  An exact complex symplectic manifold carries a sheaf of stable categories,
locally equivalent to a microlocalization of a category of constructible
sheaves. This sheaf of categories admits a t-structure, whose heart is locally
equivalent to a microlocalization of a category of perverse sheaves. The
abelian category of local systems on a spin conic complex Lagrangian embeds
fully faithfully in the heart. The sheaf of homs between two objects in the
heart is itself a perverse sheaf, shifted by half the dimension of the ambient
manifold. Analogous results hold for complex contact manifolds.
  The correspondence between microsheaves and Fukaya categories yields
t-structures on Fukaya categories of conic complex symplectic manifolds, with
holomorphic Lagrangians in the heart.
