Deformation-Quantization of Complex Involutive Submanifolds

The sheaf of rings of WKB operators provides a deformation-quantization of the cotangent bundle to a complex manifold. On a complex symplectic manifold $X$ there may not exist a sheaf of rings locally isomorphic to a ring of WKB operators. The idea is then to consider the whole family of locally defined sheaves of WKB operators as the deformation-quantization of $X$. To state it precisely, one needs the notion of algebroid stack, introduced by Kontsevich. In particular, the stack of WKB modules over $X$ defined in Polesello-Schapira (see also Kashiwara for the contact case) is better understood as the stack of modules over the algebroid stack of deformation-quantization of $X$. Let $V$ be an involutive submanifold of $X$, and assume for simplicity that the quotient of $V$ by its bicharacteristic leaves is isomorphic to a complex symplectic manifold $Z$. The algebra of endomorphisms of a simple WKB module along $V$ is locally (anti-)isomorphic to the pull-back of WKB operators on $Z$. Hence we may say that a simple module provides a deformation-quantization of $V$. Again, since in general there do not exist globally defined simple WKB modules, the idea is to consider the algebroid stack of locally defined simple WKB modules as the deformation-quantization of $V$. In this paper we start by defining what an algebroid stack is, and how it is locally described. We then discuss the algebroid stack of WKB operators on a complex symplectic manifold $X$, and define the deformation-quantization of an involutive submanifold $V$ by means of simple WKB modules along $V$. Finally, we relate this deformation-quantization to that given by WKB operators on the quotient of $V$ by its bicharacteristic leaves.