In this Appendix, we will try to explain the constructions of the paper in a slightly more general context of D-modules and “formal geometry” of Gelfand and Kazhdan. To save space, we only sketch the proofs, and wework in the algebraic setting (generalization to complex-analytic varieties is immediate, exactly the same arguments work).

Let S be a smooth rational curve on a complex manifold M. It is called ample if its normal bundle is positive: NS=⨁O(i_k),i_k<0. We assume that M is covered by smooth holomorphic deformations of S. The basic example of such a manifold is a twistor space of a hyperkähler or a 4–dimensional anti-selfdual Riemannian manifold X (not necessarily compact). We prove “a holography principle” for such a manifold: any meromorphic function defined in a neighbourhood U of S can be extended to M, and any section of a holomorphic line bundle can be extended from U to M. This is used to define the notion of a Moishezon twistor space: this is a twistor space admitting a holomorphic embedding to a Moishezon variety M′. We show that this property is local on X, and the variety M′ is unique up to birational transform. We prove that the twistor spaces of hyperkähler manifolds obtained by hyperkähler reduction of flat quaternionic-Hermitian spaces by the action of reductive Lie groups (such as Nakajima’s quiver varieties) are always Moishezon.

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow} X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a grassmannian or an isotropic grassmannian (possibly mixing grassmannians and isotropic grassmannians), and the embeddings $\phi_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one of certain standard ind-grassmannians and that the latter are pairwise non-isomorphic ind-varieties.

In this paper we study wall-crossing functors between categories of modules over quantizations of symplectic resolutions. We prove that wall-crossing functors through faces are perverse equivalences and use this to verify an Etingof type conjecture for quantizations of Nakajima quiver varieties associated to affine quivers. In the case when there is a Hamiltonian torus action on the resolution with finitely many fixed points so that it makes sense to speak about categories O over quantizations, we introduce new standardly stratified structures on these categories O and relate the wall-crossing functors to the Ringel duality functors associated to these standardly stratified structures.