Curved L-infinity algebras and lifts of torsors

TL;DR

This paper develops a curved $L_$-algebra framework to study the problem of lifting torsors through extensions of unipotent algebraic groups, connecting algebraic geometry with homotopical algebra.

Contribution

It introduces a curved $L_$-algebra structure on the Cech complex to model torsor lifts, extending the Deligne-Getzler groupoid to curved settings.

Findings

Curved $L_$-algebra structure models torsor lifting problems.

The curved Deligne-Getzler groupoid is isomorphic to the groupoid of lifts.

Provides a new algebraic approach to torsor extension problems.

Abstract

Consider an extension of finite dimensional nilpotent Lie algebras $0 \to \mathfrak{h} \to \tilde{\mathfrak{g}} \to \mathfrak{g} \to 0$ (over a field $k$ of characteristic zero) corresponding to an extension of unipotent algebraic groups $1 \to H \to \tilde{G} \to G \to 1$. For a $G$-torsor $P$ on an algebraic variety $X$ over $k$, we study the problem of lifting $P$ to $\widetilde{G}$-torsor $\widetilde{P}$. Fixing a trivialization of $P$ on open subsets of an affine cover, we give the Cech complex of $\mathfrak{h}$-valued functions the structure of a curved $L_\infty$-algebra and define a curved version of the Deligne-Getzler groupoid. We show that this groupoid is isomorphic the groupoid of cocycle level $\tilde{G}$-lifts of $P$.