Canonical ring

In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring

of sections of powers of the canonical bundle K. Its nth graded component (for ) is:

that is, the space of sections of the n-th tensor product Kⁿ of the canonical bundle K.

The 0th graded component is sections of the trivial bundle, and is one-dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model, is called the Kodaira dimension of V.

One can define an analogous ring for any line bundle L over V; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.

Properties

Birational invariance

The canonical ring and therefore likewise the Kodaira dimension is a birational invariant: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings. As a consequence one can define the Kodaira dimension of a singular space as the Kodaira dimension of a desingularization. Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.