Atomic sheaves on hyper-K\"ahler manifolds via Bridgeland moduli spaces

In this paper, we provide new examples of 1-obstructed and atomic sheaves on an infinite series of locally complete families of projective hyper-K\"ahler manifolds. More precisely, (1) we prove that the fixed loci of the natural anti-symplectic involutions on the moduli spaces of stable objects in the Kuznetsov component $\mathcal{K}u(X)$ of a Gushel--Mukai fourfold $X$ are 1-obstructed Lagrangian submanifolds, (2) we construct a family of immersed atomic Lagrangian submanifolds on each moduli space of stable objects in $\mathcal{K}u(X)$, and (3) we construct non-rigid projectively hyperholomorphic twisted bundles on any hyper-K\"ahler manifold of $\mathrm{K3^{[n]}}$-type for infinitely many $n$. Additionally, we discuss examples of atomic Lagrangian submanifolds satisfying $b_1=20$ in a family of hyper-K\"ahler manifolds of $\mathrm{K3^{[2]}}$-type, as well as atomic sheaves supported on non-atomic Lagrangians.