We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface.
These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone.
Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.

§ INTRODUCTION

Consider an algebraic variety $X$ equipped with an automorphism or endomorphism $f$.
Suppose also that $f$ has some hyperbolicity, e.g. the action of $f^*$ on cohomology $H^\bullet(X)$ has spectral radius larger than $1$.
When $X$ is defined over $\bC$, a basic first step in complex dynamics is to construct closed positive currents which are expanded or contracted by $f$.
When $X$ is defined over a global field, such as a number field, one is led to arithmetic dynamics considerations where it is often possible to construct height functions that have analogous equivariance properties under $f$.
The closed positive currents, or rather their potentials, can be viewed as the local heights at the archimedean place associated to the global height function.

This paper studies the case when there is a large group of automorphisms.
While it is possible to apply the standard constructions to each individual group element, taking into consideration the whole group leads to a substantially larger set of currents and heights, which also assemble into a well-behaved object.

§.§ Main results

We consider K3 surfaces since these have some of the most interesting automorphisms groups which are “large” in an appropriate sense.
Our assumptions are spelled out in <ref>, but in short, we assume that the Picard rank $\rho$ is at least $3$ and the automorphism group is a lattice in the corresponding $\SO_{1,\rho-1}(\bR)$.
Because we consider automorphisms and not endomorphisms, the canonical currents and heights are defined only for cohomology classes of self-intersection zero and on the boundary of the ample cone.
Furthermore, to assemble them into a well-behaved family, we must introduce a slightly larger space denoted $\partial^\circ \Amp_c(X)$, see <ref> for precise definitions.
With these preparations, here are our main two results.
In both cases, $X$ is a K3 surface satisfying the assumptions in <ref>.

[Canonical currents]

There exists a unique $\Aut(X)$-equivariant map

\[
\eta\colon \partial^\circ\Amp_c(X)\to \cZ_{1,1}(X)
\]

which is continuous and takes values in closed positive currents.
It is compatible with taking cohomology classes, the currents in the image have $C^0$ potentials and the map is also continuous for the $C^0$-topology of potentials on currents.

See <ref> for further details and the proof.
Furthermore the currents in the image also have laminarity properties, see <ref>.

[Canonical heights]

Suppose that $X$ is defined over a number field $k$ and let $\ov k$ be an algebraic closure.
There exists a unique $\Aut(X)$-equivariant map

\[
h^{can}\colon \partial^\circ\Amp_c(X)\to \cH(X)
\]

taking values in height functions and compatible with the map to the Picard group, such that for any $p\in X\left(\ov k\right)$ the function

\[
h^{can}(-;p)\colon \partial^\circ \Amp_c(X)\to \bR
\]

is non-negative, continuous, and equivariant for the scaling action of $\bR_{>0}$ on both sides.

See <ref> for further details and the proof.
To a given point $p\in X\left(\ov k\right)$ we can associate the star-shaped set in $\partial^\circ \Amp(X)$ where the height is bounded above by $1$.
<ref> presents some numerical simulations.
In <ref> we associate further invariants to this canonical height, which are in particular constant along $\Aut(X)$-orbits.

The star-shaped sets associated to canonical heights (in blue).
The red circles are cohomology classes of normalized mass $1$.
See <ref>.

§.§.§ Uniqueness

The currents constructed in <ref> are the unique closed positive representatives of their cohomology classes, when the cohomology class is not proportional to a rational one.
This is a result of Sibony–Verbitsky, included here as <ref>.
This then implies that the map $\eta$ in <ref> is unique with its stated properties.
For rational classes, there is always the current that comes from degenerating the Ricci flat metrics.
However, as soon as the Picard rank $\rho$ is at least $4$, one needs to “fill-in” a positive-dimensional space of currents for the rational classes, and these are in general distinct from the currents obtained via Ricci-flat metrics.
See <ref> for some examples, and <ref> for a further discussion.

The heights constructed in <ref> are also uniquely characterized by equivariance and continuity for fixed $p\in X(\ov k)$.
We expect that a different characterization, in the spirit of the Sibony–Verbitsky argument (<ref>) is possible.
It would involve proving the same uniqueness theorem for positive currents at the non-archimedean places of $k$.

§.§.§ Elliptic fibrations

To establish the above results, we need to obtain accurate estimates for the dynamics of parabolic automorphisms of the K3 surface, i.e. those that preserve an elliptic fibration.
The needed estimates are established with the help of the following intermediary results, which might be of independent interest.

Suppose that $X\xrightarrow{\pi}B$ is a genus one fibration, where $B$ is a curve and $X$ is a surface, all defined over the field $k$.
For currents we take $k=\bC$ and for heights $k$ is a number field.

[Preferred currents]

Under the assumptions of <ref> on $X\to B$, suppose that $\alpha$ is a smooth closed $(1,1)$-form on $X$ such that $\int_{X_b}\alpha=0$ i.e. $\alpha$ integrates to zero on some (hence every) fiber.

Then there exists $\phi\in L^{\infty}(X)$ such that $\alpha+dd^c\phi$ restricts to zero on every smooth fiber of the fibration.

For the proof and a more refined statement, see <ref>.
In fact $\phi$ can be arranged smooth over the open locus where the fibers are smooth, but the key point is global boundedness of $\phi$ on $X$.
We also obtain a “current-valued pairing”, see <ref>.
As a consequence of this construction, we obtain a “current-valued pairing” analogous to the results of <cit.>.

Here is also the height version of <ref>:

[Preferred heights]

Under the assumptions of <ref>, suppose that $L^0$ is a line bundle on $X$ that restricts to have degree $0$ on every irreducible component of any fiber of the elliptic fibration.

Then there exists a height function $h^{pf}_{L^0}$ associated to $L^0$ such that $h^{pf}_{L^0}$ restricts on each fiber $X_b$ as a canonical height associated to $L^0\vert_{X_b}$.

Note that we say “a” canonical height, because a canonical height depends on a choice of basepoint in each fiber, so the assertion is that $h^{pf}_{L^0}\vert_{X_b}$ equals such a canonical height up to some constant which is allowed to depend on $b\in B(\ov k)$.
For the proof and a more refined version, see <ref>.
Just like for the preferred currents in <ref>, the main difficulty is to obtain a height function defined on all of $X$.

§.§ Applications of the main results

§.§.§ Regularity of boundary currents

It is well-known that cohomology $(1,1)$-classes on the boundary of the Kähler cone of a compact Kähler manifold always contain closed positive currents, whose potentials are quasi-plurisubharmonic functions, and an important question is to understand when these currents can be chosen to be sufficiently regular.

This question is in general very hard, even in the case of K3 surfaces. For example, while one might naively expect that classes on the boundary of the Kähler (or even ample) cone of a K3 surface should contain a smooth semipositive form, this was recently shown to be false by the authors [32] using a Kummer rigidity result in dynamics [16, 33]. In these examples the class is an eigenclass for a hyperbolic automorphism of the surface.

However, in these dynamical counterexamples the currents have Hölder continuous (but not smooth) potentials.  A direct consequence of <ref> is then that every K3 surface that satisfies the assumptions in <ref> has the
property that every class on the boundary of the ample cone contains a closed positive current with continuous potentials.
This confirms the following conjecture of the second-named author <cit.> for these surfaces:

Every class $[\alpha]\in H^{1,1}(X,\mathbb{R})$ on the boundary of the Kähler cone of a compact Calabi-Yau manifold $X$ contains a closed positive current with bounded potentials.

On K3 surfaces, this conjecture is known when the selfintersection of the class is positive  <cit.>, or when it vanishes and the class is rational (up to scaling) <cit.>. For the remaining irrational classes with vanishing selfintersection, the conjecture was only previously known for the aforementioned eigenclasses of hyperbolic automorphisms.

§.§.§ Ricci-flat metrics

Let $X$ be a K3 surface that satisfies the assumptions in <ref>, equipped with a Kähler metric $\omega$ and a closed real $(1,1)$-form $\alpha$ whose class $[\alpha]\in \NS(X)_{\mathbb{R}}$ is on the boundary of the ample cone and has vanishing selfintersection. Thanks to Yau's Theorem [77], for any $t>0$ the cohomology class $[\alpha]+t[\omega]$ contains a unique Ricci-flat Kähler metric $\omega_t$, which is of the form $\omega_t=\alpha+t\omega+i\partial\overline{\partial}\vp_t$ for smooth potentials $\vp_t$, which can be normalized to have average zero. These metrics are degenerating as $t\to 0$, and understanding their behavior is a problem that has received much attention recently, see for example the recent survey [72].

While the picture is by now quite clear when $[\alpha]$ is a rational class (up to scaling, or equivalently it is pulled back from the base of an elliptic fibration), the behavior of the metrics $\omega_t$ remains mysterious when the class is irrational.
As an application of our results, we are able to shed some light:

Suppose $X$ is a K3 surface that satisfies the assumptions in <ref>, let $\alpha,\omega,\omega_t$ be as above, and suppose $[\alpha]$ is irrational. Then as $t\to 0$ the Ricci-flat metrics $\omega_t$ weakly converge to the canonical current in $[\alpha]$ given by  <ref>, and its normalized potentials $\vp_t$ satisfy a uniform $L^\infty$ estimate independent of $t$. Furthermore, the Gromov-Hausdorff limit of $(X,\omega_t)$ is a point.

This settles the second author's <cit.> for these surfaces. The $L^\infty$ bound also holds when $[\alpha]$ is rational, but in this case it was already known. On the other hand, when $[\alpha]$ is rational (and nontrivial), the Gromov-Hausdorff limit is a certain metric on $\mathbb{P}^1$, and the weak limit as currents need not coincide with the canonical current from <ref>, as we shall explain in <ref> and <ref>.

§.§.§ Small points and bounded orbits

In <ref> we include some simple applications of the canonical heights from <ref>.
Here is one, see <ref>:
[Bounded orbit equivalent to vanishing height]

Suppose that $p\in X(\ov k)$ and $[\alpha]\in \partial^\circ \Amp_c(X)$ is an irrational direction (see <ref>).
Let $\{\gamma_i\}_{i\geq 1}$ be the sequence of automorphisms associated to $[\alpha]$ and the generators, by the construction in <ref>.

Then $h^{can}([\alpha];p)=0$ if and only if the set $\{\gamma_n\cdots \gamma_1 p\}_{n\geq 1}$ is finite.

We also include in <ref> some results about finiteness of orbits along elliptic fibrations, with some mild adaptations to our case where the existence of a section is not assumed.
We will return to further arithmetic applications of these facts in another text.

