Equivalent definitions of Kodaira dimension

Question

The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways:

1. The maximal dimension of the image of the rational maps $φ_{|mL|} : X \dashrightarrow \mathbb{P}(H^0(X, m L)^*)$
2. The unique integer $k$ such that $h^0(m L) = O(m^k)$.
3. $\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right) - 1$

I know why the first two definitions are equivalent but I'm struggling to find a reference treating the third. Several people cite Ueno's book Classification Theory of Algebraic Varieties and Compact Complex Spaces for this, but as far as I can see he only proves that first two are equivalent. Can anyone provide a proof of $(1) \iff (3)$ or $(2) \iff (3)$ or point me to a reference?

Answer 1

For $(1) \iff (3)$, we have the following chain of identities:

1. $\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to

2. $\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to

3. $\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to

4. $\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to

5. $\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to

6. the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to

7. the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)

8. the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).