Kaehler structures on spin 6-manifolds

TL;DR

This paper investigates the topological and geometric properties of spin 6-manifolds, showing many cannot admit Kaehler structures and analyzing the finiteness of deformation and Chern number types.

Contribution

It demonstrates that many spin 6-manifolds share homotopy types with Kaehler manifolds but differ in homeomorphism types, and establishes finiteness results for deformation types and Chern numbers.

Findings

Many spin 6-manifolds have the same homotopy type as Kaehler manifolds but are not homeomorphic.

Finitely many deformation types of smooth complex projective spin threefolds of general type exist for given Betti numbers.

On a fixed spin 6-manifold, Chern numbers are finitely many across all Kaehler structures.

Abstract

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kaehler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projectve spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kaehler structures.