Calibrated fibrations of compact manifolds with special holonomy

Thesis

We study compact manifolds of special holonomy G₂ ⊂ SO(7) and Spin(7) ⊂ SO(8) and their calibrated submanifolds, the coassociative and Cayley submanifolds. These are minimal submanifolds arising from the holonomy restriction. Calabi–Yau fourfolds appear as special examples of Spin(7)-manifolds.

Physicists expect that a Calabi–Yau threefold X admits a mirror X̂ (where the complex geometry of X is equivalent to the symplectic geometry of X̂ and vice-versa). A proposed geometric explanation, the SYZ conjecture, stipulates that both fiber over the same base B³ with (possibly singular) calibrated torus fibres that are dual to one another.

We study the analogous existence problem of calibrated fibrations in the Spin(7) case and prove that Cayley fibrations of compact Spin(7)-manifolds, where fibres may admit certain types of conical singularities, are stable under small deformations of the Spin(7)-structure. More precisely, we require all the fibres to be unobstructed in their respective moduli spaces and the cones to have well-behaved critical rates. Furthermore, the singular locus should be of codimension at least 2 in the base and the asymptotically conical Cayleys required for the desingularisation of singular fibres should have deformations of a unique asymptotic rate. Complex fibrations of Calabi–Yau fourfolds with at worst Morse-type singularities satisfy all of these conditions.

As an application, we prove the existence of coassociative Kovalev-Lefschetz fibrations of G₂-manifolds arising as twisted connected sums. We present an explicit example of a coassociative fibration on the twisted connected sum G₂-manifold obtained from two quartic building blocks. This completes the program initiated by Kovalev to find examples of coassociative fibrations using gluing methods [24].

Along the way we revisit the deformation theory of compact Cayley submanifolds (McLean, Clancy, Moore) and conically singular Cayley submanifolds (Moore) and describe the deformation theory of asymptotically conical Cayley submanifolds of ℝ⁸. We do this in the unifying framework of families of almost Cayley submanifolds (whose tangent bundles are close to a bundle of Cayley planes) in not necessarily torsion-free Spin(7)-manifolds, and define a canonical deformation operator even for submanifolds that are not Cayley. This generalises a number of results in the existing literature.

Furthermore, we study the desingularisation theory of conically singular Cayley sub-manifolds by attaching asymptotically conical submanifolds at the singularities and prove a general gluing theorem. As an application, we determine when immersed points of Cayley submanifolds may be smoothed by gluing in a Lawlor neck.

Authors: Englebert, G

• DOI: 10.5287/ora-2zepj08az
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: differential geometry; special holonomy; G2; Calyley submanifold; Spin(7)
• Subjects: Holonomy groups; Geometry, Differential; Calibration