In <ref> we associate an invariant to $\Aut(X)$-orbits of rational points:
[Total height]

For any $p\in X(\ov k)$ the natural volume of

\[
\{[\alpha]\in \partial^\circ \Amp_c(X)\colon h^{can}([\alpha];p)\leq 1\}
\]

is constant on the $\Aut(X)$-orbit of $p$.

In <ref> the set in question is bounded by the blue curve.

§.§.§ Suspension space

Associated to an algebraic K3 surface is a natural finite-volume hyperbolic manifold.
We introduce in <ref> a “suspended space” over this hyperbolic manifold and consider associated dynamical systems.
Some elementary results are included in <ref> and <ref>.
Some more challenging questions are formulated there as well; the one we find most interesting concerns unipotent flows, see <ref>.
Let us note that the results of Cantat–Dujardin [15] are very closely related to this suspended space, although they do not seem to translate directly to this setting (see however <ref>).

The currents provided by <ref> can be used to construct the measure of maximal entropy associated to the K3 surface.
This will be treated in another text.

§.§ Context and related works

§.§.§ Relation to the work of Cantat–Dujardin

The two recent papers of Cantat and Dujardin are concerned with closely related topics.
In <cit.> they construct closed positive currents on the boundary of the ample cone, for some irrational classes, obtained as limit points of random walks on $\Aut(X)$.
They also obtain uniqueness, by a method similar to that of Sibony–Verbitsky, as well as potentials.

It is convenient to compare our results using the language of hyperbolic geometry and the word metric on the group $\Aut(X)$.
See <ref> for some of the notation.
It is shown in [36] that, provided there is at least one parabolic automorphism, for a Lebesgue-typical geodesic the cusp excursions are quite long.

Specifically they prove that

\[
\lim \frac{\dist_{word}(1,\gamma)}{\dist_{hyp}([\omega_0],\gamma[\omega_0])} = +\infty
\]

where $[\omega_0]$ is a basepoint in hyperbolic space, and $\dist_{word},\dist_{hyp}$ denote the word length distance on $\Aut(X)$ and hyperbolic distance respectively.
On the other hand, for random walks the above limit is finite.
This implies that on the boundary, harmonic measure for random walks is singular for Lebesgue measure.
In particular, the set of cohomology classes covered by <cit.> is of zero Lebesgue measure.

In order to access the Lebesgue-generic points on the boundary of the ample cone, one needs to separately treat the parabolic automorphism and obtain sharper estimates.
This is in fact the bulk of the work in our case, both for currents (<ref>) as well as for heights (<ref>).

§.§.§ Relation to the work of Baragar

Let us additionally remark that Baragar [4] has considered some related constructions related to heights on K3 surfaces.
With van Luijk [11] they provide numerical evidence that a much stronger notion of canonical height does not exist.
This was established by Kawaguchi [46] for certain K3 surfaces and in greater generality by Cantat and Dujardin in [14] for a large class of surfaces.
Specifically, their definition of canonical heights would require an $\Aut(X)$-equivariant assignment of heights to any class in $\NS(X)$, not just the ones in the isotropic cone, and which is furthermore linear for the additive structure on $\NS(X)$.

Let us note that because of the necessity to consider the blown-up boundary $\partial^\circ\Amp_c(X)$, i.e. the non-uniqueness of canonical heights in rational boundary directions if $\rk \NS(X)\geq 4$, under this rank assumption and provided the existence of at least one elliptic fibration it follows from the construction in this text that canonical heights in this strong sense cannot exist.

§.§.§ Work of Sibony–Verbitsky

On K3 surfaces (and manifolds) Sibony and Verbitsky [70] have announced a quite general uniqueness result for positive currents of self-intersection zero.
We include a special case which is relevant for our work in <ref>, proving it using the methods described to the first-named author by Verbitsky.

§.§.§ Relation to work of DeMarco–Mavraki

The basic results on how the canonical height moves in a family of elliptic curves were obtained by Tate [71] and Silverman [64, 62].
In their recent work, DeMarco and Mavraki [24] further enhanced those results, showing in particular that the relevant height functions come from adelically metrized line bundles and enjoy equidistribution properties.

In the present text, we adapted the classical results of Tate and Silverman to our situation of a family of genus one curves with automorphisms, lacking in general a section.
One distinction is our <ref> yielding preferred heights on the total space of the fibration.
These objects could be of independent interest and we expect they also come from adelically metrized line bundles.

§.§.§ Betti map

Recall that given an elliptic fibration $X\to B$ over $\bC$ (or more generally a family of abelian varieties), one has a canonical “Betti form” $\omega_{Betti}$ on the locus of smooth fibers $X^{\circ}$ which in particular has the property that it restricts to the flat metric on each fiber.
This object, which apparently originated in <cit.>, was later rediscovered in <cit.> in the context of noncompact Calabi-Yau manifolds, and has since been extensively used in the study of degenerating Calabi-Yau metrics where it is dubbed a “semi-flat form” (see e.g. [42, 39, 44], and also <cit.> which gives a characterization of this form that also shows that it coincides with the Betti form), as well as in arithmetic considerations, see e.g. [2].

However, it is typically not possible to extend the Betti form to an object with useful regularity on all of $X$.
Analogously to the Betti form, one also has the –Tate cannonical heights on each fiber of the fibration, which again typically do not extend to a height function on all of $X$.

Our constructions of preferred currents and heights in <ref> and <ref> can be viewed as one way to address this issue.
If we consider forms, or heights, that have degree $0$ on the fibers, then it is possible to obtain objects global on $X$ with some regularity.

§.§ Outline

§.§.§ Standing assumptions

Throughout this paper, we make the following simplifying assumptions.
Our K3 surface $X$ is algebraic, of Picard rank $\rho\geq 3$, and contains no $(-2)$ curves.
A result of Sterk [69] implies that in this case $\Aut(X)$ maps (with finite kernel) to a lattice in the orthogonal group $\Orthog_{1,\rho-1}(\bR)$.
This also implies that all singular fibers of elliptic fibrations on $X$ are of Kodaira type $I_1$ or $II$.

In <ref>, which treats currents in elliptic fibrations, these assumptions are relaxed to allow any elliptically fibered Kähler surface with singular fibers of Kodaira type $I_1$ or $II$ (see <ref>).
In <ref>, which treats heights in elliptic fibrations, we allow quite general elliptic fibrations (see <ref>).
In particular, the results in these two sections do not assume that $X$ is a K3 surface.

In the results on heights we assume that the K3 surface is defined over a finite extension of $\bQ$.

§.§.§ Relaxing the standing assumptions

As it will become apparent from the method of proof, there is no need for most results to make the assumption on the absence of $(-2)$ curves on the K3 surface.
We expect that the main results on the existence of equivariant currents and heights continue to hold for any projective K3 surface, and will consider the general case in subsequent work.

Let us note also that there are plenty of examples satisfying our standing assumptions, for instance Wehler surfaces [76] defined by an equation of bidegree $(2,2,2)$ in $\bP^1\times \bP^1\times \bP^1$.

§.§.§ Acknowledgments

We are grateful to Misha Verbitsky for telling us about his proof of uniqueness of closed positive currents.
Ian Agol informed us about the notion of truncated hyperbolic space, appearing in <ref>, whose boundary is the relevant object for many of our constructions.
We have also benefited from conversations with Serge Cantat, Romain Dujardin, Bertrand Deroin, and Hans-Joachim Hein.
We are grateful to Serge Cantat for extensive feedback that significantly improved the exposition, and to Jeff Diller and Curt McMullen for further feedback.

This research was partially conducted during the period SF served as a Clay Research Fellow.
Both authors gratefully acknowledges support from the Institute for Advanced Study for an excellent research environment and for hosting SF and VT when a major part of this work was completed. VT would also like to thank the Department of Mathematics and the Center for Mathematical Sciences and Applications at Harvard University for their hospitality.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-2005470, DMS-1638352 (SF) and DMS-1903147, DMS-1610278 (VT).

§ HYPERBOLIC GEOMETRY BACKGROUND

§.§.§ Outline of section

We describe some of the geometric properties of the ample cone in the language of hyperbolic geometry, since we find it more convenient.
Specifically, in <ref> we introduce the groups and several flavors of hyperbolic spaces that we consider.
In <ref> we introduce the several flavors of boundaries that will appear in our results.
Finally, in <ref> we include some of the linear algebra estimates that are equivalent to geometric properties in hyperbolic space.
For a general introduction to hyperbolic geometry, the reader can consult the introductory notes [17] and the more comprehensive survey [74].

§.§ Groups and spaces

A fundamental domain with horoballs at the cusps, and a reference point in the fundamental domain.

§.§.§ Setup

Denote by $N:=\NS(X)$ the –Severi group of the K3 surface $X$, with $N_{\bZ}$ the corresponding lattice and $N_{\bR}$ the extension of scalars to $\bR$.
The intersection pairing on it has signature $(1,\rho-1)$ where $\rho=\rk N$.
By our standing assumptions on $X$ in <ref>, the group $\Aut(X)$ is a lattice in $\Orthog(N_{\bR})$.
We denote by $\Amp(X)\subset N_{\bR}$ the ample cone of $X$, which again by our standing assumptions is one of the connected components of the set of vectors $v\in N_{\bR}$ satisfying $v.v>0$.
We will further denote by $\Amp^1(X)$ the subset of ample classes satisfying $v.v=1$.
Then equipped with the (negative of) the induced metric, $\Amp^1(X)$ is isometric to hyperbolic $(\rho-1)$-space.

Define next $	\cM = \leftquot{\Aut(X)}{\Amp^1(X)}$ which is a finite-volume hyperbolic orbifold of dimension at least $2$.
The arguments in the rest of the text are insensitive to passing to a finite-index subgroup, so we assume from now on that $\Aut(X)$ is torsion-free.
In particular, cusp stabilizers (see below) can be assumed to be free abelian groups of rank $\bZ^{\rho-2}$.

§.§.§ Cusps of $\cM$

Under our standing assumptions, by <cit.>, see also <cit.>, given a vector $[E]\in N_{\bZ}$ which is primitive and satisfies $[E].[E]=0$, precisely one of $\pm [E]$ can be represented by a smooth genus one curve $E$ which furthermore determines a genus one fibration on $X$ with base $\bP^1$.
We always take $[E]$ to denote the class represented by a genus one curve.

