Atomic objects on hyper-K\"ahler manifolds

TL;DR

This paper introduces atomic sheaves and complexes on hyper-K"ahler manifolds, revealing their properties and similarities to simple sheaves on K3 surfaces, including formality of derived endomorphism dg algebras and non-existence of spherical objects.

Contribution

It defines atomic objects on hyper-K"ahler manifolds and proves their key properties, extending concepts known from K3 surfaces to higher dimensions.

Findings

Formality of the dg algebra of derived endomorphisms for stable atomic bundles

Characterization of atomic Lagrangian submanifolds

Non-existence of spherical objects on hyper-K"ahler manifolds

Abstract

We introduce and study the notion of atomic sheaves and complexes on higher-dimensional hyper-K\"ahler manifolds and show that they share many of the intriguing properties of simple sheaves on K3 surfaces. For example, we prove formality of the dg algebra of derived endomorphisms for stable atomic bundles. We further demonstrate the characteristics of atomic objects by studying atomic Lagrangian submanifolds. In the appendix, we prove non-existence results for spherical objects on hyper-K\"ahler manifolds.