In mathematics, the pluricanonical ring of an algebraic variety V (which is nonsingular), 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
[1] 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.
Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.
Caucher Birkar, Paolo Cascini, and Christopher D. Hacon et al. (2010) proved this conjecture.
The dimension is the classically defined n-th plurigenus of V. The pluricanonical divisor