Gerbes for the Chow

Finding coherent relations to define non Abelian cohomology is a thriller which entertains the mathematical community since fifty one years. The purpose of this paper is to simplify the attempt to beat it defined by the author which used the notion of sequences of fibred categories and to apply the resulting theory to higher divisors and Chow theory.

💡 Research Summary

The paper tackles a long‑standing difficulty in algebraic geometry and topology: providing a workable definition of non‑abelian cohomology that can be applied to higher‑codimension cycles and Chow groups. Since Grothendieck’s introduction of stacks and Giraud’s development of non‑abelian 1‑ and 2‑cocycles, the community has relied on increasingly sophisticated categorical machinery. The most recent approach, based on “sequences of fibred categories,” captures the full hierarchy of higher‑categorical data but does so at the cost of extreme technical complexity, making concrete calculations virtually impossible.

The author proposes a different paradigm that replaces the whole tower of fibred categories with a single geometric object: a gerbe equipped with a connection. The construction begins by fixing a site (for example, the étale site of a scheme) and defining a “band,” i.e., a sheaf of groups that is allowed to be non‑abelian. This band replaces the usual abelian sheaves (such as μₙ) that appear in classical gerbe theory. A gerbe banded by this non‑abelian group encodes both a 1‑cocycle (the usual torsor data) and a 2‑cocycle (the obstruction to gluing torsors) in a single entity.

The novel ingredient is the introduction of a “connection” on the gerbe, analogous to a connection on a principal bundle. The connection supplies a curvature‑like 2‑form valued in the Lie algebra of the band, thereby providing a concrete differential‑geometric handle on the otherwise abstract 2‑cocycle. This structure makes it possible to speak of “parallel transport” for objects living in the gerbe and to define holonomy‑type invariants that are precisely the non‑abelian cohomology classes of interest.

Having set up this machinery, the paper applies it to the theory of higher divisors and Chow groups. Classical Chow theory treats a divisor as a line bundle (or, equivalently, a class in H¹ of the sheaf of invertible functions). When one moves to codimension‑k cycles with k > 1, the natural objects are no longer line bundles but higher‑rank or even non‑linear objects, and the associated cohomology becomes non‑abelian. The author defines a “gerbe divisor” as a gerbe banded by a non‑abelian group together with a connection whose curvature represents the cycle class. In this picture, the addition of cycles corresponds to the tensor product of gerbes, while push‑forward of cycles corresponds to the gerbe‑theoretic push‑forward (the direct image functor on the underlying stacks). The resulting “gerbe Chow group” therefore contains the classical Chow group as a subgroup (the abelian part) but also records higher‑order non‑abelian data.

Three main theorems are proved. First, for any chosen non‑abelian band there exists a connection‑equipped gerbe, and its automorphism 2‑group is canonically isomorphic to the second non‑abelian cohomology set H²(X, Band). Second, there is a bijection between gerbe divisors and codimension‑k cycle classes, establishing that every higher cycle can be uniquely represented by a gerbe with connection. Third, the usual intersection product on Chow groups lifts to a well‑defined operation on gerbe Chow groups, compatible with tensor product and push‑forward of gerbes. The proofs are carried out in full detail, with explicit examples on projective spaces, K3 surfaces, and certain quotient stacks illustrating how the abstract theory reduces to concrete calculations.

In the final sections the author compares this gerbe‑centric approach with the existing stack‑based non‑abelian cohomology frameworks of Giraud, Căldăraru, Murray, and Stevenson. While those frameworks are powerful, they require handling full 2‑categories and a plethora of higher morphisms. By contrast, the gerbe‑plus‑connection model condenses all relevant data into a single geometric object, dramatically simplifying both conceptual understanding and computational practice. The paper also outlines several avenues for future work: extending the construction to 3‑gerbes (capturing H³‑type invariants), exploring applications to moduli spaces of sheaves and derived categories, and investigating physical interpretations where gerbes with connection model higher‑form gauge fields (e.g., B‑fields in string theory).

Overall, the work offers a clear, technically robust, and computationally accessible framework for non‑abelian cohomology, bridging the gap between abstract higher‑categorical theory and concrete problems in algebraic geometry such as the study of higher divisors and Chow groups.