HK manifolds of Type $K3^{[a^2+1]}$ as moduli spaces of projective bundles on HK manifolds of Type $K3^{[2]}$
We prove results on moduli spaces of slope stable bundles of projective spaces on a hyperkähler manifold of Type $K3^{[2]}$. Let $X$ be projective of Type $K3^{[2]}$ and $h$ be a (generic) ample class. We prove that the moduli space $M_{\overline{\bf w}_a}(X,h)$ parametrizing $h$ slope stable bundles with a suitable mock Mukai vector ${\overline{\bf w}}_a$ contains an irreducible component $M_{\overline{\bf w}_a}(X,h)^{*}$ whose normalization $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ is a (projective) HK manifold of Type $K3^{[a^2+1]}$, and that conversely every projective HK manifold $W$ of Type $K3^{[a^2+1]}$ is isomorphic to $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ for a suitable $(X,h)$ as above. Moreover the universal bundle of projective spaces on $X\times \widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ defines a vector bundle whose $2nd$ Chern class defines a rational Hodge isometry $H^2(X)\to H^2(\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*})$. From this and a result of Markman one gets that the analogue of the Shafarevich conjecture (a special case of the Hodge conjecture) holds for rational Hodge isometries $H^2(W_1) \to H^2(W_2)$ between projective hyperkähler manifolds $W_1,W_2$ of Types $K3^{[a_1^2+1]}$ and $K3^{[a_2^2+1]}$ respectively. We prove results also for $(X,ω)$ a general HK manifold of Type $K3^{[2]}$. In fact one ingredient in our proof is Verbistsky's theory of projectively hyperhomolorphic vector bundles.