  In this paper we propose a noncommutative generalization of the relationship
between compact K\"ahler manifolds and complex projective algebraic varieties.
Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson
tensor to deform the underlying complex structure into a generalized complex
structure, such that the prequantum line bundle and its tensor powers deform to
a sequence of generalized complex branes. Taking homomorphisms between the
resulting branes, we obtain a noncommutative deformation of the homogeneous
coordinate ring. As a proof of concept, this is implemented for all compact
toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson
structure, resulting in what could be called noncommutative toric varieties. To
define the homomorphisms between generalized complex branes, we propose a
method which involves lifting each pair of generalized complex branes to a
single coisotropic A-brane in the real symplectic groupoid of the underlying
Poisson structure, and compute morphisms in the A-model between the Lagrangian
identity bisection and the lifted coisotropic brane. This is done with the use
of a multiplicative holomorphic Lagrangian polarization of the groupoid.
