Canonical ring

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

${\displaystyle R(V,K)=R(V,K_{V})\,}$

of sections of powers of the canonical bundle K. Its nth graded component (for ${\displaystyle n\geq 0}$) is:

${\displaystyle R_{n}:=H^{0}(V,K^{n}),\ }$

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

The 0th graded component ${\displaystyle R_{0}}$ 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.^[1]