Constructions of derived equivalent hyper-Kähler fourfolds

We study twisted derived equivalences of hyper-Kähler fourfolds. We describe when two hyper-Kähler fourfolds of $K3^{[2]}$-type of Picard rank $1$ with isomorphic transcendental lattices are derived equivalent. Then we present new constructions of pairs of twisted derived equivalent hyper-Kähler manifolds of Picard rank $\geq 2$.