Classifying affine line bundles on a compact complex space

TL;DR

This paper investigates the classification of affine line bundles on compact complex spaces by studying an affine Picard functor and establishing conditions for its representability, linking it to cohomological properties.

Contribution

It introduces an affine Picard functor for classifying affine line bundles and characterizes its representability in terms of the constancy of a specific map, connecting deformation theory and cohomology.

Findings

The affine Picard functor is representable if and only if a certain map is constant.

When representable, the functor's space is a linear space over the Picard variety.

The proof relates the functor's representability to Bingener's deformation theory results.

Abstract

The classification of affine line bundles on a compact complex space $X$ is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. For a fixed Chern class $c$, we introduce the affine Picard functor $Picaff_{X,x_0}^c:An^{op}\to Set$ which assigns to a complex space $T$ the set of families of linearly $x_0$-framed affine line bundles on $X$ with Chern class $c$ parameterized by $T$. Our main result states that this functor is representable if and only if the map $h^0:Pic^c(X)\to\mathbb{N}$ is constant. If this is the case, the space which represents this functor is a linear space over $Pic^c(X)$ whose underlying set is $\coprod_{l\in Pic^c(X)} H^1(\mathcal{L}_{\{l\}\times X})$, where $\mathcal{L}$ is a Poincar\'e line bundle normalized at $x_0$. The main idea idea of the proof is to compare the representability of our functor to the representability of a functor considered by Bingener related to the deformation theory of $p$-cohomology classes. Our arguments show in particular that, for $p=1$, the converse of Bingener's representability criterion holds.