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$K3^{[n]}$
We prove existence and unicity of slope-stable vector bundles on a general polarized hyperkähler (HK) variety of type 
$K3^{[n]}$ with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but, in fact, we might have listed almost all slope-stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type 
$K3^{[n]}$ with 
$20$ moduli.
We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional 
$f : X \rightarrow \mathbb {A}_{k}^{1}$, the derived critical locus has a natural Lagrangian fibration 
$\textbf {Crit}(f) \rightarrow X$. In the case where f is nondegenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.
The base surface 
$B$ of a Lagrangian fibration 
of a projective, irreducible symplectic fourfold 
$X$ is shown to be isomorphic to 
$\mathbb{P}^{2}$.
