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