# Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

## TL;DR

This paper proves that the deformation theory of pairs consisting of a variety with trivial canonical bundle and a line bundle is unobstructed because the controlling DG-Lie algebra is homotopy abelian.

## Contribution

It establishes that the DG-Lie algebra controlling deformations of such pairs is homotopy abelian when the variety has trivial canonical bundle, ensuring unobstructed deformations.

## Key findings

- Deformations of pairs with trivial canonical bundle are unobstructed.
- The controlling DG-Lie algebra is homotopy abelian in this setting.
- Deformation theory simplifies for these pairs due to homotopy abelianity.

## Abstract

We investigate the deformations of pairs $(X,L)$, where $L$ is a line bundle on a smooth projective variety $X$, defined over an algebraically closed field $\mathbb{K}$ of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair $(X,L)$ is homotopy abelian whenever $X$ has trivial canonical bundle, and so these deformations are unobstructed.