The hyperbolic manifold $\cM$ need not be compact and its cusps are in bijection with $\Aut(X)$-orbits of vectors $[E]$ representing genus one fibrations.
Call any such class $[E]$ a parabolic boundary point.
It determines for each $c>0$ an (open) horoball in $\Amp^1(X)$ defined by

\[
H_{[E],c}:=\{[\omega]\in \Amp^1(X)\colon [\omega].[E]<c\}.
\]

Taking the complements of these horoballs yields

\begin{align}
\label{eqn:amp_c_defn}
\begin{split}
\Amp^1_c(X) & :=\{[\omega]\in \Amp^1(X)\colon [\omega].[E]\geq c\\
& \qquad \qquad \forall [E]\text{ parabolic boundary point}\}\\
\cM_c & = \leftquot{\Aut(X)}{\Amp^1_c(X)}
\end{split}
\end{align}

Take $c>0$ sufficiently small such that the inclusion $\cM_c\into \cM$ induces an isomorphism $\pi_1(\cM_c)\toisom \pi_1(\cM)$ and the complement $\cM\setminus \cM_c$ is a finite disjoint union of “standard hyperbolic cusps”, i.e. quotients of a horoball by an abelian subgroup of cofinite volume.
We regard $c>0$ as fixed from now on, and picked sufficiently small for later parts of the argument.

For each parabolic class $[E]$ let $\Gamma_{[E]}\subset \Aut(X)$ be its stabilizer.
It is a group commensurable to $\bZ^{\rho-2}$.
Since we passed to a finite index subgroup of $\Aut(X)$ to remove torsion, we can assume that $\Gamma_{[E]}\isom \bZ^{\rho-2}$, which we now do.

§.§.§ Fundamental domain, basepoint

We fix a basepoint $[\omega_0]\in \Amp^1_c(X)$ and its associated Dirichlet fundamental domain $F_{[\omega_0]}$.
This fixes also representatives of $\Aut(X)$-equivalences class of cusps, given by finitely many parabolic points $[E_1],\ldots, [E_p]$, and their corresponding parabolic automorphism groups.
Denote by $F_{[\omega_0],c}:=F_{[\omega_0]}\cap \Amp^1_c(X)$ the compact part of the fundamental domain.

§.§.§ Recurrent and divergent rays

Let $s$ be any hyperbolic ray starting in the fundamental domain $F_{[\omega_0],c}$.
When projected to the finite volume hyperbolic manifold $\cM$, either $s$ returns infinitely often to the compact set $\cM_c$, or not.
Rays that return to compact sets are called recurrent, and divergent otherwise.
If the endpoint of the ray $s$ is a parabolic boundary point, i.e. proportional to a rational vector, then the ray is divergent, and otherwise the ray is recurrent.

A recurrent geodesic.
The centers of fundamental domains along the way, and cusp excursions.
The parameters $\lambda_i$ as the distances between the projections of the centers of fundamental domains to the geodesic.

We will be interested in parametrizing the list of translates of the compact part of the fundamental domain $\gamma F_{[\omega_0],c}$, that are encountered by geodesic rays.

§.§.§ Fixed finite set of generators

Recall that we have the fixed representatives for each cusp and parabolic groups $\Gamma_{[E_i]}$ with $i=1\ldots k$.
We fix a finite set of generators $S\subset \Aut(X)$ (or rather, its finite index subgroup that we are considering) such that the fundamental domain $F_{[\omega_0],c}$ can be taken to any of its adjacent translates by one of the generators.
Then we have the following geometric construction, which writes the elements of the fundamental group as products of generators and parabolic elements in $\Gamma_{[E_i]}$.

Let $s$ be any hyperbolic ray starting in the fundamental domain $F_{[\omega_0],c}$.
Any point on the geodesic $s$ is contained in some translate of the fundamental domains $F_{[\omega_0]}$, and this divides up $s$ into segments.
Given one such segment, one possibility is that it, as well as the next one, are contained in translates of the compact part $F_{[\omega_0],c}$.
Then we will use a generator from $S$ to relate the two adjacent fundamental domains.
The other possibility is that there is a sequence of fundamental domains that $s$ encounters, such that the corresponding segments in $s$ are contained in translates of $F_{[\omega_0]}\setminus F_{[\omega_0],c}$.
The union of those segments is entirely contained in a horoball associated to a specific parabolic boundary point $[E]$.
If the ray $s$ eventually exits the horoball, then we can connect the first and the last fundamental domains by an element of $\Gamma_{[E]}$.

To summarize, attached to $s$ there exists a sequence $\{\gamma_i\}$ with $\gamma_{i}\in S\bigcup_{i=1}^p \Gamma_{[E_i]}$, where $[E_1],\ldots,[E_p]$ are representatives of the $\Aut(X)$-orbits on parabolic points, with the following properties:

* The sequence of points $\pi_s(\gamma_1\cdots \gamma_n[\omega_0])\in s$ on $s$ are a bounded hyperbolic distance from $\gamma_1\cdots \gamma_n[\omega_0]$.
Denote by $\lambda_i$ the (oriented) distance between the successive projections (see <ref>).

\[
\lambda_i:=\pm\dist(\pi_s(\gamma_1\cdots \gamma_i[\omega_0]), \pi_s(\gamma_1\cdots \gamma_{i+1}[\omega_0]))
\]

where the sign is $+$ if the points come in the written order as seen from the origin of the ray $s$ and $-$ otherwise.
The displacements $\lambda_i$ are typically positive (see <ref>).

* If the endpoint of $s$ is rational (the geodesic is divergent), then the sequence $\{\gamma_i\}$ is finite (with $n$ elements) and the geodesic $(\gamma_1\cdots \gamma_n)^{-1}s$ goes into the rational endpoint corresponding to some $[E_{i(s)}]$ among the fixed representatives.

* Furthermore, for any geodesic ray $s$, given any $n\geq 1$ there exists $\ve>0$ such that if the distance between the endpoints of $s,s'$, as seen in the visual metric from $[\omega_0]$ is less than $\ve$, then we can choose the corresponding sequence $\{\gamma_i(s')\}$ such that $\gamma_i(s)=\gamma_i(s')$ for all $i\leq n$.

In the case when $s$ is rational, say with sequence $\gamma_1,\cdots, \gamma_n$, the last requirement is to be understood as saying that $\gamma_{n+1}(s')\in \Gamma_{[E_{i(s)}]}$ and given any $R>0$, we can pick $\ve>0$ such that $\norm{\gamma_{n+1}(s')}_{\NS}\geq R$ (see <ref> for $\norm{\bullet}_{\NS}$).
In other words, by picking $\ve>0$ sufficiently small, we can arrange that the cusp excursion is sufficiently long.
This last assertion can be verified in the upper half-space model $\bR^{\rho-2}\times \bR_{+}$ of hyperbolic space, by placing the parabolic point at infinity.
Then if $s$ is going into the parabolic point, it must be a vertical geodesic, and a neighborhood of the parabolic boundary point is given by the set of points $p\in \bR^{\rho-2}$ satisfying $\norm{p}\geq A$.
By taking $A$ sufficiently large, i.e. the neighborhood sufficiently small, we can ensure that the geodesic excursion into the cusp is sufficiently long.

There exist constants $C_0, \delta>0$ such that for any geodesic ray $s$ and sequence of displacements $\lambda_1,\ldots, \lambda_n,\ldots$ as above, we have the estimate for any $a,N\geq 1$:

\[
\delta \cdot N - C_0 \leq \lambda_a + \cdots + \lambda_{a+(N-1)}
\]

Let $\gamma_1,\ldots, \gamma_n,\ldots$ be the generators corresponding to the fundamental domains visited by the ray $s$.
On the automorphism group $\Gamma$, let $d^{orb}$ denote the distance function obtained from embedding it to $\Amp^1$ via the orbit of $[\omega_0]$, and restricting the hyperbolic distance function.
Then we have

\[
d^{orb}(\gamma_1\cdots \gamma_a , \gamma_1\cdots \gamma_{a+N}) \leq \lambda_a+\cdots \lambda_{a+(N-1)} - C'
\]

for some fixed constant $C'$, since the centers of the fundamental domain and their projections to $s$ are a bounded distance away.

On the other hand we have the distance function $d^{orb,c}$ on $\Gamma$ obtained from the embedding into $\Amp_1^c$ with its path metric, so we clearly have $d^{orb}\geq d^{orb,c}$.
Finally, we have the estimate

\[
d^{orb,c}(\gamma_1\cdots \gamma_a , \gamma_1\cdots \gamma_{a+N})
\geq
\frac{1}{D}N - C''
\]

where $D,C''$ depends on the diameter of the compact part of the fundamental domain.
Concatenating the bounds yields the result.

§.§ Boundaries

§.§.§ Truncated hyperbolic space

The space $\Amp^1_c(X)$ defined in <ref> is a truncated hyperbolic space, see e.g. <cit.> and [57].
We will quote some useful results from <cit.>.
First, it is a complete, $\CAT(0)$, uniquely geodesic metric space.
The geodesics are described in Cor. 11.34 of loc.cit. as a concatenation of hyperbolic geodesics in the interior of $\Amp^1_c(X)$, and flat geodesics in the boundary horospheres, such that the successive pieces meet tangentially.

Observe that the horoball associated to a class $[E]$ is defined by $[\omega].[E]<c$ and its boundary is naturally an affine space over $[E]^{\perp}/[E]$, where $[E]^{\perp}\subset N$ denotes the orthogonal to $[E]$ for the intersection pairing, and hence contains $[E]$ since $[E]$ is isotropic.
In particular the quotient $[E]^{\perp}/[E]$ is of dimension $\rho-2$ and carries a non-degenerate negative-definite inner product (for which we flip the sign to make it positive-definite).
In particular, the visual boundary of the horosphere is naturally the unit sphere $\bS([E]^{\perp}/[E])$.

§.§.§ The boundary of truncated hyperbolic space

A description of geodesics also gives a description of the visual $\CAT(0)$-boundary, see II.8 in loc.cit. for the definition of visual boundary.
According to <cit.> the boundary $\partial \Amp^1_c(X)$ is homeomorphic to a sphere $\bS^{\rho-2}$ with a countable collection of dense pairwise disjoint open discs removed, one disc for each cusp.
Recall that under our assumptions on the K3 surface, each cusp is identified with a rational point on the boundary of the full hyperbolic space $\Amp^1(X)$.
Thus set-theoretically we have that

\[
\partial \Amp^1_c(X) = \partial \Amp^1(X)^{irr} \coprod_{[E]}\bS\left([E]^{\perp}/[E]\right)
\]

where $\partial \Amp^1(X)^{irr}$ denotes the irrational points on the boundary of the full hyperbolic space, and $\bS\left([E]^\perp/[E]\right)$ denotes the unit sphere associated to the quotient space with its natural nondegenerate (negative) definite inner product.
This set-theoretical decomposition is equivariant for the action of $\Aut(X)$.

