Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences

In this thesis we use the Beauville-Bogomolov decomposition to compute the LLV algebra of smooth projective complex varieties admitting a holomorphic symplectic form, generalizing known results from hyperk\"ahler and abelian varieties. Using this explicit computation, we prove for many such varieties that Orlovs conjecture holds, which states that for two derived equivalent smooth projective varieties there exists an isomorphism of rational cohomology preserving the grading and Hodge structure. Moreover, we prove that this conjecture holds for all four-dimensional smooth projective varieties admitting a holomorphic symplectic form.