Proposition 11.21.5F38A90B57ECC

Proof. Let $\alpha $ be the form induced by $\mathrm{dd}^{\mathrm{c}}\log |z|^2$. By the calculation in example 11.21.4, it is semipositive of rank $n-1$, with radial kernel. The direct sum metric has diagonal curvature, and the normalized curvature evaluated at $v$ is

If every $|a_ j|<1$, then every $b_ j<0$. The coefficient is then strictly positive for $v\ne 0$, so theorem 11.18.2 with $s=n-1$ shows that $U_ F$ is $(r+1)$-convex. On any embedded elliptic fiber $C$ of $X_ q\to \mathbb P^{n-1}$, each line metric has flat Chern connection by the logarithmic calculation in example 11.21.4, and so does their direct sum. Hence corollary 11.21.3 gives least index $r+1$ for $U_{F|_ C}$. This is a closed complex submanifold of $U_ F$, being its inverse image over $C$, and lemma 11.17.3 gives the lower bound.

If every $|a_ j|\geq 1$, then every $b_ j\geq 0$. For $v_ j\ne 0$, the function $|z|^{2b_ j}|v_ j|^2$ is plurisubharmonic by the calculation with the logarithm and the exponential in example 11.21.4. It is also smooth at $v_ j=0$, because $z\ne 0$, and by continuity its complex Hessian is nonnegative there as well. The sum $G$ is therefore plurisubharmonic on the covering space of the total space and descends to $F$. It is positive on $F^\times $ and homogeneous of degree two under complex scalars, so proposition 11.21.1 applies. $\square$