19.8.12 Homogeneous sections on a Hopf manifold82379C382500

On a scalar Hopf manifold, a holomorphic section of a line bundle lifts to a function on $\mathbb C^ n\setminus \{ 0\} $ with an exact homogeneity property under the contraction. Hartogs extension reduces the section spaces to spaces of homogeneous polynomials, and the invariance of the metric makes the Bergman kernel constant. The resulting growth is of order $m^{n-1}$, reflecting the dimension of the projective base rather than that of the manifold.

Let $n\geq 2$, let $z=(z_1,\ldots ,z_ n)$ be the coordinates on $\mathbb C^ n$, and set

On the covering space put

The contraction changes $s$ by the constant $-2$, so these forms descend to $X_ n$. The calculation proving (23.1.3), written in section 23.1 for $n=3$ and with $\beta $ in place of $\zeta $, applies in every dimension and gives

Thus $\omega $ is a Gauduchon form. Projectivization defines a holomorphic map $\pi :X_ n\to \mathbb P^{n-1}$ with elliptic fibers, and $\alpha $ is the pullback of the Fubini–Study form normalized to have total mass one. Since $\zeta $ restricts to twice the normalized area form on each fiber, as in the proof of proposition 23.1.4, and $\omega ^ n=n\alpha ^{n-1}\wedge \zeta $, we obtain

Set $\mathrm{d}V=\omega ^ n/(2n)$, a probability measure.

For an integer $k\geq 0$, let $L_ k$ be the line bundle on $X_ n$ obtained from $(\mathbb C^ n\setminus \{ 0\} )\times \mathbb C$ through the identification

Give it the metric $|t|_{h_ k}^2=|t|^2|z|^{-2k}$. Under the identification the two factors change by $\mathrm{e}^{-2k}$ and $\mathrm{e}^{2k}$, so this metric descends. In the frame $t=1$ the squared norm is $|z|^{-2k}$, and the curvature form is $k\alpha $. We apply the constructions of subsection 19.8.1 with $X=X_ n$, $L=L_ k$, $h=h_ k$ and this probability volume form.

For $n=3$, $X_3$ is the Gauduchon threefold of theorem 23.1.1, on which Hermitian Monge–Ampère masses are unbounded; for $k>0$, the line bundles $L_ k$ nevertheless have positive degree and quadratic section growth. In the notation of example 11.21.4, $L_ k$ is the line bundle $E_ a$ on $X_ q$ with $q=\mathrm e^{-1}$ and $a=\mathrm e^{-k}$. Hence, for every $n\geq 2$ and $k>0$, the complement $\mathbb P(\mathcal O\oplus L_ k)\setminus \mathbb P(\mathcal O)$ of 11.21 has least smooth convexity index two, that is, it is $2$-convex but not $1$-convex in the sense of 11.17. When $k=0$, the line bundle is trivial and the least smooth convexity index of its complement is $n+1$, by the same example.