On toric generators in the unitary and special unitary bordism rings

TL;DR

This paper constructs new families of toric and special unitary quasitoric manifolds that generate the unitary and special unitary bordism rings, respectively, using complex projectivisations and modifications to achieve desired Chern class properties.

Contribution

It introduces explicit constructions of toric and special unitary quasitoric manifolds that serve as generators for their respective bordism rings, expanding the known algebraic generators.

Findings

Constructed a family of toric manifolds generating the unitary bordism ring.

Developed a family of special unitary quasitoric manifolds with polynomial generators.

Achieved manifolds with vanishing first Chern class through complex structure modifications.

Abstract

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.