[Paper Review] On the twistor space of a (co-)CR quaternionic manifold

Radu Pantilie|arXiv (Cornell University)|Jan 18, 2012

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper provides a deformation-theoretic characterization of twistor spaces for (co-)CR quaternionic manifolds, establishing that a complex manifold is a twistor space if it carries a locally complete family of rational curves with positive normal bundles and constant first cohomology dimension. The key contribution is a geometric correspondence linking complex manifolds with conjugations and embedded spheres to co-CR quaternionic structures, enabling the construction of new examples via dimensional reduction and twistor retractions.

ABSTRACT

We characterise, in the setting of the Kodaira-Spencer deformation theory, the twistor spaces of (co-)CR quaternionic manifolds. As an application, we prove that, locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold is endowed with a natural co-CR quaternionic structure. Also, for any positive integers $k$ and $l$, with $kl$ even, we obtain the geometric objects whose twistorial counterparts are complex manifolds endowed with a conjugation without fixed points and which preserves an embedded Riemann sphere whose normal bundle is $l$ times the line bundle of Chern number $k$. We apply these results to prove the existence of natural classes of co-CR quaternionic manifolds.

Motivation & Objective

• To characterize the twistor spaces of (co-)CR quaternionic manifolds using Kodaira–Spencer deformation theory.
• To establish a correspondence between complex manifolds with conjugations and embedded spheres and co-CR quaternionic structures.
• To prove that the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold inherits a natural co-CR quaternionic structure locally.
• To construct geometric objects whose twistorial counterparts are complex manifolds with conjugations preserving embedded spheres of specified normal bundle.

Proposed method

• Use of Kodaira–Spencer deformation theory to characterize twistor spaces via conditions on the normal bundle and cohomology of twistor lines.
• Identification of necessary and sufficient conditions for a complex manifold to be a twistor space: split tangent exact sequence and constant h^1 dimension for normal bundles.
• Application of the theory to show that the leaf space of a nowhere zero quaternionic vector field on a quaternionic manifold carries a natural co-CR quaternionic structure.
• Construction of examples via rational ruled surfaces and holomorphic sections with prescribed normal bundles under conjugations.
• Use of twistor retractions to relate generic submanifolds in quaternionic manifolds to co-CR structures via pullback of twistor data.
• Explicit construction of twistor spaces for Grassmannians and jet spaces to realize all possible (k,l) pairs in the correspondence.

Experimental results

Research questions

• RQ1 Under what conditions is a complex manifold the twistor space of a co-CR quaternionic manifold?
• RQ2 How can the leaf space of a nowhere zero quaternionic vector field on a quaternionic manifold be endowed with a co-CR quaternionic structure?
• RQ3 What is the geometric correspondence between complex manifolds with conjugations and embedded spheres and co-CR quaternionic structures?
• RQ4 For which integers k and l does there exist a complex manifold with conjugation preserving a sphere with normal bundle l𝒪(k)?
• RQ5 How do twistor retractions and holomorphic sections relate to the construction of co-CR quaternionic manifolds?

Key findings

• A complex manifold Z is the twistor space of a co-CR quaternionic manifold if and only if it carries a locally complete family of rational curves whose normal bundles are positive and the tangent exact sequence splits, with constant h^1 dimension of Tt ⊗ N* for each curve t.
• Locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold naturally inherits a co-CR quaternionic structure.
• For any positive integers k and l with kl even, there exists a geometric correspondence between complex manifolds with conjugation preserving a sphere with normal bundle l𝒪(k) and quadruples (M,N,x,φ) where N is quaternionic, M is a generic submanifold of type (k,l), and φ is a twistorial retraction.
• The twistor space of the Grassmannian of 3-planes in R^{n+3} is a hyperquadric in CP^{n+2}, and its normal bundle on twistor lines is n𝒪(2), providing a concrete example of the correspondence.
• Examples of twistor spaces are constructed via jet spaces of maps from CP^1 to itself and via rational ruled surfaces S_n, yielding co-CR quaternionic structures for all k and l with kl even.
• The construction yields hyper co-CR quaternionic manifolds as products of the twistor spaces from Examples 4.4 and 4.5, covering all admissible (k,l) pairs.