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Corollary 12.19.16940F7BB9270C

Corollary 12.19.16.940F7BB9270C In every dimension $n\geq 6$ there is a connected compact complex manifold with

\[ v_-(h)=0,\qquad v_+(h)=+\infty \]

for every Hermitian form $h$.

Proof. Take the product of the threefold above with the Hopf threefold of section 23.1, with a split metric. Potentials pulled back from the first factor make the masses tend to zero, while the smooth potentials of theorem 23.1.1, pulled back from the second factor, make them tend to infinity. In each case the product integral is $\binom 63$ times the product of the two factor masses. The metric comparison in the preceding proof handles the lower mass for every Hermitian form, and the metric independence in proposition 13.14.15 handles the upper mass. Products with projective spaces give every higher dimension by the same expansion of wedge powers. $\square$

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