Canonical ring

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. proved this conjecture. ==The plurigenera==

The dimension :P_n = h^0(V, K^n) = \operatorname{dim}\ H^0(V, K^n) is the classically defined n-th plurigenus of V. The pluricanonical divisor K^n, via the corresponding linear system of divisors, gives a map to projective space \mathbf{P}(H^0(V, K^n)) = \mathbf{P}^{P_n - 1}, called the n-canonical map. The size of R is a basic invariant of V, and is called the Kodaira dimension. ==Notes==