D-orbifolds and d-bordism

Thesis

The purpose of this thesis is to study d-manifolds and d-orbifolds and their bordism groups. D-manifolds and d-orbifolds were recently introduced by Joyce as a new class of geometric objects to study moduli problems in algebraic and symplectic geometry. In the spirit of Joyce we will introduce the notion of (stable) nearly and homotopy complex structures on these 2-categories and study their unitary bordism groups. Fukaya and Ono proved that the moduli space of n-pointed, genus g, J-holomorphic curves M_g,n(M,J,β) carries a so called stably almost complex structure, and as Kuranishi spaces are closely related to d-orbifolds, the introduction of complex structures will be essential in studying symplectic Gromov-Witten invariants using d-orbifolds.

We furthermore introduce the notion of representable d-orbifolds and prove that these representable d-orbifolds can be embedded into an orbifold. We will then explain how a result of Kresch can be used to show that many important moduli spaces in algebraic geometry, are representable and thus embeddable d-orbifolds. Moreover we will sketch how one could prove an analogous result in the symplectic case.

We then prove as one of our main results, that for a compact manifold the unitary d-bordism group is isomorphic to its ‘classical’ unitary bordism group. This result extends a result by Joyce who proved a similar statement for oriented manifolds and d-manifolds.

Furthermore we will introduce the notion of blowups in the 2-category of d-manifolds and prove that these d-blowups satisfy a universal property. Finally, we sketch how our results may be used to make a step towards a proof of the Gopakumar–Vafa integrality conjecture and a “resolution of singularities” theorem for d-orbifolds.

Authors: Volk, B; Benjamin Volk

• Publication date: 2014
• DOI: 10.5287/ora-agd21ggmy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English