Plurigenera

Plurigenera are the dimensions P_m(X) = dim H^0(X, mK_X) of the spaces of global sections of multiples of the canonical bundle K_X on a smooth projective variety X over the complex numbers. The sequence (P_m) encodes the growth of sections of multiples of the canonical bundle and, together with the graded ring R(X, K_X) = ⊕_{m≥0} H^0(X, mK_X), provides a compact numerical handle on the canonical geometry of X.

The finite generation of the/pluricanonical ring, a consequence of the minimal model program, implies the existence of a canonical model for X and underpins the birational classification of higher-dimensional varieties. The asymptotic growth of P_m defines the Kodaira dimension: κ(X) = limsup_{m→∞} log P_m / log m, taking values in { -∞, 0, 1, ..., dim X }. If κ(X) = dim X, X is of general type; in this case P_m grows roughly like a constant times m^{dim X}, and the pluricanonical maps φ_m associated to |mK_X| are birational for all sufficiently large m.

Special cases: for a smooth projective curve C of genus g ≥ 2, P_1 = g and P_m = (2m − 1)(g − 1) for m ≥ 2. For genus 0 or 1, the behavior differs because K_C is negative for P^1 and trivial for elliptic curves, affecting P_m accordingly.

Plurigenera are central in birational geometry and the classification of algebraic varieties, reflecting the positivity of the canonical bundle and guiding the geometry of higher-dimensional canonical maps.