14.26.2 Three rank-one directions0CFD28034339

On $U=\{ A\ne 0,B\ne 0\} $ put

The logarithms are computed in a local homogeneous lift, but their Hessians do not depend on the lift. The forms $P,Q$ are pullbacks of Fubini–Study forms from the two projective lines $\mathbb P(\mathbb C^2)$. All three forms are semipositive of rank one, and

Indeed, in local coordinates $A=(1,a)$ and $B=\tau (1,b)$, $P$ and $Q$ are positive multiples of $\mathrm{i}\, \mathrm{d}a\wedge \mathrm{d}\bar a$ and $\mathrm{i}\, \mathrm{d}b\wedge \mathrm{d}\bar b$, while

Its $\mathrm{d}\tau $ component is nonzero, so the three rank-one forms correspond to independent directions. In particular, for nonnegative $x,y,z$,

and the form is positive exactly when all three coefficients are positive.

Since $\mathrm{dd}^{\mathrm{c}}r=P-Q$, the chain rule gives, for a smooth function $G:\mathbb R\to \mathbb R$,

Let $\sigma (r)=\mathrm{e}^ r/(1+\mathrm{e}^ r)$. Since $\log (|A|^2+|B|^2)=\log |B|^2+\log (1+\mathrm{e}^ r)$, we have

A function of $r$ that is constant near each end of $\mathbb R$ extends smoothly across $X\setminus U$. A form $xP+yQ+zR$ whose coefficients equal those of (14.26.6) for $|r|\geq 2$ extends smoothly by $\omega _ X$ across $X\setminus U$.