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 $\overline L\_{\alpha}$.



Arithmetic ampleness and scalings of the metric
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.