Let us finally note that for each point $p\in\partial \Amp^1_c(X)$, we have a sequence of translates $\gamma_1\cdots \gamma_n[\omega_0]$ of the basepoint which approach it, such that the elements $\gamma_i$ belong to a fixed finite set $S$ or to one of finitely many parabolic groups $\Gamma_{[E]}$, as in <ref>.
If $p$ is not on one of the spheres, then the index $n$ goes to infinity.
If $p$ is on one of the spheres, then the index eventually remains constant equal to $n$ and only the last element changes, say $\gamma_{n,i}$ with $n$ fixed and $i\to +\infty$.
Furthermore, for each parabolic boundary point $[E]$ we will construct in <ref> an injection $\xi\colon \Gamma_{[E]}\into [E]^{\perp}/[E]$ and denote by $\norm{-}_{\NS}$ the positive-definite norm on $[E]^\perp/[E]$ induced from the intersection pairing and flipping the sign.
With this identification we have that $\frac{\xi(\gamma_{n,i})}{\norm{\xi(\gamma_{n,i})}_{\NS}}$ approaches $\left(\gamma_1\cdots \gamma_{n-1}\right)^{-1}p$ on the sphere of $[E]^{\perp}/[E]$, and furthermore $\norm{\xi(\gamma_{n,i})}_{\NS}\to +\infty$.

§.§.§ The projectivized visual boundary

We will need a variant of the above boundary, where we identify antipodal points on spheres, namely at the set-theoretical level we have:

\begin{align}
\label{eqn:boundary_stratification_projectivized}
\partial^{\circ}\Amp^{1}_c(X):= \partial \Amp^1(X)^{irr} \coprod_{[E]}\bP\left([E]^{\perp}/[E]\right)
\end{align}

To give this space a topology, we describe it alternatively as a projective limit of blow-ups of the rational points on the boundary:

\[
\partial^{\circ}\Amp^{1}_c(X) = \varprojlim_{R\to +\infty}  \Bl_{\cC_R} \partial \Amp^1(X)
\]

where $\cC_R$ denotes the cusp points whose representatives fiber classes $[E]$ satisfy $[\omega_0].[E]\leq R$.
This projective limit is independent of the choice of basepoint $[\omega_0]$ since the function on $\partial \Amp(X)$ given by $v\mapsto [\omega_0].v$ is comparable, up to multiplicative constants, with one given by another class $[\omega_0']$.
This description of the boundary is also closer in spirit to the one from <cit.>.
Note that to give a similar description of the visual boundary $\partial \Amp^1_c(X)$, we must simply replace blow-ups by real-oriented blowups.

§.§.§ Blown-up boundary of the ample cone

In order to state our main result, we must introduce a version of the space $\partial^\circ \Amp^1_{c}(X)$ from <ref> that also includes scalings.
Denote by $\partial \Amp(X)\subset \NS(X)_{\bR}$ the boundary of the ample cone (with the origin removed).
Under our assumptions on the K3 surface, this is equal to one of the components of $\{v\in \NS(X)_{\bR}\setminus 0\colon v.v=0\}$, i.e. a cone formed by null rays.

Let us define then

\[
\partial^\circ \Amp_c(X):= \varprojlim_{R\to +\infty} \Bl_{\cE_R}  \partial \Amp(X)
\]

where $\cE_R$ denotes the (finitely many) rays spanned by a primitive integral vector $[E]$ satisfying $[\omega_0].[E]\leq R$ and $\Bl_{\cE_R}$ denotes the blowup of the boundary of the ample cone along the given rays.
Note that the projective limit and induced topology are independent of the choice of $[\omega_0]$ and the induced map

\begin{align}
\label{eqn:fullbdy_to_projectivized}
\partial^\circ \Amp_c(X) \xrightarrow{\bullet/\bR_{>0}}
\partial^\circ \Amp^1_c(X)
\end{align}

obtained by quotienting by positive rescaling is continuous.
Finally, note that every point in $\partial^\circ\Amp_c(X)$ determines a cohomology class via the natural projection to $\partial\Amp(X)\subset \NS(X)_\bR$.

We also have the stratification analogous to <ref>

\begin{align}
\label{eqn:boundary_stratification}
\partial^{\circ}\Amp_c(X):= \partial \Amp(X)^{irr} \coprod_{[E]}
\Big(\bR_{>0}[E]\Big)\times\bP\Big([E]^{\perp}/[E]\Big)
\end{align}

where $\bR_{>0}[E]$ denotes the ray spanned by $[E]$.
To specify what it means for a function on $\partial^{\circ}\Amp_c(X)$ to be continuous, it suffices to specify what it means for a sequence of points $p_i$ in it to converge to another one $p$.
Let us denote by $[p_i],[p]$ their projections under $\partial^\circ\Amp_c(X)\to \partial \Amp(X)$, i.e. the corresponding cohomology classes.
If $p$ is in the irrational part, for convergence it suffices to have that the projections $[p_i]$ converge to the projection $[p]$.
If the point is in one of the rational strata, of the form say $p=[E]\times [\xi]$ (or some rescaling of it), then we must have that $[p_i]$ converge to $p$ in the blow-up $\Bl_{\bR_{>0}[E]}\partial \Amp(X)$ of the ray spanned by $[E]$.
Note that $p$ is by definition a point on the exceptional divisor of the blow-up.

To prove continuity of various quantities on the boundary, we will fix an ample class $[\omega_0]$ and use it to identify $\partial^\circ \Amp_c^1(X)$ with the subset of $\partial^\circ \Amp_c(X)$ formed of classes that have cup-product $1$ with $[\omega_0]$.
In that case, convergence of points on the boundary translates directly to geometric statements about geodesic rays in hyperbolic space, see <ref>.

§.§.§ Some examples

We describe the boundaries discussed above in some special cases, namely $\rho=3,4$.
Since the boundaries for the ample cone (without a normalization) are topologically just $\bR$ times the boundaries associated to hyperbolic space, we will only describe the latter.
We always assume that there are at least some parabolic points, otherwise all constructions reduce to usual boundaries.

When $\rho=3$, so $\Amp^1(X)$ is hyperbolic $2$-space, the usual boundary $\partial \Amp^1(X)$ is just a circle.
The oriented blown-up boundary $\partial \Amp^1_c(X)$ is a Cantor set, obtained from the circle by inserting an open set at the countable many rational points on the circle.
The projectively blown-up boundary $\partial^\circ \Amp^1_c(X)$ identifies pairs of points separated by a gap in the Cantor set, recovering the circle usual boundary.

When $\rho=4$, so $\Amp^1(X)$ is hyperbolic $3$-space, the usual boundary $\partial \Amp^1(X)$ is a $2$-sphere.
The oriented blown-up boundary $\partial \Amp^1_c(X)$ is a Sierpinski carpet, obtained from the $2$-sphere by inserting an open ball at the countable many rational points on the circle.
The projectively blown-up boundary $\partial^\circ \Amp^1_c(X)$ identifies opposite points on the boundary spheres to yield $\bR\bP^1$'s.
Now $\partial \Amp^1_c(X)$ and $\partial^\circ \Amp^1_c(X)$ are no longer manifolds.

§.§ Linear algebra and hyperbolic geometry

§.§.§ Setup

We now interpret some of the geometric constructions in <ref> in terms of linear-algebraic properties of the matrices corresponding to the group elements $\gamma_i$.
First, because the compact part of the fundamental domain $F_{[\omega_0],c}$ has bounded diameter, all projected basepoints are a uniform bounded distance from their projections in <ref> above.

The fixed basepoint $[\omega_0]\in \Amp^1(X)$ determines a positive-definite metric on $\NS(X)$, which we use from now on and denote $\norm{\bullet}$.
The operator norm for this metric, for a matrix $g\in \SO(N_{\bR})$, is related to the hyperbolic distance by

\begin{align}
\label{eqn:operator_norm_hypb_dist}
\norm{g}_{op} = e^{\dist([\omega_0],g[\omega_0])}
\end{align}

and using the notation from <ref> we have that

\[
\frac 1C \leq
\frac{\norm{\gamma_i\gamma_{i+1}\cdots \gamma_{i+j}}_{op}}
{e^{\lambda_i + \phantom{\cdots} \cdots \phantom{\cdots} + \lambda_{i+j}}}
\leq C
\]

for some fixed constant $C$ and for all $i\geq 1, j\geq 0$.

§.§.§ Mass in cohomology

For $[\omega]\in \Amp(X)$ denote by $M([\omega]):=[\omega_0].[\omega]$, the mass with respect to the reference class $[\omega_0]$.
This will be a useful normalization factor to bring all cohomology classes to a bounded set.
Additionally, there exists a constant $C>0$ (depending on $[\omega_0]$ or equivalently the fixed inner product) such that

\[
\frac{1}{C}\norm{[\omega]} \leq M([\omega]) \leq C\norm{[\omega]} \text{ for all }[\omega] \text{ s.t. }[\omega]^2\geq 0 \text{ and }[\omega].[\omega_0]\geq 0
\]

Indeed, the comparison follows immediately on the compact set of classes with $M([\omega])=1$ and inside the ample cone, and the inequalities are invariant by rescaling.

§.§.§ Transversality and growth

Any $g\in \SO(N_{\bR})$ has a KAK (or Cartan, or polar), decomposition as $g=k_1ak_2$ where $k_i$ preserves the fixed norm $\norm{\bullet}$ on $N_{\bR}$ (equivalently, the basepoint $[\omega_0]$) and the element $a$ expands a null-vector by $\norm{g}_{op}$ and contracts another null-vector by $\norm{g}_{op}^{-1}$.
The span of the two null-vectors contains $[\omega_0]$.

We will say that $v\in N_\bR$ is $C$-transverse for $g\in \SO(N_{\bR})$ if we have that

\[
\norm{gv} \geq \frac{1}{C}\norm{g}_{op}\norm{v}
\]

If the constant $C$ only depends on some parameters fixed in terms of the hyperbolic manifold $\cM$, we will say that $v$ is transverse for $g$ without specifying the parameter $C$.

