Deformation quantization of projective schemes and differential operators

The present talk is devoted to noncommutative projective schemes within Kapranov's model of noncommutative algebraic geometry. We classify all noncommutative projective schemes obtained from the differential chains in the universal enveloping algebra of the free nilpotent Lie algebra of index q. The construction proposed allows us to propose a new method of deformation quantization of the commutative projective schemes within the considered model. In the beginning of my talk I mention some key facts from the projective geometry, thereafter a general approach to NC-deformations and their morphisms will be discussed, close to the end of my talk some examples of quantizations of the projective curves, surfaces will be provided, and Feigin-Shoikhet construction turns out to be an example of a non-geometric quantization.