Quantization of complex symplectic manifolds

We introduce a canonical quantization algebroid of a complex
symplectic manifold.
Here, unlike in the usual deformation-quantization, the deformation
parameter h-bar is not central.
Its centralizer contains Polesello-Schapira's deformation-quantization
algebroid,
up to a twist by the gerbe parameterizing the primitives of the
symplectic 2-form.

Regular holonomic quantization modules along a Lagrangian subvariety
are equivalent to regular holonomic modules along its contactification,
with coefficients in the algebroid of classical microdifferential
operators.
If the manifold is compact, the derived category of regular holonomic
quantization modules is Calabi-Yau.

This is joint work with Masaki Kashiwara.