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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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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 Þór Magnússon Feb 8 '12 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/siu_grauert_volume_paper.pdf –  YangMills Feb 8 '12 at 15:31
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). –  Ruadhaí Dervan Sep 13 '13 at 8:20

2 Answers 2

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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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? –  Gunnar Þór Magnússon Jul 5 '13 at 15:46
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. –  Misha Verbitsky Jul 12 '13 at 20:38
up vote 1 down vote accepted

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 of general type. 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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