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P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have trouble understanding how he conludes that his ring is not finitely generated.

I'll briefly describe his construction. First, Wilson fabricates a projective surface $\widetilde{\mathbb{P}}$ with a divisor $D$ such that the ring $$ R(\widetilde{\mathbb{P}} ,D) := \bigoplus_{m \geq 0} H^0(X,mD) $$ is not finitely generated by following ideas of Zariski: Let $C \subset \mathbb P^2$ be an elliptic curve and $H$ a line. Blow up 12 points in general position on $C$ and a point outside of $C$ to get $\widetilde{\mathbb{P}}$. Let $C'$ be the proper transform of $C$, $E$ the exceptional divisor of the point outside of $C$ and $H' = f^*H - E$. If $D := C' + H'$, then $R(\widetilde{\mathbb{P}},D)$ is not finitely generated.

Second, Wilson makes a double cover $S$ of $\widetilde{\mathbb{P}}$ ramified over a general element of $|6C' + 6H'|$ and desingularizes to get $\alpha : \tilde S \to \widetilde{\mathbb{P}}$. Then $K_{\tilde S} \sim \alpha^*(2C' + 3H' + E)$.

Third, he makes a nontrivial torus bundle $\pi : W \to \widetilde{\mathbb{P}}$, that is therefore non-Kahler. The fibers are of dimension 2, so $W$ is a fourfold. One remarks that $K_{W / \widetilde{\mathbb{P}}} = \pi^*\mathcal O(-2H')$.

Fourth, take the fibered product $V = W \times_{\widetilde{\mathbb{P}}} \tilde S$ and let $g : V \to \widetilde{\mathbb{P}}$ be the induced morphism. As $\alpha : \tilde S \to \widetilde{\mathbb{P}}$ is finite of degree 2, this is a compact non-Kahler fourfold. We see that $K_V = g^* \mathcal O(2C' + H' + E)$.

Until now, I'm mostly fine with his construction and can either see or am willing to take on faith how each step is necessary. But here Wilson claims that $R(V,K_V)$ is not finitely generated, and I don't see how. I assume there's some link between $R(V,K_V)$ and $R(\widetilde{\mathbb{P}},D)$, but it has escaped me so far. How do we see this?

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