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edited Sep 2, 2022 at 20:49

Let $\overline{L}$ be any arithmetically ample line bundle. In the way you have written it down, $\overline{L}_{\alpha}$ is not arithmetically ample for $\alpha$ sufficiently large, more precisely for $\alpha\ge\alpha_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L_{\mathbb{C}}^{\dim X-1}}\right)$:

Let $s\in H^0(X,L^{\otimes n})$ be any global section of any $n$-th tensor power of $L$. Then we have the following equality of arithmetic intersection numbers $$n\overline{L}^{\dim X}=\overline{L}\cdot\ldots\cdot\overline{L}\cdot\overline{L}^{\otimes n}=\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}-\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}.$$ By the arithmetic ampleness $\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}\ge 0$. Thus $$\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\ge -n\overline{L}^{\dim X}.$$ Let us write $|s|_{\alpha}$ for the norm of $s$ with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $|s|=\sqrt{h^{\otimes n}(s,s)}$, this means $|s|_{\alpha}=\sqrt{\alpha^n h^{\otimes n}(s,s)}=\alpha^{n/2}|s|$. Thus \begin{align*} \int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}&=\frac{n}{2}\log \alpha\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}+\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\\ &\ge \frac{n}{2}(\log \alpha) L_{\mathbb{C}}^{\dim X-1}-n\overline{L}^{\dim X}=\frac{n}{2}\left((\log\alpha)L_{\mathbb{C}}^{\dim X-1}-2\overline{L}^{\dim X}\right) \end{align*}\begin{align*} \int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}&=\frac{n}{2}\log \alpha\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}+\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\\ &\ge \frac{n}{2}(\log \alpha) L_{\mathbb{C}}^{\dim X-1}-n\overline{L}^{\dim X}\\ &=\frac{n}{2}\left((\log\alpha)L_{\mathbb{C}}^{\dim X-1}-2\overline{L}^{\dim X}\right) \end{align*} If $\alpha\ge\alpha_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L_{\mathbb{C}}^{\dim X-1}}\right)$, then $$\log\sup|s|_{\alpha}\ge\frac{\int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}}{\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}}\ge0.$$ Thus, $\sup|s|_{\alpha}\ge 1$ such that $s$ is not strictly small with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $s$ and $n$ were arbitrary, $\overline{L}_{\alpha}^{\otimes n}$ does not have any strictly small global sections for any $n$. In particular, $\overline{L}_{\alpha}$ is not arithmetically ample for every $\alpha\ge\alpha_0$. Note, that by the ampleness of $L_{\mathbb{C}}$ we always have $L_{\mathbb{C}}^{\dim X-1}>0$ such that $\alpha_0$ is always well-defined.

Let $s\in H^0(X,L^{\otimes n})$ be any global section of any $n$-th tensor power of $L$. Then we have the following equality of arithmetic intersection numbers $$n\overline{L}^{\dim X}=\overline{L}\cdot\ldots\cdot\overline{L}\cdot\overline{L}^{\otimes n}=\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}-\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}.$$ By the arithmetic ampleness $\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}\ge 0$. Thus $$\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\ge -n\overline{L}^{\dim X}.$$ Let us write $|s|_{\alpha}$ for the norm of $s$ with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $|s|=\sqrt{h^{\otimes n}(s,s)}$, this means $|s|_{\alpha}=\sqrt{\alpha^n h^{\otimes n}(s,s)}=\alpha^{n/2}|s|$. Thus \begin{align*} \int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}&=\frac{n}{2}\log \alpha\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}+\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\\ &\ge \frac{n}{2}(\log \alpha) L_{\mathbb{C}}^{\dim X-1}-n\overline{L}^{\dim X}=\frac{n}{2}\left((\log\alpha)L_{\mathbb{C}}^{\dim X-1}-2\overline{L}^{\dim X}\right) \end{align*} If $\alpha\ge\alpha_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L_{\mathbb{C}}^{\dim X-1}}\right)$, then $$\log\sup|s|_{\alpha}\ge\frac{\int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}}{\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}}\ge0.$$ Thus, $\sup|s|_{\alpha}\ge 1$ such that $s$ is not strictly small with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $s$ and $n$ were arbitrary, $\overline{L}_{\alpha}^{\otimes n}$ does not have any strictly small global sections for any $n$. In particular, $\overline{L}_{\alpha}$ is not arithmetically ample for every $\alpha\ge\alpha_0$. Note, that by the ampleness of $L_{\mathbb{C}}$ we always have $L_{\mathbb{C}}^{\dim X-1}>0$ such that $\alpha_0$ is always well-defined.

