2012/08/14 by Helge Øystein Maakestad, Maakestad, Helge Øystein
Mathematics · Physics and Astronomy · #14A05 #14C25 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Combinatorics #Complex Variables (math.CV) #Connection (principal bundle) #FOS: Mathematics #Geometry #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Mathematics #Matrix (chemical analysis) #Nabla symbol #Nonlinear Waves and Solitons #Omega #Pure mathematics
paper · pdf · doi:10.48550/arxiv.1208.2806
openalex publication_date 2012/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The aim of this paper is to give a new method to construct explicit formulas for algebraic differential operators of any order on a finitely generated projective module E on a commutative unital ring A. We moreover give explicit formulas for algebraic connections on a class of finitely generated projective modules on ellipsoid surfaces. The connections we construct are non-flat with trace of curvature equal to zero. We construct these formulas using an idempotent matrix M defining the module E. Such an idempotent matrix M is constructed from a "projective basis" B defining the module E. Associated to a projective basis B for E we construct a connection ∇B. The curvature R∇B of the connection ∇B is given by a Lie product: R∇B(x,y):=[∇B(x)(M), ∇B(y)(M)] involving the matrix M, and this Lie product is non-zero in general. Hence the curvature formula indicates that most projective finite rank modules do not have a flat algebraic connection. We also give an explicit formula for a non-flat algebraic connection on the cotangent bundle Ω of the real 2-sphere. The cotangent bundle Ω is topologically non-trivial and it is not clear if it has a flat algebraic connection. All higher Chern classes in deRham cohomology are zero: ci(Ω)=0 for all i ≥ 1. We relate the construction to non-abelian extensions and a refined characteristic class c(Ω) introduced in another paper on the subject. The class c(Ω) is defined using the connection ∇B but it is independent of choice of connection. The class c(-) lives in a torsor. The methods introduced in the paper prove that the underlying complex manifold of any complex affine regular hypersurface is a Calabi-Yau manifold. This is because its canonical bundle is trivial.