Note that being $C$-transverse is invariant under scaling $v$.
Given a KAK decomposition $g=k_1a k_2$, the condition that the unit vector $v$ is transverse for $g$ means that $k_2v$ is not too close to the unit eigenvector contracted by $a$, with distance depending on the amount of transversality.
Relative to the reference basepoint $[\omega_0]$, a vector $v$ is $C$-transverse for $g$ if $v$ and $g^{-1}[\omega_0]$ are not too close, projectively.
If $v$ satisfies $v.v=1$ (so it determines a point in hyperbolic space), then $C$-transversality is equivalent to $v$ not being contained in a neighborhood of the endpoint at infinity of the geodesic ray starting at $[\omega_0]$ and going to $g^{-1}[\omega_0]$.

Because the generators $\gamma_i$ in <ref> are chosen in terms of the visits of a hyperbolic geodesic to fundamental domains, it follows that we have the following properties:

* The vector $[\omega_0]$ is $C$-transverse for $\gamma_i\cdots \gamma_j$ for any $j\geq i\geq 1$.

* The vectors $\gamma_{j+1}\cdots \gamma_{j+k}[\omega_0]$ are $C$-transverse for $\gamma_i\cdots \gamma_j$ for any $k\geq 1$ and $j\geq i\geq 1$

for some constant $C$ that only depends on $\cM$ and $[\omega_0]$.
These estimates are simply restatements of the geometric properties of the geodesics, translated via the identification of operator norms and hyperbolic distances in <ref>.

From now on, to avoid making constants explicit, we will use the notation $A\leqapprox B$ to mean that there exists some constant $c>0$, depending on $\cM$ and $[\omega_0]$, such that $A\leq c\cdot B$.

Suppose that $x,y\in N_\bR$ are two vectors that are transverse for an element $g\in \SO\left(N_\bR\right)$.
Then we have the estimate

\[
\norm{
\frac{gx}{M(gx)}
\frac{gy}{M(gy)}
} \leqapprox \frac{1}{\norm{g}_{op}}
\]

Let us write the KAK decomposition $g=k_1ak_2$ and let $v_-,v_+$ be the vectors contracted by $a$, each of mass $1$, so that $2[\omega_0]=v_- + v_+$.
The transformation $k_1$ does not affect the estimates so we can assume that $k_1=\id$.
We then have that

\[
k_2 x = c_-(x)v_- + c_+(x)v_+ + v_0(x)
\]

with $v_0(x)$ orthogonal to both $v_-$ and $v_+$.
By the transversality of $x$ for $g$ we know that $c_+\geq c>0$ where $c$ is some constant determined by the amount of transversality.
By a direct calculation it follows that $\frac{ak_2x}{M(ak_2x)}=v_+ + \frac{O(1)}{\norm{g}_{op}}$.
A similar argument applies to $y$ and the result follows.

§ ELLIPTIC FIBRATIONS AND CURRENTS

§.§.§ Outline

The main result of this section is <ref>, which establishes the existence of canonical currents associated to parabolic automorphisms of an elliptic fibration. In this section we can relax considerably our standing assumptions in <ref> to the following more general setup.

§.§.§ Standing assumptions for this section

Throughout this section, $X$ denotes a compact surface with a genus one fibration

\[
X \xrightarrow{\pi} B
\]

where $B$ is a compact complex curve, such that the following holds:

The singular fibers of $\pi$ are reduced and irreducible, i.e. the singularities have Kodaira type $I_1$ or $II$.

Denote by $\Aut_\pi(X)$ the automorphisms of $X$ that preserve the fibers of $\pi$.
In most cases, including K3 surfaces, this group is of finite index in the group of all automorphisms of $X$ taking fibers to (possibly other) fibers.
Indeed, in the K3 case (where $B=\bP^1$), the induced group of automorphisms of $\bP^1$ must preserve $\pi_*\dVol$ and hence be of finite order.

Observe also that when $X$ is a K3 surface, the standing assumption of this section is implied by our earlier standing assumptions in <ref>, since by <cit.> any singular fiber which is not of type $I_1$ or $II$ contains a $(-2)$-curve.

§.§ Calculations in cohomology

§.§.§ Setup

For a point $b\in B$ denote by $X_b:=\pi^{-1}(b)$ the corresponding fiber and let $[E]\in \NS(X)$ denote the class of a general fiber of the fibration.
Importantly, in this section we consider the action of $\Aut(X)$ on $\NS(X)$ by pushforward; this is relevant for keeping track of some signs.
The following filtration of the –Severi group is preserved by $\Aut_\pi(X)$:

\begin{align}
\label{eqn:cohomology_filtration_elliptic_fibration}
0\subsetneq [E] \subsetneq [E]^{\perp} \subsetneq \NS(X)
\end{align}

where $[E]^{\perp}$ denotes the orthogonal complement for the intersection pairing.
The induced intersection pairing on $[E]^\perp/[E]$ is strictly negative-definite and we let $\Aut_\pi^\circ(X)$ denote the subgroup of $\Aut_\pi(X)$ whose induced action on $[E]^\perp/[E]$ is trivial.
This is a finite-index subgroup of $\Aut_\pi(X)$.

§.§.§ Linearizing automorphisms

Because $\Aut_\pi^\circ(X)$ preserves the filtration in <ref> and acts as the identity on subquotients, there is a group homomorphism

\begin{align}
\label{eqn:aut_pi_linearization_cohomology}
\begin{split}
\Aut_\pi^\circ(X) & \to \Hom\left(\rightquot{[E]^\perp}{[E]},[E]\right)\\
\gamma & \mapsto \xi(\gamma)(v)= \gamma_*(v)-v
\end{split}
\end{align}

Via duality and identifications of duals using the intersection pairing, we have equivalently

\[
\xi(\gamma)^t \in \Hom\left(\NS(X)/[E]^{\perp},\rightquot{[E]^\perp}{[E]}	\right)
\]

Since the class of the fiber $[E]$ is canonically defined, both spaces on the right-hand side above can be canonically identified with $[E]^\perp/[E]$.
By abuse of notation, we occasionally identify here and below a free $\bZ$-module of rank $1$ with its generator (and in our situation we have a distinguished choice of one of the two generators).

* The map from <ref> injects $\Aut_\pi^\circ(X)$ into $[E]^\perp/[E]$.

* Given $\xi\in [E]^\perp/[E]$, define

\[
n_{\xi}(v) := (v.[E])\xi - (v.\xi)[E] \quad \forall v\in \NS(X)
\]

This is a nilpotent transformation, well-defined and independent of the lift of $\xi$ from $[E]^\perp/[E]$ to $[E]^\perp$, and strictly decreasing the filtration from <ref>.
Furthermore, we have that $n_\xi\left(\NS(X)/[E]^\perp\right)\isom (\xi+[E])/[E]$ and $n_\xi(\xi)=-(\xi.\xi)[E]$, and $n_\xi$ is an element of the orthogonal Lie algebra, i.e. $n_\xi(v).w + v.n_\xi(w)=0$.

* The action of $\gamma\in \Aut_\pi^{\circ}(X)$ on $\NS(X)$ is given by

\[
1 + n_{\xi} + \tfrac12 {n_{\xi}^2} \quad \text{ with } n_\xi:=n_{\xi(\gamma)}.
\]

In particular, we have for a vector $v\in \NS(X)$ and $\xi=\xi(\gamma)$ associated to $\gamma\in \Aut_\pi^{\circ}(X)$ that the action is given by

\begin{align}
\label{eqn:cohomology_action_explicit}
\gamma(v) = v + \Big[(v.[E])\xi - (v.\xi)[E]\Big] + (v.[E])\frac{-\xi.\xi}{2}[E],
\end{align}

a well-known formula, see e.g. <cit.>.
Note that $\xi.\xi<0$ since the intersection pairing on $[E]^\perp/[E]$ is strictly negative-definite.

The map from <ref> is injective, because more generally the kernel of the map $\Aut(X)\to \Orthog(\NS(X))$, when $h^{1,0}=0$, is a group of automorphisms of $X$ preserving an ample line bundle (so they are naturally a subgroup of some $\PGL_n$).
The subgroup of elements in $\Orthog(\NS(X))$ which preserve $[E]$ and act as the identity on $[E]^{\perp}/[E]$ is then naturally identified with $[E]^\perp/[E]$, as can be seen for instance by choosing an explicit basis.

For (ii) the stated properties of $n_{\xi}$ are checked directly from the formula defining it.

To establish (iii) observe that the properties of $n_\xi$ match those that define the logarithm of the action of $\gamma$ on $\NS(X)$.
Indeed $\log \gamma$ is a nilpotent transformation in the orthogonal Lie algebra that can be defined as $(\gamma-1) + \tfrac 12(\gamma-1)^2$.
The term $(\gamma-1)^2$ maps $\NS(X)/[E]^{\perp}$ to $[E]$, so does not contribute to the expression $(\gamma-1) v \mod [E]$ for a vector $v$.
Therefore $\log \gamma$ and $n_{\xi}$ are nilpotent transformations in the orthogonal Lie algebra which agree on $\NS(X)/[E]$, and hence agree.
Therefore $\gamma=\exp(n_{\xi})$ as claimed.

§.§.§ A remark on signs

The cup-product on $[E]^\perp/[E]$ is naturally negative-definite.
Because of this, in order to have convenient algebraic expressions, and to work with positive currents, we will have to introduce some minus signs when defining certain currents.
One such example is in <ref>, where the pairing $\eta(\gamma,[\alpha])$ gives a current in the cohomology class $-[\xi(\gamma)].[\alpha]$ on $B$.

§.§ Analytic properties of parabolic automorphisms

§.§.§ Setup

For the fibration $X\xrightarrow{\pi}B$ let $B^\circ\subseteq B$ denote the image of the smooth fibers and $X^\circ:=\pi^{-1}\left(B^{\circ}\right)$.
The fiber over $b\in B$ is denoted $X_b$ as before.
For this section, fix a metric $\omega$ on $X$.

Recall that an usc function $\phi:X\to\mathbb{R}\cup\{-\infty\}$ is called quasi-psh if locally it equals the sum of a psh function plus a smooth function.
Quasi-psh functions thus satisfy $dd^c\phi\geq \gamma$ weakly on $X$, for some smooth real $(1,1)$-form $\gamma$. More generally, a quasi-positive $(1,1)$ current is defined as $T=\alpha+dd^c\phi\geq \gamma$, where $\phi$ is quasi-psh and $\alpha,\gamma$ are smooth $(1,1)$-forms.
We can define the restriction of $T$ to any fiber $X_b,b\in B$ for which $\phi|_{X_b}\not\equiv -\infty$ simply by setting $T|_{X_b}=\alpha|_{X_b}+dd^c(\phi|_{X_b})$, where in the case when $X_b$ is singular we refer e.g. to <cit.> for details about smooth forms and currents on singular complex analytic spaces.

