Theorem 11.9.4C222D1BB3C65

Proof. Step 1: the split structure. Fix $\mu $ and use lemma 11.9.1 to choose $k_ s$ with $R_ s\wedge \Theta _ s=\lambda _ s\mu $, where $\lambda _ s>0$. Put

For a squared norm $G$, define $I_0(G)=\sum _ s\int _ X\beta _ s(G)\wedge \Theta _ s$. By lemma 11.9.3, the term $\mathrm{dd}^{\mathrm{c}}U_ s$ integrates to zero against $\Theta _ s$, and the axis estimates of lemma 11.9.2 then give

Choose $\varepsilon $ so that $\varepsilon C<\lambda _*/2$, and keep all weights fixed.

Step 2: nearby holomorphic structures. Write a nearby operator as $\bar\partial _ A=\bar\partial _0+A$, with $\bar\partial _ A^2=0$. Its Chern connection for $k$ is $D_ A=D_0+A-A^{*k}$, where the adjoint changes the type from $(0,1)$ to $(1,0)$. Let $J_ A$ be the corresponding complex structure on the total space and $H_{A,v}$ the horizontal lift of $D_ A$. These depend smoothly on $A$, and $H_{A,v}$ is complex linear and tangent to the fixed sphere bundle $\Sigma $.

For $g\in C^\infty (\Sigma ,\mathbb R)$, define its homogeneous extension by

and define a real differential operator on $\Sigma $ by

It has order at most two. By the identity $\mathrm d^{\mathrm c}_ Jf=-(4\pi )^{-1}\mathrm df\circ J$, its coefficients depend on $A$ and its first derivatives, as well as on the fixed smooth data. Consequently they converge in $C^2$ as $A\to 0$ in $C^4$.

Step 3: the adjoint operator. Take the formal adjoint $\mathcal P_ A^*$ for the fixed density $\mathrm d\nu =\mathrm d\sigma _ x\, \mu (x)$ on the compact sphere bundle, and put $b_ A=\mathcal P_ A^*1$. Locally, if $\mathrm d\nu =m(y)\, \mathrm dy$ and $\mathcal P_ A=\sum _{|\alpha |\leq 2}c_{\alpha ,A}\partial ^\alpha $, then

Thus $\| b_ A-b_0\| _{C^0}\to 0$ as $A\to 0$ in $C^4$. For $g=G|_\Sigma $, homogeneity gives $\mathcal Eg=G$ and

Since $0<g\leq N\circ \pi $ and the spherical measures have mass one,

Choose the neighborhood so that $\| b_ A-b_0\| _{C^0}<\lambda _*/4$ in it. Then $I_ A(G)\leq -\frac14\lambda _*\int _ XN\mu <0$ for every smooth squared complex norm $G$. If $G$ were plurisubharmonic for $J_ A$, all horizontal pullbacks of $\mathrm{dd}^{\mathrm{c}}_{J_ A}G$ would be semipositive, and their pairings with $\Theta _ s$, weighted by $\rho _ s\geq 0$, would give $I_ A(G)\geq 0$, a contradiction. $\square$