14.29.11 Extension over the projective endsEA62A44D2D9F

The coefficients just constructed have the right product but need not extend across the projective ends. Away from the zero levels of $f_*(t)$, we interpolate the coefficients of $P$ and $Q$ to those of $f_*(t)^{1/3}\omega _ X$, and determine the coefficient of $R$ by the prescribed product.

Choose a smooth even cutoff $0\leq \eta \leq 1$, equal to one on $|r|\leq 3/2$ and zero on $|r|\geq 2$. On $|r|<3/2$ set $(x,y,z)=(x_ b,y_ b,z_ b)$. On $|r|>5/4$, where $f_*(\sigma (r))>0$, put

These coefficients are smooth and positive on the outer region. On the overlap $5/4<|r|<3/2$, we have $\eta =1$, $p=q=0$ and $x_ by_ bz_ b=d_*$, so the two definitions agree. For $|r|\geq 2$ they equal $\kappa (\sigma (r),\sigma (-r),\sigma '(r))$. Thus

extends smoothly across both omitted projective lines, being equal near them to $f_*(t)^{1/3}\omega _ X$, which is smooth and positive there since $f_*(0)=f_*(1)=a^2b^2>0$. On $X$, the form $\omega _0$ is semipositive, positive away from $r=\pm 1$, and of rank two on these levels.

The function

is constant on each of the intervals $(-\infty ,-1]$ and $[1,\infty )$, and so $F(r)$ extends to a smooth nonconstant function on $X$. By (14.26.5), half of its complex Hessian is $H_0=\mathrm{dd}^{\mathrm{c}}F/2=p(P-Q)+qR$. By (14.29.10) and (14.29.3), the forms $A_0=\omega _0-H_0$, $B_0=\omega _0+H_0$ satisfy

on $U$, and hence on $X$. Their ranks and positivity were determined above, after (14.29.9). The construction is symmetric: $x(-r)=y(r)$, and $z$ is even.