Lemma 11.30.1.8B6FA358A78D The manifold $X$ is smooth compact projective. The line bundle $L$ is globally generated, $A$ is very ample, and for every integer $k\geq 0$,
\[ \begin{aligned} H^0(X,L^ k)& = \bigl (H^0(\mathbb P^1,\mathcal O(k)) \oplus H^0(\mathbb P^1,\mathcal O(k-q))\bigr ) \\ & \qquad \otimes H^0(\mathbb P^{n-1},\mathcal O(k)). \end{aligned} \]
In the second summand, the sections are multiplied by the coordinate $w$ of the cover. In particular,
\[ h^0(X,L^ k)=\bigl (k+1+\max (k-q+1,0)\bigr ) \binom {n-1+k}{k}. \]
For a basis $\sigma _1,\ldots ,\sigma _ N$ of $H^0(X,A)$, the Fubini–Study weight and the corresponding root weight, in a local frame of $L$ and the induced frame of $A=L^ q$, are
\[ \phi _ A=\log \sum _{i=1}^ N|\sigma _ i|^2, \qquad \phi =\phi _ A/q. \]
The metric $\mathrm e^{-\phi }$ on $L$ has strictly positive curvature. Thus it is strictly Griffiths-positive, and $G(v)=\mathrm e^{\phi }|v|^2$ is strictly plurisubharmonic on $F$ off its zero section.
Proof.
The direct image of the structure sheaf of $C$ is $\mathcal O\oplus \mathcal O(-q)\, w$, with multiplication of the second summand given by $w^2=\beta $. Tensoring by $\mathcal O(k)$, the local basis $1,w$ gives the stated decomposition of sections on $C$. Simple zeros of $\beta $ make the cover smooth: near a branch point the equation gives a coordinate $w$ with the base coordinate a nonzero multiple of $w^2$ to leading order. The cover is connected because the rational function representing $\beta $ has a simple zero and therefore is not a square. Finiteness over compact $\mathbb P^1$ makes $C$ compact.
On $C$, the complete sections of $L_ C^ q$ are the degree-$q$ base monomials together with $w$. The map to projective space defined by these sections separates the base points by the Veronese coordinates, separates the two sheets by $w$, and has injective differential. At the ramification points the coordinate $w$ gives the missing derivative, and elsewhere the base coordinates do. Compactness makes this map an embedding. This shows that $C$ is projective and that $L_ C^ q$ is very ample.
For the decomposition of sections on the product, expand a section on each $\mathbb P^{n-1}$ fiber in its complete degree-$k$ monomial basis. The coefficients depend holomorphically on $C$ and transform as sections of $L_ C^ k$. This proves the displayed tensor identity without a cohomological vanishing assumption. The dimensions follow from the monomial counts, since line bundles of negative degree on $\mathbb P^1$ have no nonzero sections. The degree-one base sections generate $L_ C$, hence $L$. The product of the curve embedding and the degree-$q$ Veronese embedding, followed by the Segre embedding, shows that $A$ is very ample.
Its complete linear system therefore defines an immersion, so the Fubini–Study metric on $A$ has strictly positive curvature. Dividing its weight by $q$ defines a metric on $L$, since the transition functions of $A=L^ q$ are the $q$th powers of those of $L$, and its curvature is $1/q$ times that positive form. The final assertion follows from lemma 11.29.1.
$\square$
Pome
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