Corollary 12.32.337D7893F0652

Proof. The isomorphism of theorem 12.32.2 and theorem 12.31.3 give

where $\binom {2}{3}=0$. The surface $M$ is projective, and by coherent GAGA the holomorphic bundle $V$ is algebraic. Thus $\mathbb P(V)$ is a projective fourfold, and the volume of its tautological line bundle is the limit of $24h^0(M,\operatorname {Sym}^ rV)/r^4$, which equals $d>0$. Hence this line bundle is big. If the section algebra of $V$ were finitely generated, so would be its quotient by $w$, contradicting theorem 12.31.4.

For the canonical extension $0\to \mathcal O_ M\to V\to T_ M\to 0$ of a Kähler class, the dual extension of $\mathcal O_ M$ by $\Omega ^1_ M$ has nonzero ruling component, as shown in the proof of theorem 12.30.4. Its class therefore has $\rho \ne 0$ in the conventions used here, and the same argument applies. $\square$