## Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu 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 (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?

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As far as I know this is open and I haven't heard of anyone working on this. The not completely unrelated problem of invariance of plurigenera is expected to hold for Kahler manifolds, though. (I.e. Demailly expects it to hold.) – Gunnar Magnusson Feb 8 2012 at 7:54
Thanks for the comment. The invariance of plurigenera in the Kahler case is also stated explicitly as a conjecture by Siu here (Conjecture 0.4) math.harvard.edu/~siu/siu_reprints/… – YangMills Feb 8 2012 at 15:31