TITLE: Non finitely generated graded ring of a divisor in dimension >2
QUESTION [8 upvotes]: I spent some time looking for an example of non finitely generate graded ring
$R=\oplus H^0(X,mD)$ where $D$ is a divisor on a variety $X$ of dimensison $>2$.
I know there are several such examples (e.g Zariski's), but they are all on surfaces.
I believe there must exist many examples. Do you know any?

REPLY [9 votes]: I give Siu's analytical method  to test that the ring $$R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$$  is not finitely generated
Let $X$ be a compact complex manifold and the ring $R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$ is finitely generated and let $s^{(m)}_1, … , s^{(m)}_{q_m}\in H^0 (X,mL) $
be a basis over $\mathbb C$. Let
$$ \Phi =\sum_{m=1}^\infty \epsilon_m( ∑_{j=1}^{q_m} |s^{(m)}_j|^2)^{1/m} $$
where $\epsilon_m$ is some sequence of positive numbers decreasing fast enough to guarantee convergence of the series. Then all the Lelong numbers of the closed positive $(1,1)$-current $$T=\frac{\frak{\sqrt{-1}}}{2\pi}\partial\bar\partial \log Φ$$ are rational numbers. 
So one of ways to find some examples is to show that the Lelong number of $T$ is not rational at some point,  hence the ring $R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$ is not finitely generated
Definition: Let
$W\subset \mathbb C^n$
be a domain, and $\Theta$ a positive current of degree $(q,q)$ on
$W$. For a point $p\in W$
one defines
$$\mathfrak v(\Theta,p,r)=\frac{1}{r^{2(n-q)}}\int_{|z-p|<r}\Theta(z)\wedge (dd^c|z|^2)^{n-q}$$
The
Lelong number
of $\Theta$ at
$p$
is defined as
$$\mathfrak v(\Theta,p)=\lim_{r \to 0}\mathfrak v(\Theta,p,r)$$
Let $\Theta$ be the curvature of singular hermitian metric $h=e^{-u}$, one has
$$\mathfrak v(\Theta,p)=\sup\{\lambda\geq 0: u\leq \lambda\log(|z-p|^2)+O(1)\}$$