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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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    $\begingroup$ 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.) $\endgroup$
    – Gunnar Þór Magnússon
    Commented Feb 8, 2012 at 7:54
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    $\begingroup$ 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/siu_grauert_volume_paper.pdf $\endgroup$
    – YangMills
    Commented Feb 8, 2012 at 15:31
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    $\begingroup$ I haven't read the paper, so this is just a comment, but a paper of Fujino on the arXiv today claims to show that the canonical ring of a compact Kaehler manifold is finitely generated: arXiv:1309.3015 (Corollary 4.2). $\endgroup$
    – Ruadhaí Dervan
    Commented Sep 13, 2013 at 8:20

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As pointed out by Ruadhai Dervan in the comments, a paper by Fujino 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 finite generation upstairs can be deduced from finite generation downstairs (with a boundary divisor term), and this latter statement follows from BCHM. The details are in the paper of Fujino cited above.

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    $\begingroup$ The base of the Iitaka map is not always of log general type, this is only true if the variation (in the sense of Viehweg) of the Iitaka map is maximal. $\endgroup$
    – user105074
    Commented Sep 21, 2020 at 0:34
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Most likely the canonical ring in Kahler situation is also finitely generated. You can check the paper http://arxiv.org/abs/1304.4013 "Minimal models for Kaehler threefolds" (Andreas Hoering, Thomas Peternell) where the MMP for Kaehler threefolds is constructed. I would expect that their results would easily imply that the canonical ring in dimension=3 is finitely generated.

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    $\begingroup$ Hi Misha! Do you have an idea of how that implication would work? I imagine MMP lets us reduce to $X$ with $K_X$ nef, but is it clear that such a manifold has finitely generated canonical ring if it is not projective? $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jul 5, 2013 at 15:46
  • $\begingroup$ non-projective Kahler 3-manifolds are in fact rare: they all admit a holomorphic 2-form; its radical gives a 1-dimensional foliation on a manifold. Its curvature by Brunella is represented by a positive current, unless each leaf of this foliation has closure isomorphic to a rational line. In the second case we already have a good idea about the Mori fibration. In the first case, the canonical bundle is pseudoeffective, which is not far from K_x being nef. $\endgroup$
    – Misha Verbitsky
    Commented Jul 12, 2013 at 20:38

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