A celebrated recent theorem of [Birkar-Cascini-Hacon-McKernan](http://dx.doi.org/10.1090/S0894-0347-09-00649-3) and [Siu](http://arxiv.org/abs/math/0610740) says that the canonical ring $R(X)=\oplus\_{m\geq 0}H^0(X,mK\_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely generated $\mathbb{C}$-algebra.


On the other hand [P.M.H. Wilson](http://www.numdam.org/item?id=CM_1981__43_3_365_0) (using a construction of Zariski) gave an example of a compact complex manifold $X$ with $R(X)$ not finitely generated. However his manifold $X$ is not Kähler.


Does anyone know an example of a compact Kähler manifold $X$ with $R(X)$ not finitely generated? Or is this an open problem?



Example of a compact Kähler manifold with non-finitely generated canonical ring?
As pointed out by Ruadhai Dervan in the comments, [a paper by Fujino](http://arxiv.org/abs/1309.3015) contains the answer to this question: the canonical ring of any compact Kähler manifold is finitely generated. By bimeromorphic invariance of this ring, the result even holds for compact complex manifolds in Fujiki's class $\mathcal{C}$.


The idea is to consider the Iitaka fibration of the manifold, which has the obvious property that its base is always a projective variety. Thanks to [Fujino-Mori](http://projecteuclid.org/euclid.jdg/1090347529) finite generation upstairs can be deduced from finite generation downstairs (with a boundary divisor term), and this latter statement follows from [BCHM.](http://dx.doi.org/10.1090/S0894-0347-09-00649-3) The details are in the paper of Fujino cited above.