For a compact complex manifold $M^n$ we will use the notation $\cA_{k,k}(M)$ to denote the space of currents dual to smooth $(k,k)$-forms on $M$ (i.e. currents of bidegree $(n-k,n-k)$). Thus, the space of $(1,1)$ currents on the base $B$ is equal to $\cA_{0,0}(B)$.

If $T$ is a $(1,1)$ current on $M$ and $\omega$ is a Kähler metric, we will use the standard notation $\mathrm{tr}_\omega T$ for the distribution defined by

$$\mathrm{tr}_\omega T=\frac{n \omega^{n-1}\wedge T}{\omega^n}.$$

In particular, for a smooth function $u$ we define its $\omega$-Laplacian by $\Delta_\omega u=\mathrm{tr}_\omega(dd^c u)$.

A smooth $(1,1)$-form $\alpha$ on $X$ is called $\pi$-trivial if for all $b\in B$ we have $\alpha\vert_{X_b}\equiv 0$.
Similarly, a quasi-positive $(1,1)$-current $T$ on $X$ is $\pi$-trivial if for all $b\in B$ the restriction $T|_{X_b}$ is well-defined and is zero.
We also define $\pi$-triviality on $X^\circ$ if the above condition holds only for all $b\in B^\circ$.

* Note that $\pi$-trivial forms need not be pulled back from $B$.
However, if a $\pi$-trivial form $\alpha$ is of the form $\alpha=dd^c\phi$ for a function $\phi$ then there exists some $\phi'$ on $B$ with $\pi^{*}\phi'=\phi$ since the condition $dd^c\left(\phi\vert_{X_b}\right)\equiv 0$ implies that $\phi$ is constant on fibers.

* However, if $\eta$ is a $\pi$-trivial closed $(1,1)$-current which is in the class of a fiber, then $\eta=\pi^*\eta_B$ for a closed $(1,1)$-current $\eta_B$ on $B$.
Indeed, pick some smooth $\omega_B$ on $B$ such that $[\eta]=\pi^*[\omega_B]$ in cohomology and write $\eta = \pi^*\omega_B + dd^c \phi$ for some $\phi$.
For every fiber we have $\eta\vert_{X_b}= 0$ (since $\eta$ is $\pi$-trivial) and $\pi^*\omega_B\vert_{X_b}\equiv 0$ so it follows that $dd^c\left(\phi\vert_{X_b}\right)\equiv 0$, which in turn implies $\phi$ is constant on fibers.

* Additionally, if $\eta$ is a closed positive $(1,1)$-current in the class of the fiber, then it is necessarily $\pi$-trivial and pulled back from $B$.
Indeed, writing $\eta=\pi^*\omega_B + dd^c\phi$ it follows that $dd^c\phi\geq 0$ on each fiber and hence $\phi$ is constant on fibers.

Assume all singular fibers of $\pi$ are reduced and irreducible (i.e. of Kodaira types $I_1$ and $II$). Let $\alpha$ be a closed smooth real $(1,1)$-form on $X$, which satisfies

\begin{equation}
\label{eqn:alpha_zero_integral}
\int_{X_b}\alpha=0,\quad \text{for all }b\in B
\end{equation}

or equivalently $[\alpha].[E]=0$.

Then there exists $\phi\in L^\infty(X)\cap C^{\infty}\left(X^\circ\right)$, unique up to a function pulled back from $B$, such that

\[
\alpha + dd^c \phi \text{ is }\pi\text{-trivial on }X,
\]

\[
\norm{\phi}_{L^\infty(X)} \leq C(X,\omega)\norm{\alpha}_{L^\infty(X,\omega)}.
\]

Using the arguments in the proof of <ref> and more work, it is possible to show that the function $\phi$ in <ref> is smooth on $X$ minus the (finitely many) singular points of the singular fibers.
Since this information is not needed here, we will not belabor this point.

We first construct the desired function $\phi$ on $X^\circ$. For this, given any $b\in B^\circ$, the condition that $\alpha+dd^c\phi$ is $\pi$-trivial restricts to $X_b$ to

and this equation on $X_b$ is certainly solvable since <ref> gives that $\int_{X_b}\alpha=0$ and so $\alpha|_{X_b}$ is null-cohomologous and hence $dd^c$-exact. Since the fiberwise solution is unique up to a constant, we see that

for every $b\in B^{\circ}$ there is a unique function $\phi_b\in C^\infty(X_b)$ solving

\begin{equation}\label{fiberwise}
dd^c\phi_b=-\alpha|_{X_b},\quad \int_{X_b}\phi_b\omega_b=0,
\end{equation}

on $X_b$.
It follows from standard Hodge theory with parameters (i.e. elliptic estimates on the fixed smooth manifold underlying $X_b$ with respect to the smoothly varying family of Riemannian metrics defined by $\omega_b$) that the functions $\phi_b$ vary smoothly in $b\in B^\circ$.
Thus, taken together, they define a smooth function $\phi$ on $X^\circ$ (i.e. $\phi|_{X_b}=\phi_b$) which satisfies

$$(\alpha+dd^c\phi)|_{X_b}=0,\quad \text{for all }b\in B^{\circ},$$

and so $\alpha+dd^c\phi$ is $\pi$-trivial on $X^\circ$.

Next let us discuss what happens at the singular fibers, so let $0\in B\setminus B^\circ$ be a critical value on the base, with singular fiber $X_0\xrightarrow{\iota} X$ and singular point $x_0\in X_0$.	Let $\wtilde{X_0}\xrightarrow{\nu}X_0$ be its normalization, which maps a finite set $S\subset\wtilde{X_0}\isom \bP^1$ to $x_0 \in X_0$ and is biholomorphic on the complement, and denote by $\mu=\iota\circ\nu$. The set $S$ consists of two points when $X_0$ is of type $I_1$ and of one point when it is of type $II$.

Assumption <ref> implies that $\int_{\wtilde{X_0}}\mu^*\alpha=0,$ and so $\mu^*\alpha$ is null-cohomologous on $\wtilde{X_0}$ and hence $dd^c$-exact. Since $\mu^*\omega$ is a semipositive $(1,1)$-form on $\wtilde{X_0}$ which is Kähler away from a finite set (in particular $\int_{\wtilde{X_0}}\mu^*\omega>0$), we can find a  unique function $\ti{\phi}_0\in C^\infty(\wtilde{X_0})$ satisfying

\begin{equation}\label{hodge}
\int_{\wtilde{X_0}}\ti{\phi}_0\mu^*\omega = 0 \qquad \text{and} \qquad dd^c\ti{\phi}_0=-\mu^*\alpha,
\end{equation}

Then $\phi_0:=\nu_*\ti{\phi}_0$ is a bounded function on $X_0$ which satisfies $\alpha|_{X_0}+dd^c \phi_0=0$, and so extending $\phi$ to $X_0$ by $\phi|_{X_0}=\phi_0$, and repeating this procedure at all singular fibers, we obtain a function $\phi$ on $X$, smooth on $X^\circ$, such that $\alpha+dd^c\phi$ is $\pi$-trivial on $X$.

Every other function $\phi'$ with this property will satisfy $dd^c(\phi'-\phi)|_{X_b}=0$ and so will differ from $\phi$ by the pullback of a function from $B$.

It remains to show that $\phi$ is bounded on $X$, and since by construction $\phi$ is bounded on the singular fibers we only need to prove boundedness on $X^\circ$, or in other words that there is a constant $C$ such that for all $b\in B^{\circ}$ we have

\begin{equation}
\label{eqn:phi_is_bounded}
\sup_{X_b}|\phi_b|\leq C.
\end{equation}

To see this, let $G_b$ be the Green's function of the Laplacian of $(X_b,\omega_b)$, which is symmetric in its two variables and is normalized by $\int_{X_b}G_b(\cdot,x)\omega_b=0$ for all $x\in X_b$, and which is defined by the property that for every smooth function $u$ on $X_b$ with $\int_{X_b}u\omega_b=0$ we have

$$u(x)=-\int_{X_b}(\Delta_{\omega_b}u) G_b(\cdot,x)\omega_b \quad \text{for all $x\in X_b$.}$$

The key claim then is that we can find $C_0>0$ such that for all $b\in B^{\circ}$ we have

\begin{equation}
\label{eqn:lower_bound_Green}
G_b(\cdot,\cdot)\geq -C_0,
\end{equation}

i.e. we have a uniform lower bound for the Green's function of the Laplacian of the degenerating Riemann surfaces $(X_b,\omega_b)$.
To establish this, we employ a classical argument of Cheng–Li [18], which is clearly explained in <cit.> (see also the recent exposition in <cit.>), which shows that to prove <ref> it suffices to obtain a uniform upper bound for the Poincaré constant of $(X_b,\omega_b)$, which is uniform for all $b\in B^{\circ}$ (using here that $X_b$ has real dimension $2$ and that  the volume of $(X_b,\omega_b)$ is constant as we vary $b$).

More precisely, the argument of Cheng–Li gives a uniform bound lower bound for the Green's function of $(X_b,\omega_b)$ in terms of a positive lower bound the volume $\int_{X_b}\omega_b$ (which holds trivially in the present case) and of an upper bound for the Poincaré constant $C_{P,b}$ of $(X_b,\omega_b)$, which is characterized by

$$\int_{X_b} u^2 \omega_b \leq C_{P,b} \int_{X_b}|d u|^2_{\omega_b}\omega_b,$$

for all functions $u\in C^\infty(X_b)$ with $\int_{X_b}u\omega_b=0$. In other words, $C_{P,b}$ is the reciprocal of the lowest positive eigenvalue of the Laplacian of $\omega_b$.

Thus, to complete the proof of the claim <ref> it remains to obtain a uniform upper bound for $C_{P,b}$ independent of $b\in B^\circ$.
This bound is proved by  Yoshikawa [79] (see also the shorter and clear exposition in <cit.>), using the assumption that all singular fibers are reduced and irreducible (and in general the Poincaré constant blows up near singular fibers which do not satisfy this). Combining all these results we obtain <ref>.

