It is proved in this [paper](http://www.jstor.org/stable/1971417) by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the $\mathcal{O}\_X$-algebra $$\mathcal{R}\_X(D):=\oplus\_{m\geq 0}\mathcal{O}\_{X}(mD)$$ is finitely generated.


I want to know:


(1) If the above result still true for higher dimensional varieties(with mild singualrities or even a variety of Fano type)?


(2) My interest in this problem comes from using its "Corollary" in higher dimensional case (see Corollary 4.5 loc.cit):


Let $X$ be a 3-dimensional variety with terminal singularities, then there exists a projective birational morphism $\phi: Y \to X$ such that:
(1) $Y$ has at most $\mathbb{Q}$-factorial terminal singularities, and (2) $\phi$ is small.


I want to know if in the higher dimensional case, there still exists such small modification of a variety to be a $\mathbb{Q}$-factorial variety?



Finite generation of certain $\mathcal{O}_X$-algebra
Indeed, this is fine. In fact it is an exercise in Kollár's arXiv notes (of course utilizing the MMP):


<http://arxiv.org/pdf/0809.2579.pdf>


see in particular exercises 90-110.


The upshot is that everything is fine for KLT ambient spaces.