Let $s\in H^0(X,L^{\otimes n})$ be any global section of any $n$-th tensor power of $L$. Then we have the following equality of arithmetic intersection numbers $$n\overline{L}^{\dim X}=\overline{L}\cdot\ldots\cdot\overline{L}\cdot\overline{L}^{\otimes n}=\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}-\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}.$$ By the arithmetic ampleness $\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}\ge 0$. Thus $$\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\ge -n\overline{L}^{\dim X}.$$ Let us write $|s|_{\alpha}$ for the norm of $s$ with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $|s|=\sqrt{h^{\otimes n}(s,s)}$, this means $|s|_{\alpha}=\sqrt{\alpha^n h^{\otimes n}(s,s)}=\alpha^{n/2}|s|$. Thus \begin{align*} \int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}&=\frac{n}{2}\log \alpha\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}+\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\\ &\ge \frac{n}{2}(\log \alpha) L_{\mathbb{C}}^{\dim X-1}-n\overline{L}^{\dim X}\\ &=\frac{n}{2}\left((\log\alpha)L_{\mathbb{C}}^{\dim X-1}-2\overline{L}^{\dim X}\right) \end{align*} If $\alpha\ge\alpha_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L_{\mathbb{C}}^{\dim X-1}}\right)$, then $$\log\sup|s|_{\alpha}\ge\frac{\int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}}{\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}}\ge0.$$ Thus, $\sup|s|_{\alpha}\ge 1$ such that $s$ is not strictly small with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $s$ and $n$ were arbitrary, $\overline{L}_{\alpha}^{\otimes n}$ does not have any strictly small global sections for any $n$. In particular, $\overline{L}_{\alpha}$ is not arithmetically ample for every $\alpha\ge\alpha_0$. Note, that by the ampleness of $L_{\mathbb{C}}$ we always have $L_{\mathbb{C}}^{\dim X-1}>0$ such that $\alpha_0$ is always well-defined.

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created Sep 2, 2022 at 18:45

Robert Wilms

Let $s\in H^0(X,L^{\otimes n})$ be any global section of any $n$-th tensor power of $L$. Then we have the following equality of arithmetic intersection numbers $$n\overline{L}^{\dim X}=\overline{L}\cdot\ldots\cdot\overline{L}\cdot\overline{L}^{\otimes n}=\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}-\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}.$$ By the arithmetic ampleness $\left(\overline{L}|_{\mathrm{div}(s)}\right)^{\dim X-1}\ge 0$. Thus $$\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\ge -n\overline{L}^{\dim X}.$$ Let us write $|s|_{\alpha}$ for the norm of $s$ with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $|s|=\sqrt{h^{\otimes n}(s,s)}$, this means $|s|_{\alpha}=\sqrt{\alpha^n h^{\otimes n}(s,s)}=\alpha^{n/2}|s|$. Thus \begin{align*} \int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}&=\frac{n}{2}\log \alpha\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}+\int_{X(\mathbb{C})}\log |s|c_1\left(\overline{L}\right)^{\dim X-1}\\ &\ge \frac{n}{2}(\log \alpha) L_{\mathbb{C}}^{\dim X-1}-n\overline{L}^{\dim X}=\frac{n}{2}\left((\log\alpha)L_{\mathbb{C}}^{\dim X-1}-2\overline{L}^{\dim X}\right) \end{align*} If $\alpha\ge\alpha_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L_{\mathbb{C}}^{\dim X-1}}\right)$, then $$\log\sup|s|_{\alpha}\ge\frac{\int_{X(\mathbb{C})}\log |s|_{\alpha}c_1\left(\overline{L}\right)^{\dim X-1}}{\int_{X(\mathbb{C})}c_1\left(\overline{L}\right)^{\dim X-1}}\ge0.$$ Thus, $\sup|s|_{\alpha}\ge 1$ such that $s$ is not strictly small with respect to $\overline{L}_{\alpha}^{\otimes n}$. As $s$ and $n$ were arbitrary, $\overline{L}_{\alpha}^{\otimes n}$ does not have any strictly small global sections for any $n$. In particular, $\overline{L}_{\alpha}$ is not arithmetically ample for every $\alpha\ge\alpha_0$. Note, that by the ampleness of $L_{\mathbb{C}}$ we always have $L_{\mathbb{C}}^{\dim X-1}>0$ such that $\alpha_0$ is always well-defined.