Brauer groups and derived equivalence for Lagrangian fibrations

Moduli spaces of sheaves on K3 surfaces give some of the most important examples of hyperkähler manifolds. We study their geometry using Hodge theory and derived categories. We are especially interested in Beauville--Mukai systems.
We first investigate when a given moduli space of sheaves is fine, using a certain Brauer class called the obstruction class. These were defined and computed by Caldararu in the case that the moduli space is itself a K3 surface. We extend his results to higher-dimensional moduli spaces using similar methods. An interesting new ingredient is the Caldararu class. Caldararu classes were used by Mukai and Caldararu, but were named in this thesis.
We apply Caldararu classes to the study of derived equivalence for K3 surfaces. More precisely, we answer the question of whether every Fourier--Mukai partner of an elliptic K3 surface is isomorphic to a Jacobian. This question was asked by Hassett--Tschinkel. The answer to the question is negative in general. The main ingredients for the proof are Caldararu classes, Ogg--Shafarevich theory, and the Derived Torelli Theorem.
We use our explicit description of the obstruction class for a higher-dimensional moduli space to study birational and derived equivalence for Beauville--Mukai systems, and to generalise Ogg--Shafarevich theory to this setting.
For moduli spaces of sheaves on elliptic K3 surfaces, we provide a complete description of birational equivalence in terms of Caldararu classes. We also use the results of Beckmann to show there exist moduli spaces that are not derived equivalent but which have Hodge isometric transcendental lattices, giving counterexamples to the Derived Torelli conjecture.