Next, observe that by definition we have

$$-\norm{\alpha}_{L^\infty(X,\omega)}\omega\leq \alpha\leq \norm{\alpha}_{L^\infty(X,\omega)}\omega,$$

holds on all of $X$. Restricting to $X_b$, tracing with respect to $\omega_b$, and using <ref> we get that

$$-\norm{\alpha}_{L^\infty(X,\omega)}\leq \mathrm{tr}_{\omega_b}(\alpha|_{X_b})=-\Delta_{\omega_b}\phi_b\leq \norm{\alpha}_{L^\infty(X,\omega)},$$

for all $b\in B^{\circ}$.
We can then use the Green's formula for $\phi_b$ on $X_b$, which has fiber integral zero, to get that for all $x\in X_b$ and all $b\in B^{\circ}$ we have

and using the $L^\infty$ bound for $\Delta_{\omega_b}\phi_b$ and the facts that $G_b(\cdot,x)+C_0\geq 0$ and $G_b(\cdot,x)$ integrates to zero on $X_b$, we immediately obtain that <ref> holds.

Suppose $\alpha$ is a closed smooth $(1,1)$-form on $X$ which satisfies <ref> and $\phi\in L^\infty(X)\cap C^\infty(X^\circ)$ is provided by <ref>, i.e. so that $\alpha + dd^c\phi$ is $\pi$-trivial on $X$.
For any $\gamma \in \Aut_\pi^{\circ}(X)$ there exists a closed $(1,1)$ current $\eta=\eta(\gamma,\alpha)\in \cA_{0,0}(B)$ such that

\[
\gamma_*\left(\alpha+dd^c\phi\right) = \left(\alpha+dd^c\phi\right) + \pi^* \eta.
\]

Furthermore $\eta$ has continuous potentials on $B$.

Later on in <ref> we will establish that $\eta$ depends only on the classes $\xi(\gamma)$ and $[\alpha]$ in $[E]^\perp/[E]$.

Thanks to <ref>, we have

Fix a smooth form $\beta$ on $B$ with the correct total integral so that $\pi^*\beta$ is a smooth representative of $-([\alpha].\xi)[E]$. We can then write $\gamma_*\alpha-\alpha=\pi^*\beta+dd^cf$ for some smooth function $f$ on $X$ (observe here that $\gamma_*\alpha-\alpha$ is smooth on $X$ since $\gamma_*\alpha=(\gamma^{-1})^*\alpha$). Let $\phi\in L^\infty(X)\cap C^\infty(X^\circ)$ be as in <ref>, then
the current $\gamma_*(\alpha+dd^c\phi)- (\alpha+dd^c\phi)$ is $\pi$-trivial on $X$ and in the class of $-([\alpha].\xi)[E]$,
so by <ref> it follows that this current is of the form $\pi^*\eta$ for some $\eta\in \cA_{0,0}(B)$. On $X$ we thus have

\begin{equation}\label{grz}
\pi^*\eta=(\gamma_*\alpha-\alpha) + dd^c(\gamma_*\phi-\phi) = \pi^*\beta+dd^c(f+\gamma_*\phi-\phi),
\end{equation}

so that $f+\gamma_*\phi-\phi$ is constant on all fibers and hence it is the pullback of a function $u$ on $B$, with $\eta=\beta+dd^c u$. Since $\phi\in L^\infty(X)\cap C^\infty(X^\circ)$ and $f\in C^\infty(X)$, we see that $u\in L^\infty(B)\cap C^\infty(B^\circ)$. We are thus claiming that $u$ is a continuous function on $B$.

Let thus $0\in B\backslash B^\circ$ with singular fiber $X_0=\pi^{-1}(0)$. 	Using the same notation as in the construction of $\phi$ in <ref>, given any sufficiently small open set $U\subseteq \left(X_0\setminus x_0\right)$ the elliptic fibration map $\pi$ can be holomorphically trivialized near $U$ as $U\times D\to D$ where $D\subset B$ is an open disc containing $0$, and the map $\mu$ is an isomorphism over $U$.

The key claim is then that on some nonempty open set $U$ as above, in the local trivialization we have that $(f+\gamma_*\phi-\phi)|_{U\times\{b\}}$ converges uniformly on $U$ to $(f+\gamma_*\phi-\phi)|_{U\times\{0\}}$ as $b\to 0$. Since these functions are all fiberwise constants (equal to $\pi^*u$), the desired continuity of $u$ follows from this, and since $f$ is smooth on all of $X$, it suffices to prove the claim for $\gamma_*\phi-\phi$.

To do this,
fix a Kähler metric $\ti{\omega}$ on $\wtilde{X_0}$, and take any sufficiently small $U\subseteq \left(X_0\setminus x_0\right)$ with trivialization as above. For $b\in D\backslash\{0\},$ we get smooth functions $\phi_b=\phi|_{U\times\{b\}}$ on $U$ which satisfy $dd^c\phi_b=-\alpha_b$ where $\alpha_b=\alpha|_{U\times\{b\}}$, and which vary smoothly in $b\in D\backslash\{0\}$. Therefore they also satisfy

$$\Delta_{\mu_*\ti{\omega}}\phi_b=-\mathrm{tr}_{\mu_*\ti{\omega}}\alpha_b,\quad \text{on }U,$$

where the right hand side here is a family of smooth functions that varies smoothly in $b\in D$,
and the functions $\phi_b$ have a uniform $L^\infty$ bound by <ref>. We can then apply standard Schauder estimates for the Laplacian of $\mu_*\ti{\omega}$, which show that given any $U'\Subset U$ and $k\in\mathbb{N}$ and $0<a<1$ there is a constant $C$ (independent of $b$) such that

$$\|\phi_b\|_{C^{k+2,a}(U')}\leq C(\|\Delta_{\mu_*\ti{\omega}}\phi_b\|_{C^{k,a}(U)}+\|\phi_b\|_{L^\infty(U)})= C(\|\mathrm{tr}_{\mu_*\ti{\omega}}\alpha_b\|_{C^{k,a}(U)}+\|\phi_b\|_{L^\infty(U)})\leq C',$$

where again $C'$ is independent of $b$.

Thus, shrinking $U$ slightly, Ascoli-Arzelà then implies that for some sequence $b_i\to 0$ the corresponding functions $\phi_{b_i}$ converge in the smooth topology on $U$ to a smooth function $\hat{\phi}_{U,0}$ on $U$ which satisfies

$$\Delta_{\mu_*\ti{\omega}}\hat{\phi}_{U,0}=-\mathrm{tr}_{\mu_*\ti{\omega}}\alpha|_U,\quad \text{on }U.$$

We can repeat this procedure on a different open set $U'$, taking a subsequence of $b_i$, and obtain smooth convergence on $U\cup U'$. Continuing this way, taking a countable collection of such $U_j$'s which cover $X_0\setminus x_0$ (with discs $0\in D_j\subset B$ with shrinking radii), and passing to a diagonal sequence, we obtain some new sequence $b_i\to 0$ and a bounded smooth function $\hat{\phi}_0$ on $X_0\setminus x_0$ such that $\phi_{b_i}$ converge to $\hat{\phi}_0$ locally smoothly on $X_0\setminus x_0$ (in the obvious sense) and so we have

$$\Delta_{\mu_*\ti{\omega}}\hat{\phi}_{0}=-\mathrm{tr}_{\mu_*\ti{\omega}}\alpha|_{X_0\setminus x_0},\quad \text{on }X_0\setminus x_0.$$

We also have that as $b\to 0$ the part of $X_b$ which is not covered by the union of the $U_j\times\{b\}$'s with $b\in D_j$ has measure that is going to zero.
Thanks to this, and to the uniform $L^\infty$ bound for $\phi_b$ from <ref>, the relation $\int_{X_b}\phi_b\omega_b=0$ passes to the limit to

$$\int_{X_0\setminus x_0}\hat{\phi}_0 \omega=0.$$

Pulling back via $\mu$, which is an isomorphism over $X_0\setminus x_0$, we see that

$$\Delta_{\ti{\omega}}\mu^*\hat{\phi}_{0}=-\mathrm{tr}_{\ti{\omega}}\mu^*\alpha,\quad \text{on }\wtilde{X_0}\setminus S,\quad \int_{\wtilde{X_0}\setminus S}\mu^*\hat{\phi}_0 \mu^*\omega=0.$$

On the other hand the function $\ti{\phi}_0$ defined by (<ref>) satisfies

$$\Delta_{\ti{\omega}}\ti{\phi}_{0}=-\mathrm{tr}_{\ti{\omega}}\mu^*\alpha,\quad \text{on }\wtilde{X_0},\quad \int_{\wtilde{X_0}}\ti{\phi}_0 \mu^*\omega=0,$$

so the difference $\mu^*\hat{\phi}_0-\ti{\phi}_0$ is a smooth function on $\wtilde{X_0}\setminus S$ which is bounded, belongs to the kernel of $\Delta_{\ti{\omega}}$ and it also integrates trivially against $\mu^*\omega$. By a classical Riemann-type extension theorem (“bounded harmonic functions extend across point singularities”, see e.g. <cit.>) $\mu^*\hat{\phi}_0-\ti{\phi}_0$ extends smoothly across $S$ and the extension is then forced to be zero by the integral condition. Thus $\hat{\phi}_0=\phi_0$, and in particular if we picked a different subsequence $b_i\to 0$, the limit is forced to be the same by the uniqueness of $\phi_0$, and so the functions $\phi_b$ converge locally smoothly in our trivialization to $\phi_0$ without having to pass to any subsequence.

The same argument applies to $\gamma_*\phi_b:=(\gamma_*\phi)|_{X_b}$, which satisfies

$$dd^c\gamma_*\phi_b=-(\gamma_*\alpha)|_{X_b},\quad \int_{X_b}\gamma_*\phi_b\gamma_*\omega=0,$$

and so we see that  $\gamma_*\phi_b$ converges locally smoothly on $U$ to $\gamma_*\phi_0$, and this completes the proof of the key claim.

When $\pi$ has general singular fibers, it is clear from the proof of <ref> that we can still find $\phi\in C^\infty(X^\circ)$ such that $\alpha+dd^c\phi$ is $\pi$-trivial on $X^\circ$. However, in general we see no reason for $\phi$ to be bounded on $X^\circ$.

