20.3.41 Exact traces on Hopf manifolds06493A31E440

The projective base of a scalar Hopf manifold has higher dimension than the curve in subsection 20.3.37. A divisor on a modification of the Hopf manifold can therefore map to a set of codimension greater than one in the base. Extracting its image as a divisor on a modified base reduces the trace constraint to a closed-background problem. The elliptic action then identifies the full trace envelope with the pullback of the base envelope. Thus exact realization is possible for these data even though the target has unbounded mass.

Fix $n\geq 3$ and put

where $Y$ is defined by $z_ n=0$ and $B\subset \mathbb P^ n$ is the same hyperplane. Both maps are holomorphic submersions with connected elliptic fibers. Set $d=n-1$, and let $\kappa $ be the Fubini–Study form on $B$, normalized by $\int _ B\kappa ^ d=1$. With $s=\log |z|^2$ on the covering space of $Y$, the forms of subsection 23.1.32 satisfy

For $a,b>0$ write $\Omega _{a,b}=a\alpha +b\beta $; this is a Hermitian form on $Y$.

Take $U\in \mathrm{QPSH}(\mathbb P^ n)$ with $\nu (U,B)=0$, set $u=U\circ p$, and choose $C\geq 1$ with $C\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}U\geq 0$. Throughout the construction assume $a\geq 2C$ and $b>0$. The submersion identity of lemma 5.37.7, followed by restriction, gives

Local product coordinates also give $\nu (u,Y)=0$. The ideals $\mathfrak b_ t$ are nonzero, and their trace data $\tau _{B,U}$ is defined as in equation 20.3.3. We do not require $U|_ B$ to be a genuine quasi-plurisubharmonic function.