Pome1DD06E8B2862Write points of $\mathbb P^3$ as $[z:w]$, with $z,w\in \mathbb C^2$, and set
Thus $g$ is an automorphism of $\Omega $; let $\langle g\rangle $ be the cyclic group it generates. This group acts freely. Indeed, if $g^ k$ fixed a point, the same scalar would have to equal $1$ on the nonzero block $z$ and $q^ k$ on the nonzero block $w$, which forces $k=0$.
The function
is well defined on $\Omega $ and satisfies $t\circ g=t+4\pi $. Consequently the action is properly discontinuous: $t$ is bounded on compact subsets, so only finitely many translates of one compact set can meet another. Moreover, the strip $\{ 0\leq t\leq 4\pi \} $ is compact. In homogeneous coordinates of total norm one, its defining inequalities bound both blocks away from zero, so it is a closed subset of $\mathbb P^3$ contained in $\Omega $. Since the strip meets every orbit,
is a connected compact complex threefold.
The map $[z:w]\mapsto ([z],[w])$ induces a holomorphic submersion $X\to \mathbb P^1\times \mathbb P^1$. Using a logarithm of the scalar relating two points with the same image, one sees that its fibers are the elliptic curves $E=\mathbb C/(2\pi \mathbb Z+2\pi \mathrm{i}\mathbb Z)$. Let $\alpha ,\beta $ be the pullbacks of the Fubini–Study forms of the two factors, each normalized to have integral one. Although $t$ does not descend to $X$, its differentials $\mathrm dt$ and $\mathrm d^{\mathrm c}t$ do. The form
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is Hermitian: $\alpha +\beta $ is semipositive with kernel the vertical line, while $\mathrm{d}t\wedge \mathrm d^{\mathrm c}t$ is semipositive and strictly positive on the vertical line.
On the open set $\{ z_0w_0\ne 0\} $ we use the coordinates
Near $0\in E$, the last coordinate is represented by an ordinary complex number. For any one of these coordinates, denoted by $v$, put $\sigma _ v=\frac{\mathrm{i}}{2\pi }\mathrm{d}v\wedge \mathrm{d}\bar v$. Choose a smooth probability density $p$ relative to $\sigma _ v$ that is positive on $\{ |v|<1\} $ and vanishes on $\{ |v|\geq 1\} $. Choose also a smooth function $\chi _0$ with values in $[0,1]$, supported in $\{ |v|<1\} $ and equal to one on $\{ |v|\leq 1/2\} $, and write $\chi (x,y)=\chi _0(x)\chi _0(y)$. For $0<\delta <1/100$ define
extended by zero. This is a smooth semipositive $(1,1)$-form on $X$, because the chosen functions are smooth with compact support in their coordinate domains. As $\delta \downarrow 0$ it concentrates on the hypersurface $\{ \zeta =0\} $.
The first step shows that normalized positive $\mathrm{dd}^{\mathrm{c}}$-closed $(2,2)$-forms can pair arbitrarily strongly with $B_\delta $.
Lemma 12.19.4.669B72559DF9 For $\delta >0$ put
the supremum being taken over smooth real $(2,2)$-forms $\Theta $. These numbers are finite and tend to infinity as $\delta \downarrow 0$.
Proof. For fixed $\delta $, choose a constant $M_\delta >0$ with $B_\delta \leq M_\delta \omega $; this shows that $C_\delta \leq M_\delta $. Suppose instead that $C_{\delta _ l}\leq C$ along a sequence $\delta _ l\to 0$. Then $\int _ X(C\omega -B_{\delta _ l})\wedge \Theta \geq 0$ for every admissible $\Theta $, and in particular for the squares of Gauduchon metrics. By theorem 11.6.1, there are quasi-plurisubharmonic functions $v_ l$ such that
By theorem 5.25.2, a subsequence converges in $L^1$ to a $C\omega $-plurisubharmonic function $v$, not identically $-\infty $. Since $\delta ^{-2}p(\zeta /\delta )\sigma _\zeta $ is an approximate identity in the variable $\zeta $,
On a coordinate ball where $\chi =1$, add to $v$ a smooth strictly plurisubharmonic function whose complex Hessian dominates $C\omega $. The resulting plurisubharmonic function has a current that dominates $[\zeta =0]$, so its Lelong number is at least one at every point of an open subset of this hypersurface. Smooth corrections do not change Lelong numbers, so the same holds for $v$. Siu's analyticity theorem, recalled in 5.34.2, shows that $Z=\{ \nu (v,\cdot )\geq 1\} $ is an analytic subset of $X$. It is a proper subset because $v$ is not identically $-\infty $.
Let $\widetilde Z$ be the inverse image of $Z$ in $\Omega $. It contains an open subset of $H_0\cap \Omega $, where $H_0=\{ w_0=z_0\} $. This surface is a projective plane with two points removed, so it is connected and irreducible, and the identity principle gives $H_0\cap \Omega \subset \widetilde Z$. Since $\widetilde Z$ is invariant under $g$, also
Now use the chart $z_1=1$ near the point $[0:1:1:0]\in \Omega $. There $H_ k$ is the graph $z_0=q^{-k}w_0$. For each fixed $(w_0,w_1)$ near $(1,0)$, every local holomorphic defining function of $\widetilde Z$ vanishes at the distinct values $z_0=q^{-k}w_0$, which tend to zero as $k\to +\infty $. The one-variable identity theorem makes each defining function vanish identically on a polydisk. Thus $\widetilde Z$ contains a nonempty open subset of the connected manifold $\Omega $, contradicting properness. Hence no subsequence of $C_\delta $ stays bounded as $\delta \downarrow 0$. $\square$
By the lemma we can choose $\delta _ j\downarrow 0$, each smaller than half its predecessor, and then, by the definition of the supremum, smooth strictly positive $\mathrm{dd}^{\mathrm{c}}$-closed $(2,2)$-forms $\Theta _ j$ with
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