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?