric, are obtained from solutions of the resulting ``affine monopole equations''. The ``projective monopole equations'' encompass Hitchin's twistorial construction of selfdual Einstein metrics from three dimensional Einstein-Weyl spaces, and lead to an explicit formula for carrying out this construction directly. Examples include new selfdual Einstein metrics depending explicitly on an arbitrary holomorphic function of one variable or an arbitrary axially symmetric harmonic function. The former generically have no continuous symmetries.

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arXiv1999

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new Einstein-Weyl spaces, which we call Einstein-Weyl ``with a geodesic symmetry'', give rise to hypercomplex structures with two commuting triholomorphic vector fields.

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