PomeExample 23.2.16.BBCE95D98A62 On the scalar Hopf manifold $H_ n=(\mathbb C^ n\setminus \{ 0\} )/\langle z\mapsto z/2\rangle $, for every $n\geq 2$, there is a strictly LCK metric with indefinite Lee curvature and a smooth semipositive representative of its ordinary Bott–Chern class. The first boundary reached for increasing $t>0$ in $\omega +t\kappa _\lambda $ has positive total volume.
Example 23.2.16BBCE95D98A62
Proof. Fix $0<\epsilon <1$ and put
\[ \rho =\sum _ j|z_ j|^2,\qquad \Phi =\rho +\epsilon \operatorname {Re}(z_1^2),\qquad \omega =\frac{\mathrm{dd}^{\mathrm{c}}\rho }{\Phi },\qquad \lambda =-\mathrm d\log \Phi . \]
The bounds $(1-\epsilon )\rho \leq \Phi \leq (1+\epsilon )\rho $ give positivity. Since $\mathrm{dd}^{\mathrm{c}}\Phi =\mathrm{dd}^{\mathrm{c}}\rho $ and both $\Phi $ and $\mathrm{dd}^{\mathrm{c}}\rho $ scale by $1/4$ under $z\mapsto z/2$, the metric and Lee form descend and satisfy the LCK equation. Polar coordinates identify the quotient with $S^{2n-1}\times (\mathbb R/(\log 2)\mathbb Z)$, so it is compact and connected. The Lee period from $z$ to $z/2$ is $2\log 2$, which proves strictness.
Here $\kappa _\lambda =\mathrm{dd}^{\mathrm{c}}\log \Phi $. With $F_ i=\bar z_ i+\epsilon z_1\delta _{i1}$, its complex Hessian is
\[ (\log \Phi )_{i\bar j} =\frac{\delta _{ij}}{\Phi } -\frac{F_ i\overline{F_ j}}{\Phi ^2}. \]
Relative to $\omega $, the eigenvalues are $1$ with multiplicity $n-1$ and $h=1-|F|^2/\Phi $. Writing $z_1=x+\mathrm i y$ and $B=\sum _{j>1}|z_ j|^2$, one finds
\[ h=\frac{\epsilon ((1-\epsilon )y^2-(1+\epsilon )x^2)}{(1+\epsilon )x^2+(1-\epsilon )y^2+B}, \qquad -\epsilon \leq h\leq \epsilon . \]
At $(1,0,\ldots ,0)$ the spectrum is $-\epsilon ,1,\ldots ,1$, hence is indefinite. The bounded smooth function $u=\log \rho -\log \Phi $ is deck invariant, and
\[ \kappa _\lambda +\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}\log \rho \geq 0, \]
by the Cauchy–Schwarz inequality for the Hessian of $\log \rho $. The correction also gives the standard Hopf conformal representative: $\mathrm e^{-u}\omega =\mathrm{dd}^{\mathrm{c}}\rho /\rho $.
The first boundary occurs at $t_0=1/\epsilon $. At that value,
\[ 1+h/\epsilon =\frac{2(1-\epsilon )y^2+B}{\Phi }. \]
It vanishes only on the real $z_1$ rays with the remaining coordinates zero. The other relative eigenvalues of $\omega +t_0\kappa _\lambda $ equal $1+1/\epsilon >0$. The boundary form is therefore semipositive and its top power is positive on a nonempty open set, proving positive total volume. $\square$
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