Example 11.21.48F043288C182

Proof. Let $\alpha $ be the form on $X_ q$ induced by $\mathrm{dd}^{\mathrm{c}}\log |z|^2$; it descends because the contraction changes $\log |z|^2$ by a constant. For $u\in T_ z^{1,0}(\mathbb C^ n\setminus \{ 0\} )$, its coefficient form is, up to the positive factor $1/(2\pi )$,

where $\langle \cdot ,\cdot \rangle $ is the Euclidean Hermitian inner product. By the Cauchy–Schwarz inequality, it is positive semidefinite with kernel $\mathbb Cz$, hence of rank $n-1$. The curvature $-\mathrm{dd}^{\mathrm{c}}\log |z|^{2b}$ of the metric $G$ is $-b\alpha $. The integral of $\alpha \wedge \gamma ^{n-1}$ is strictly positive, so the degree has the sign of $-b$, which gives the three stated cases. The map $z\mapsto (z/|z|,\log |z|)$ identifies the covering space with $S^{2n-1}\times \mathbb R$, and the contraction translates the second coordinate by $\log q$. Hence $X_ q$ is diffeomorphic to $S^{2n-1}\times S^1$, and $H^2_{\mathrm{dR}}(X_ q,\mathbb R)=0$ since $n\geq 2$.

If $|a|<1$, then $b<0$, so every nonzero fiber vector gives $n-1$ positive curvature directions. Applying theorem 11.18.2 with $r=1$ and $s=n-1$, we obtain $2$-convexity. For the lower bound, consider the map $X_ q\to \mathbb P^{n-1}$, $[z]\mapsto [z]$, whose fibers are compact embedded elliptic curves $C\simeq \mathbb C^*/q^{\mathbb Z}$. On a lifted fiber $z=\zeta z_0$, the logarithm of the metric coefficient is $b\log |\zeta |^2+b\log |z_0|^2$, which is pluriharmonic in $\zeta $. Thus $E_ a|_ C$ has flat Chern connection. The complement $U_{E_ a|_ C}$ is a closed complex submanifold of $U_{E_ a}$. A $1$-convex exhaustion function of $U_{E_ a}$ would restrict to an exhaustion function of this submanifold that is strictly plurisubharmonic outside a compact set. This contradicts corollary 11.21.3, since the restricted bundle has base dimension one and rank one. The phase of $a$ affects the gluing but not these metric and curvature calculations.

If $|a|\geq 1$, then $b\geq 0$. On the covering space of $E_ a^\times $, the function

is plurisubharmonic: the first summand has semipositive Levi form and the second is pluriharmonic for $v\ne 0$. Consequently

The descended function $G$ is smooth, positive and homogeneous of degree two under complex scalars. Apply proposition 11.21.1 with rank one and total dimension $n+1$. $\square$