Kodaira dimension

In algebraic geometry, the Kodaira dimension κ(X) (or canonical dimension) measures the size of the canonical model of a projective variety X. In algebraic geometry, the Kodaira dimension κ(X) (or canonical dimension) measures the size of the canonical model of a projective variety X. Igor Shafarevich introduced an important numerical invariant of surfaces with the notation κ in the seminar Shafarevich 1965. Shigeru Iitaka (1970) extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira in Iitaka (1971). The canonical bundle of a smooth algebraic variety X of dimension n over a field is the line bundle of n-forms, which is the nth exterior power of the cotangent bundle of X.For an integer d, the dth tensor power of KX is again a line bundle.For d ≥ 0, the vector space of global sections H0(X,KXd) has the remarkable property that it is a birational invariant of smooth projective varieties X. That is, this vector space is canonically identified with the corresponding space for any smooth projective variety which is isomorphic to X outside lower-dimensional subsets. For d ≥ 0, thedth plurigenus of X is defined as the dimension of the vector spaceof global sections of KXd: The plurigenera are important birational invariants of an algebraic variety. In particular, the simplest way to prove that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus Pd with d > 0is not zero. If the space of sections of KXd is nonzero, then there is a natural rational map from X to the projective space called the d-canonical map. The canonical ring R(KX) of a variety X is the graded ring Also see geometric genus and arithmetic genus. The Kodaira dimension of X is defined to be − ∞ {displaystyle -infty } if the plurigenera Pd are zero for all d > 0; otherwise, it is the minimum κ such that Pd/dκ is bounded. The Kodaira dimension of an n-dimensional variety is either − ∞ {displaystyle -infty } or an integer in the range from 0 to n.