Let $X$ be a variety of general type.


Assume that $\dim X = 3$. In <https://eudml.org/doc/164223> it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal singularities and nef canonical bundle) is finite.


Is this statement now also known when $\dim X > 3$?


More precisely, does $X$ only have finitely many minimal models? 



Is the number of minimal models finite
By 


C. Hacon, J. McKernan, *"On the existence of ﬂips"*, math.AG/0507597.


finiteness of minimal models in dimension $n-1$ implies the existence of flips in dimension $n$.


I guess the result you are looking for is Theorem B (pag 11) of BCHM (<http://www2.imperial.ac.uk/~pcascini/Papers/0610203.pdf>).


Perhaps Section 5.5 (pag 20) of this


O. Fujino, *"Recent developments in minimal model theory"*, <https://www.math.kyoto-u.ac.jp/~fujino/recent-final.pdf>


is easier to read.


