Explicit formulas for algebraic connections on ellipsoid surfaces

Abstract

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 $\nabla_B$. The curvature $R_{\nabla_B}$ of the connection $\nabla_B$ is given by a Lie product: $R_{\nabla_B}(x,y):=[\nabla_B(x)(M), \nabla_B(y)(M)]$ 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 $\Omega$ of the real 2-sphere. The cotangent bundle $\Omega$ 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: $c_i(\Omega)=0$ for all $i \geq 1$. We relate the construction to non-abelian extensions and a refined characteristic class $c(\Omega)$ introduced in another paper on the subject. The class $c(\Omega)$ is defined using the connection $\nabla_B$ 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.