Twisted Arinkin transforms and derived categories of moduli spaces on Kuznetsov components

In this note, we generalize results of Donagi and Pantev on twisted derived equivalences between elliptically fibered surfaces to higher dimensions. First, we establish a twisted derived equivalence between torsors under abelian schemes satisfying a certain compatibility condition. Then, relying on the work of Arinkin on compactified Jacobians, we extend the equivalence to twisted compactified Jacobians associated to curves on K3 surfaces. This positively answers a question stated by Mattei and Meinsma. We then extend a result of Bottini and Huybrechts for Fano varieties of lines on cubic fourfolds to general moduli spaces of Bridgeland-stable objects on Kuznetsov components admitting rational Lagrangian fibrations.