24.4.32 Oeljeklaus–Toma manifolds68A7CB3104FE

Oeljeklaus–Toma manifolds are not Kähler, yet they carry a closed semipositive form $\theta $ whose potentials are all continuous (theorem 10.29.4). On a threefold of type $(2,1)$, the natural Gauduchon metric $\omega =\theta +y_1y_2\, \mathrm{dd}^{\mathrm{c}}|z|^2$ has an exactly quadratic mass functional, indefinite in sign, and its masses on fiber-invariant potentials are bounded above and below by positive constants (proposition 10.29.10). The known curvature criteria for finite upper and positive lower mass in dimension three require a Hermitian metric whose $\mathrm{dd}^{\mathrm{c}}$ has a sign, and these threefolds admit no such metric (proposition 10.29.14; see section 25.18 for the criteria). By (10.29.12), the remaining difficulty lies entirely in the fluctuation of a potential around its fiber average.

By proposition 13.14.15, the answer to the first question does not depend on the choice of Hermitian form. A negative answer would require potentials whose fluctuations make $\operatorname {Re}(\partial _{w_1}v\, \overline{\partial _{w_2}v})$ large, while keeping $\omega +\mathrm{dd}^{\mathrm{c}}u\geq 0$. By lemma 10.29.3, every nonzero Fourier mode on the fiber tori depends on $z$, so for a single mode $c(y)\cos (\cdots )$ the positivity of the $z$-coefficient of $\omega +\mathrm{dd}^{\mathrm{c}}u$ bounds $|c|$ by a multiple of $y_1y_2$ divided by the square of the $z$-frequency; the frequencies in the $x$-directions are not controlled pointwise. Whether compactness and the automorphy under the units bound the fluctuation term is the content of the problem. Unlike the Hopf threefold of section 23.1, the manifold contains no curves along which heat potentials could be concentrated, by the theorem of Ornea and Verbitsky recalled before corollary 10.29.7.

The rigidity in theorem 10.29.4 comes from two features of the null foliation of $\theta $: its leaves carry no nonconstant plurisubharmonic functions bounded above, and their closures are real tori on which translations act transitively. For the pullback of a Kähler form under a holomorphic fibration, the null leaves are the compact fibers, and $\theta $-plurisubharmonic functions may have poles.