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Theorem 19.9.3EA8340BC8D54

Theorem 19.9.3.EA8340BC8D54 There are constants $C_1,\ldots ,C_ n\geq 0$, depending only on $L$ and the chosen flag, such that

\[ 0\leq \nu _ i(s)\leq C_ i m \quad (1\leq i\leq n) \]

for every $m\geq 1$ and every nonzero $s\in H^0(X,L^ m)$.

Proof. For each $i$, equip $Y_{i-1}$ with a Gauduchon metric $\gamma _ i$; when $Y_{i-1}$ is a curve, any Hermitian metric will do. Put $d_ i=\dim Y_{i-1}=n-i+1$ and

\[ a_ i=\int _{Y_ i}\gamma _ i^{d_ i-1}>0, \qquad A_ i=\deg _{\gamma _ i}(L|_{Y_{i-1}}). \]

For $i=n$ the integral defining $a_ i$ is the integral of $1$ over the point $p$, so $a_ n=1$.

For $j<i$, also put

\[ B_{ij}=\deg _{\gamma _ i} \left(\mathcal O_{Y_{j-1}}(Y_ j)|_{Y_{i-1}}\right). \]

All these numbers depend only on $L$, the flag and the chosen metrics.

At the $i$th stage of the construction of $\nu (s)$, the residual section is a section of the line bundle

\[ M_{i-1}=L^ m|_{Y_{i-1}}\otimes \bigotimes _{j<i}\mathcal O_{Y_{j-1}}(-\nu _ j(s)Y_ j)|_{Y_{i-1}}. \]

Its zero divisor contains $\nu _ i(s)Y_ i$. The Poincaré–Lelong formula, tested against $\gamma _ i^{d_ i-1}$ as in the proof of theorem 19.8.5, therefore gives

E150336DE571 19.9.4
\begin{equation} a_ i\nu _ i(s)\leq mA_ i-\sum _{j<i}B_{ij}\nu _ j(s). \end{equation}

Define recursively

\[ C_ i=\frac{|A_ i|+\sum _{j<i}|B_{ij}|C_ j}{a_ i}. \]

Induction on $i$ in (19.9.4) gives $\nu _ i(s)\leq C_ i m$. $\square$

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