On tame ρ-quaternionic manifolds

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Abstract

We introduce the notion of tame ρ-quaternionic manifold that permits the construction of a finite family of ρ-connections, significant for the geometry involved. This provides, for example, the following:

• A new simple global characterisation of flat (complex-)quaternionic manifolds.

• A new simple construction of the metric and the corresponding Levi-Civita connection of a quaternionic-Kähler manifold by starting from its twistor space; moreover, our method provides a natural generalization of this correspondence.

Also, a new construction of quaternionic manifolds is obtained, and the properties of twistorial harmonic morphisms with one-dimensional fibres from quaternionic-Kähler manifolds are studied.

Introduction

A (complex) ρ-quaternionic manifold M is the parameter space of a locally complete family [8] of Riemann spheres embedded into a complex manifold Z, the twistor space of M, with nonnegative normal bundles [16].

In this paper, we introduce the notion of ‘tame’ ρ-quaternionic manifold (see Definition 2.2, below) that permits the construction, through the Ward transformation, of a finite family of ρ-connections, significant for the geometry of M (determined by Z). For example, if M is quaternionic then only one such ‘fundamental monopole’ exists (with respect to some line bundle over Z) and its flatness is equivalent to the flatness of M (Theorem 3.1). If, further, Z is endowed with a contact structure, our approach leads to a quick simple proof (Theorem 3.3) of the fact [9] that then M is quaternionic-Kähler. Moreover, our method provides a natural generalization of this fact (Theorem 3.4; compare [1]). Finally, this is related to the properties of twistorial harmonic morphisms with one-dimensional fibres which we study in Section 5, whilst a new construction of quaternionic manifolds is given in Section 4.

Section snippets

ρ-Quaternionic manifolds

For simplicity, unless otherwise stated, we work in the complex analytic category; also, the manifolds are assumed connected. A linear ρ-quaternionic structure [16] on a (complex) vector space U is given by a (holomorphic) embedding of the Riemann sphere Y into Gr(U) such that the corresponding tautological exact sequence of vector bundles0⟶F⟶Y×U⟶U⟶0 induces an isomorphism between U and the space of sections of U. Consequently, H0(F) and H1(F) are trivial and, thus, F=L⁎⊗V, for some vector

On tame ρ-quaternionic manifolds

In this section, the notations are as in Section 1. We start with a useful result which is, also, interesting in itself, as it can be seen as an extension of both the Ward transformation and the filtration given by the Birkhoff–Grothendieck theorem.

Proposition 2.1

Let M be a ρ-quaternionic manifold with twistor space Z. Let F be a vector bundle over Z whose restriction to each twistor sphere has the same isomorphism type ⨁j=1j=kajO(nj), for some k∈N∖{0}, and aj∈N∖{0}, nj∈Z, j=1,…,k, with nj≥nj+1+2, for any j=1,

Quaternionic manifolds

The quaternionic manifolds are characterised, among the ρ-quaternionic manifolds, by the fact that the Birkhoff–Grothendieck decompositions of the normal bundles of the twistor spheres contain only terms of Chern number 1. Consequently, they are tame ρ-quaternionic manifolds with only one fundamental monopole. Also, note that, the dimension of any quaternionic manifold is even. Furthermore, for any line bundle, over the twistor space of a quaternionic manifold, whose restriction to some twistor

A construction of quaternionic manifolds

In this section we consider ρ-quaternionic manifolds (of constant type and) with equivariant [19] normal sequences (of the twistor spheres). This means that the exact sequence0⟶kerdπ⟶ψ⁎(TZ)⟶TM⟶0 (obtained from 0⟶kerdπ⟶TY⟶π⁎(TM)⟶0 by taking quotients over kerdψ) is invariant under the automorphism bundle Aut Y; in particular, Aut Y is embedded into the extended automorphism bundle P of (4.1) as a section of the obvious bundle morphism from P onto Aut Y.

This implies that a distinguished ρ

On twistorial harmonic morphisms with one-dimensional fibres

In this section, although some of the results hold in more generality, for simplicity, M will denote a real(-analytic) quaternionic-Kähler or hyper-Kähler manifold. To unify the notations of this and the previous sections, one just has to replace in the latter M with MC. Further, as, now, ψ restricted to π−1(M) is a diffeomorphism, the diagram giving the twistor space simplifies, as it is well known, to a fibration whose total space and projection we will denote by Y and π, respectively

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