# Different definitions of Kodaira dimension

Let X be a smooth projective variety with canonical class K.

Let a be defined to be the maximum dimension of the image of X under the rational map induced by the linear system |nK| as n ranges over all positive integers.

Let b be defined to be the smallest positive integer c such that the sequence (dim(H^0 (X, nK))/n^c) is bounded (as n gets large).

The second definition I guess doesnt include the case with kodaira dimension smaller than 0 (ie -1 or -infinity depending on how you like to define it), so we're only talking about non-negative kodaira dimension.

Will someone please tell me why a=b?

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## 1 Answer

This is by no means a trivial result!

It is usually attributed to Iitaka, and is more or less contained in the so-called Iitaka fibration theorem. If you are looking for references, there is a book from Ueno, "Classification Theory of Algebraic Varieties and Compact Complex Spaces", and of course the book from Lazarsfeld "Positivity in Algebraic Geometry, I".

Two side remarks:

• this is not only suited for the canonical bundle, but holds for any line bundle on a smooth projective variety.

• we define $\kappa(L)=- \infty$ when $|nL|=\emptyset$ for all $n$.

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