Arithmetic ampleness and scalings of the metric

Question

Let $\overline L= (L, h)$ be a hermitian $C^ \infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & Soule' papers).

$\overline L$ is said to be arithmetically ample if:

1. $L$ is relatively ample on $X$
2. $L_{\mathbb C}$ is positive on $X_{\mathbb C}$
3. A power of $L$ is generated by small sections.

Now assume that $L$ is relatively ample. It is not difficult to show that there exist a scaling $\alpha h$, with $\alpha\in\mathbb R_{>0}$, such that $\overline L_{\alpha}=(L,\alpha h)$ is arithmetically ample.

Then I ask the following:

Are there some $X$(maybe different from $\mathbb P^n_{\mathbb Z}$) and $L$ such that $\overline L_{\alpha}$ is arithmetically ample for any $\alpha>0$.

In other words, is it possible to find an arithmetically ample line bundle such that with any other scaling of the metric, it remains arithmetically ample?

It seems to be "just" a problem regarding successive minima in the theory of lattices... If I try to picture it in my mind, it seems impossible to find a linde bundle with such properties; there should be a lower bound $\alpha_0>0$ such that below that value it is impossible to find generating small sections for the powers of $\overline L_{\alpha}$.

Answer 1

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*} 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.