In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring
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of sections of powers of the canonical bundle K. Its nth graded component (for
that is, the space of sections of the n-th tensor product Kn of the canonical bundle K.
The 0th graded component
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.
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.
Fundamental conjecture of birational geometry
A basic conjecture is that the pluricanonical ring is finitely generated. This is considered a major step in the Mori program. Caucher Birkar, Paolo Cascini, and Christopher D. Hacon et al. (2010) proved this conjecture.
The plurigenera
The dimension
is the classically defined n-th plurigenus of V. The pluricanonical divisor
The size of R is a basic invariant of V, and is called the Kodaira dimension.