On Deformation of Elliptic Quantum Planes

Elliptic Quantum Planes means here non-commutative deformations of the complex projective plane P(C). We consider deformations in the realm of non-commutative (complex) algebraic geometry. As we recall in the first section, elliptic modulus parameter enters into the game. Hence the adjective “elliptic” is used. Note also that, in that world, the complex projective line P(C), namely the Riemann sphere, does not admit even a nontrivial non-commutative deformation, unlike in the world of non-commutative differential geometry. The aim of this article is twofold. In the first two sections, we briefly review the non-commutative projective geometry mainly focusing on the case of non-commutative deformation of the projective plane. In the third section, we formulate a theorem pertaining to Hochschild cohomology of algebraic varieties, which generalizes a theorem of Keller [6].