By a *polarized arithmetical variety* I will mean a pair $(X,L)$ of a finite-type proper regular integral scheme $X$ flat and generically smooth over $\mathbb{Z}$, and a relatively ample invertible sheaf $L 
\in \mathrm{PIC}(X)$ equipped with an $F_{\infty}$-invariant hermitian metric $\|
\cdot\|$ on the associated holomorphic line bundles $L_{\mathbb{C}}$, such that $\|\cdot\|$ is the uniform limit of positive $C^{\infty}$ metrics. 

There is then an Arakelov height function $h_L$ on the algebraic points $X(\bar{\mathbb{Q}})$, given by the arithmetic intersection number of the associated multisection with $\hat{c}_1(L)$, divided by $[\mathbb{Q}(x):\mathbb{Q}]$. Let me call such a polarization $(X,L)$ *strict harmonic* if the set
$$
\{ x \in X(\bar{\mathbb{Q}}) \quad | \quad h_L(x) = \inf_{X(\bar{\mathbb{Q}})} h_L \}
$$
is Zariski-dense. Let me call $(X,L)$ *harmonic* (or *non-strict harmonic*) if 
$$
\inf_{X(\bar{\mathbb{Q}})} h_L = \liminf_{X(\bar{\mathbb{Q}})} h_L,
$$
 where the lim inf is under the Zariski topology. One may ask whether or not the two conditions are in fact equivalent.

One can show, as a consequence of the arithmetic Riemann-Roch theorem, that if $(X,L)$ is harmonic, the infimum equals the normalized arithmetic self-intersection (or arithmetic volume) $L^{\dim{X}}  \Big/  L_{\mathbb{Q}}^{\dim{X_{\mathbb{Q}}}} \cdot \dim{X}$, and that moreover, the points of minimum height have their Galois orbits equidistributed in $c_1(L)$ (which, by definition, is a uniform limit of Chern forms of smooth metrics). Examples of strict harmonic arithmetical varieties include the canonical symmetric polarizations of abelian schemes over the full ring of integers of a number field (in which case the height $h_L$ is just the Neron-Tate canonical height, and the points of minimum height are precisely the torsion points); and, on the other hand, projective space with its standard Weil height.

Two questions:

1. Is it true that semistable elliptic curves over $\mathbb{Q}$ (this case being the simplest), or more generally abelian varieties with non-integral moduli, are never harmonic in the above sense, with respect to any symmetric canonical polarization? with respect to *any* (ample, symmetric) polarization? 

2. Is it true that an arithmetical surface of genus $> 1$ is never harmonic in the above sense? In particular, is the canonical polarization $\omega = \omega_{X/\mathbb{Z}}$ ever harmonic?



*Remark on question 1.* For an elliptic curve $E/\mathbb{Q}$ with minimal discriminant $\Delta$ and the canonical polarization with $L := \mathcal{O}_E([O])$ -- note that this polarization involves the canonical compactification of the Neron model, as well as the canonical metric on $L$ from Arakelov theory -- everything reduces to the question of whether there exists a sequence of points with Arakelov height $h_L(x)$ converging to $-\log{|\Delta|}/24$. 

Obviously, I am interested in this broader question: what are the arithmetical varieties for which admit a generic sequence of points of minimum height?