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Concept# Kodaira dimension

Summary

In algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X.
Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira.
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, the
dth plurigenus of X is defined as the dimension of the vector space
of 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 > 0
is 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 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 or an integer in the range from 0 to n.
The following integers are equal if they are non-negative. A good reference is , Theorem 2.1.33.
The dimension of the Proj construction , a projective variety called the canonical model of X depending only on the birational equivalence class of X.

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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 is sections of the trivial bundle, and is one-dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model is called the Kodaira dimension of V.

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