A universal characteristic class for vector bundles with a connection
Helge Øystein Maakestad
TL;DR
The paper defines a universal fundamental class $c(E)\in \operatorname{Ext}^1(L, \operatorname{End}_A(E))$ for vector bundles with connection, presented as a pointed cohomology torsor under $\operatorname{H}^2(L, Z(\operatorname{End}_A(E)))$ and independent of the chosen connection. It proves $c(E)$ vanishes exactly when $E$ has a flat connection and uses this to obstruct algebraic parallelizability via $c(T_S)$, connecting Teleman-type invariants with classical characteristic classes. The author provides explicit curvature formulas by lifting connections to free modules via idempotents, and constructs non-flat, curvature-type examples on regular hypersurfaces and on the cotangent bundle of the real 2-sphere. The work also computes the algebraic de Rham cohomology of the complex two-sphere, showing $H^1_{DR}(S^2)$ is infinite dimensional, and develops a detailed torsor structure for Ext$^1(L, \operatorname{End}_A(E))$ with an action of $\operatorname{H}^2(L, Z(\operatorname{End}_A(E)))$, illuminating how curvature, flatness, and higher characteristic data interact in the algebraic setting. Collectively, these results provide a framework linking algebraic connections, higher invariants, and geometric questions such as the cancellation problem and algebraic Euler characteristics.
Abstract
In the paper I introduce a new characteristic class $c(E)$ for a finite rank vector bundle $E$ on an affine scheme $S:=Spec(A)$ - the fundamental class of $E$. The class $c(E)$ is not a characteristic class in the classical sense in the sense that it lives in a pointed cohomology torsor $\operatorname{Ext}^1(L, \operatorname{End}_A(E))$. Most characteristic classes lives in a cohomology group. The pointed cohomology torsor $\operatorname{Ext}^1(L, \operatorname{End}_A(E))$ is a torsor on the abelian group $\operatorname{H}^2(L, Z(\operatorname{End}_A(E)))$ where $Z(\operatorname{End}_A(E))$ is the center of the ring of endomorphisms of $E$ and where the cohomology is the Lie-Rinehart cohomology of the center. The class $c(E)$ is trivial if and only if $E$ has a flat algebraic connection. Hence the class $c(T_S)$ where $T_S$ is the tangent bundle, is an an obstruction for $S$ to be algebraically parallelizable. I use a connection $\nabla$ to define $c(E)$ and I also prove the class $c(E)$ is independent of choice of connection, hence $c(E)$ is an invariant of the vector bundle $E$. The class generalize the Chern class, the Pontryagin class, the Euler class and the Teleman characteristic class. I prove using an explicit example that the class $c(E)$ is stronger than the Chern class and the Euler class. I also give a new proof of a formula for the curvature of a connection $\nabla$ in terms of an idempotent endomorphism $φ$ defining $E$. This formula was claimed and proved in a paper put out on the arXiv in 2011, and in this paper I give a new proof that is easier to read. The class may be interesting in the study of the "cancellation problem" in affine algebraic geometry and the problem of giving algebraic formulas for the topological Euler characteristic. I also calculate the algebraic deRham cohomology of the complex two sphere and prove it is infinite dimensional.