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



Basic Question about Kodaira Dimension
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. 


