Hyper-holomorphic connections on vector bundles on hyper-K\"ahler manifolds

TL;DR

This paper investigates the deformation theory of hyper-holomorphic connections on vector bundles over hyper-K"ahler manifolds, establishing a controlling DG Lie algebra and proving formality results for derived endomorphisms.

Contribution

It introduces a DG Lie algebra framework for deformations of hyper-holomorphic connections and proves associative formality in this context, advancing understanding of their deformation theory.

Findings

Describes the DG Lie algebra controlling deformations.

Proves associative formality for derived endomorphisms.

Provides a comprehensive study of infinitesimal deformations on hyper-K"ahler manifolds.

Abstract

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-K\"ahler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation problem. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection.