Basic Question about Kodaira Dimension

Question

Greetings friends,

Let $X$ be a smooth complex projective variety with canonical divisor $K$.

For $n \in \mathbb{N}$, let $\lambda_n$ denote the rational map $X \to \mathbb{P}^M$ induced from $H^0(X, nK)$, and let $d_n = dim(im(\lambda_n))$. Let $\kappa_1$ be the maximum value $d_n$ attains as $n$ ranges over $\mathbb{N}$, where we say that $\kappa_1=-1$ if $H^0(X,nK)=\emptyset$ for all $n$.

Let $\kappa_2$ be the smallest integer so that the sequence $\frac{dim(H^0(X,nK))}{n^b}$ is bounded, where we make the same convention of $\kappa_2=-1$ for above.

I'd like to understand why $\kappa_1 = \kappa_2$; could someone please provide me with a reference that discusses this, or provide a brief sketch as to the proof?

Thank you, Robert

Answer 1

Dear Robert, a reference is Lazarsfeld, Positivity in Algebraic Geometry I, Corollary 2.1.38. Note that the statement there concerns the Iitaka dimension of any line bundle, not just the canonical bundle.

The basic idea of the proof is as follows: given a line bundle $L$ on $X$, one can find a birational morphism $f: X' \rightarrow X$ such that $X'$ fibres over another variety $Y$ of dimension $\kappa_1(L)$. One then compares the numbers $h^0(X,nL)=h^0(X',f^\ast (nL))$ with the number of sections of powers of some ample line bundles on $Y$, which one can compute with Riemann--Roch, to get the desired estimates.