**Added.** (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical variety has an isolated minimum. For example, Zagier has shown (*Algebraic numbers close to both $0$ and $1$*) that for the subvariety $Z : 1+x+y =0$ of the linear torus $\mathbb{G}_m^2$, the minimum, away from the few torsion points on $Z$, of the standard Weil height is $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big)$, with equality if and only if $x$ or $y$ is a primitive $10$th root of unity, *and this minimum is isolated*.

Another example, though of somewhat different flavor: the minimum height of a totally real algebraic number is, again, $\frac{1}{2} \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.2406059\ldots$, *and this minimum is isolated*. The lim inf of the height of a totally real algebraic number is at most $0.2732831\ldots$, and indeed it has been suggested that this is the lowest possible accumulation point for heights of totally real algebraic numbers. One realizes this accumulation point by taking, iteratively, $\xi_0 := 1$ and $\xi_n - \xi_n^{-1} := \xi_{n-1}$.

I wonder whether it is a general feature of both (a) Arakelov heights on arithmetical varieties (apart from the obvious examples); and (b) totally real or totally $p$-adic points on semiabelian varieties, to have an isolated minimum for the heights of their algebraic points.

**Original post.** 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, or more generally, with the canonical heights of Call-Silverman. A setup which generalizes both these examples is to have a self-map $f : X \to X$, an isomorphism $f^*L \cong L^{\otimes q}$ (over $\mathbb{Z}$, not just generically!) with some $q > 1$, and the height $\hat{h}_f(x) := \lim q^{-n}h(f^nx)$. This is an honest Arakelov (i.e., geometric) height precisely when both $f$ and the isomorphism $f^*L \cong L^{\otimes q}$ are defined over globally over $\mathbb{Z}$, rather than just generically over $\mathbb{Q}$.

Two questions:

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?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?