Nevertheless, we do believe that the result in <ref> does hold without the restriction on the singular fibers, which boils down to showing that while $\phi$ need not be bounded on $X^\circ$ anymore, $\gamma_*\phi-\phi$ should extend to a continuous function on $X$. Indeed the following simple argument (essentially the same as Tate's [71]) proves that $\gamma_*\phi-\phi$ is bounded on $X^\circ$ in general. Recall that $f=\pi^*u+\phi-\gamma_*\phi$ where $f$ is smooth on all of $X$. Then for $k\geq 1$ the Birkhoff sum $S_k(\gamma,f):=\sum_{i=0}^{k-1}\gamma^i_*f$ satisfies

\begin{equation*}S_k(\gamma,f)=k\pi^*u+\phi-\gamma^k_*\phi,\end{equation*}

so for every $b\in B^\circ$ and for every $k\geq 1$ we have

\begin{equation*}|u(b)|\leq \frac{1}{k}\sup_{X_b}|S_k(\gamma,f)|+\frac{1}{k}\sup_{X_b}|\phi-\gamma^k_*\phi|\leq \sup_X|f|+\frac{2}{k}\sup_{X_b}|\phi|,\end{equation*}

and letting $k\to\infty$ proves that $\|u\|_{L^\infty(B^\circ)}\leq \sup_X|f|$ as desired.

There exists a map

\[
\eta_B\colon
\Aut^\circ_\pi(X)
\times
\left(\rightquot{[E]^\perp}{[E]}\right)
\to
\cA_{0,0}(B)
\]

which is linear in each coordinate and with the following properties.

* If $\alpha$ is a closed smooth $(1,1)$-form with $[\alpha].[E]=0$, $\phi$ is provided by <ref>, and $\gamma\in \Aut^{\circ}_\pi(X)$ then

\[
\gamma_{*}(\alpha + dd^c\phi)
(\alpha + dd^c \phi) + \pi^{*}\eta_B(\gamma,[\alpha])
\]

* The map is compatible with integration and cup-product, i.e.

\[
\int_B \eta_B(\gamma,[\alpha]) = -[\xi(\gamma)].[\alpha]
\]

* For an automorphism $\gamma\in \Aut^\circ_\pi(X)$ we have

\[
\eta_B(\gamma,\xi(\gamma))\geq 0 \quad \text{ as a current on }B.
\]

* The map is symmetric, i.e. if $\xi_i=\xi(\gamma_i)$

\[
\eta_B(\gamma_1,\xi_2) = \eta_B(\gamma_2,\xi_1)
\]

By <ref>, there exists a current $\eta_B(\gamma,\alpha)$ such that

\[
\gamma_*(\alpha + dd^c\phi) = (\alpha + dd^c \phi) + \pi^*\eta_B(\gamma,\alpha)
\]

If we select a different representative in the same cohomology class, i.e. $\alpha'=\alpha + dd^c\psi$, then the current $\alpha'+dd^c\phi'$ provided by <ref> differs from $\alpha+dd^c\phi$ by $dd^c$ of a function pulled back from $B$.
Thus the expression $\gamma_*(\alpha+dd^c\phi)-(\alpha+dd^c\phi)$ depends only on the cohomology class $[\alpha]$.

The definition of $\eta_B(\gamma,[\alpha])$ immediately yields the relations

\begin{align*}
\eta_B(\gamma,[\alpha_1]+[\alpha_2])
& = \eta_B(\gamma,[\alpha_1]) + \eta_B(\gamma,[\alpha_2])\\
\eta_B(\gamma_1\circ \gamma_2,[\alpha])
& = \eta_B(\gamma_1,[\alpha]) + \eta_B(\gamma_2,[\alpha])
\end{align*}

The identity $\int_B \eta_B(\gamma,[\alpha]) = [\xi(\gamma)].[\alpha]$ follows from <ref> and the fact that $[\alpha].[E]=0$ by assumption.

Symmetry is established in <ref> and positivity is established in <ref>.
The proofs of both of these facts involve taking a large power of an element and passing to the limit, and do not depend on each other.
They do require understanding the iterates of forms that integrate nontrivially on fibers.

We now study forms that integrate nontrivially on the fiber.

Let $\omega$ be a closed smooth $(1,1)$-form with $\int_{X_b}\omega=1$ and let $\gamma\in \Aut_\pi^{\circ}(X)$ be an automorphism.
Define the closed $(1,1)$-form $\alpha$ by

\[
\gamma_*\omega = \omega + \alpha,
\]

so it satisfies $[\alpha].[E]=0$ and let $\phi$ be the function provided by <ref> applied to $\alpha$.
Then for $n\geq 0$ we have that

\[
\gamma_*^n \omega
\omega + n\cdot \alpha
+ dd^c \left[n\phi - S_n(\gamma,\phi)\right]
+ \tfrac{n(n-1)}{2}\pi^*\eta_{\gamma}
\]

where $\eta_{\gamma}:=\eta_B(\gamma,\xi(\gamma))$ is the current provided by <ref>, and $S_{k}(\gamma,\phi)=\sum_{i=0}^{k-1} \gamma_*^i\phi$ is a Birkhoff sum.

From the formula in <ref>, the class of $[\alpha]$ in $[E]^\perp/[E]$ is $\xi$, where $\xi=\xi(\gamma)$ is the element corresponding to the automorphism.
Taking $\phi$ as provided by <ref> gives

\[
\gamma_*(\alpha + dd^c\phi)
(\alpha+dd^c\phi) + \pi^*\eta_B(\gamma,\xi)
\]

where $\eta_B(\gamma,\xi)$ is the current on $B$ provided by <ref>.
Let $\eta_\gamma:=\eta_B(\gamma,\xi)$ as per the statement of the claim and note that more generally we have

\begin{align*}
\gamma_*^k(\alpha + dd^c\phi) & = (\alpha +dd^c\phi)
+ \pi^*\eta_B(\gamma^k,\xi)\\
& = (\alpha+dd^c\phi) + k\pi^*\eta_\gamma
\end{align*}

We then have by <ref>

\begin{align*}
\gamma_*^n \omega & =
\gamma_*^{n-1}(\gamma_*\omega) =
\omega + \alpha + \gamma_*\alpha + \cdots + \gamma^{n-1}_*\alpha\\
& = \omega + n\alpha + dd^c[n\phi - S_{n}(\gamma,\phi)]\\
& + (1+2+\cdots + (n-1))\pi^*\eta_\gamma
\end{align*}

as required.

From now on, to ease the notational burden and emphasize certain symmetries, we will use the notation

\[
\eta(\gamma,\gamma'):=\pi^*\eta_B(\gamma,\xi(\gamma'))
\]

viewed as a $(1,1)$-current on $X$.
To ease notation, below we write pushed-forward forms or functions as $\gamma \omega$ instead of $\gamma_*\omega$.

Later on, in <ref>, when the dependence of the pairing on the elliptic fibration will be relevant, we will write $\eta_{[E]}(\gamma,\gamma')$ for the corresponding current on $X$.

Let $\omega$ be a closed smooth $(1,1)$-form with $\int_{X_b}\omega=1$ and let $\gamma_1,\ldots, \gamma_k\in \Aut_\pi^{\circ}(X)$ be automorphism.
Define closed $(1,1)$-forms $\alpha_i$ with $[\alpha_i].[E]=0$ by

\[
\gamma_i\omega = \omega + \alpha_i,
\]

and let $\phi_i$ be the functions provided by <ref> applied to $\alpha_i$.
Then for $n_1,\ldots n_k \geq 0$ we have that

\begin{align*}
\gamma_k^{n_k}\cdots \gamma_1^{n_1} \omega
\omega + n_k \alpha_k + \cdots + n_1 \alpha_1 + \\
& + dd^c \left[
\sum_{i=1}^k n_i \phi_i - S_{n_i}\left(\gamma_i, \gamma_k^{n_k}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i\right)
\right]\\
& + \left[
\sum_{i=1}^k \frac{n_i(n_i-1)}{2}\eta(\gamma_i,\gamma_i)
\sum_{1\leq i <j \leq k}n_i n_j \eta(\gamma_i,\gamma_j)
\right]
\end{align*}

where as before $S_{n}(\gamma,\phi):=\phi + \cdots + \gamma^{n-1}\phi$ denotes a Birkhoff sum.

The case $k=1$ is <ref>.
The general formula is verified (by induction) on $k$ by direct calculation.
Let us apply $\gamma_{k}^{n_{k}}$ to the formula assumed valid for $k-1$.
Observe that the terms pulled back from $B$ are not changed, and we have, combining the previous calculations in <ref> and <ref>:

\begin{align*}
\gamma_k^{n_k}\omega & =
\omega + n_k \alpha_k + dd^c\left[n_k\phi_k - S_{n_k}(\gamma_k,\phi_k)	\right] + \frac{n_k(n_k-1)}{2}\eta(\gamma_k,\gamma_k)\\
\gamma_k^{n_k}\left(\sum_{i=1}^{k-1}n_i\alpha_i\right) & =
\sum_{i=1}^{k-1}
n_i\Big[\alpha_i +
(\phi_i - \gamma_k^{n_k}\phi_i)
+ \eta\left(\gamma_k^{n_k},[\alpha_i]\right)
\Big]
\end{align*}

Recall that $[\alpha_i]=\xi(\gamma_i)$ and so the last term above can be also expressed as $n_k\cdot \eta(\gamma_k,\gamma_i)$.
Finally we have

\begin{align*}
\gamma_k^{n_k}\sum_{i=1}^{k-1} n_i \phi_i - S_{n_i}\left(\gamma_i, \gamma_{k-1}^{n_{k-1}}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i\right)
% =\\
= \sum_{i=1}^{k-1}
n_i \gamma_k^{n_k}\phi_i -
S_{n_i}\left(\gamma_i,\gamma_k^{n_k}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i\right)
\end{align*}

where we used that the parabolic automorphisms $\gamma_i$ commute.
Adding up the contributions yields the result, noting that the terms $n_i \gamma_k^{n_i}\phi_i$ cancel out.

The pairing in <ref> is symmetric, i.e. for two automorphisms $\gamma_1,\gamma_2\in \Aut_\pi^{\circ}$ with corresponding vectors $\xi_i=\xi(\gamma_i)$ we have

\[
\eta_B(\gamma_1,\xi_2) = \eta_B(\gamma_2,\xi_1) \text{ or as we shall also write }\eta(\gamma_1,\gamma_2) = \eta(\gamma_2,\gamma_1).
\]

Since the two automorphisms commute we have that $\gamma:=\gamma_1\gamma_2 = \gamma_2\gamma_1$.
Select $\omega$ as in <ref> and apply the statement to $\gamma_1^n\gamma_2^n$ and $\gamma_2^n\gamma_1^n$.
Dividing by $n^2$ and sending $n\to+\infty$ gives

\begin{align*}
\eta(\gamma_1,\gamma_1) + \frac{1}{2}\eta(\gamma_1,\gamma_2)
+ \eta(\gamma_2,\gamma_2)
\lim_{n\to +\infty} \frac{\gamma_*^n \omega}{n^2}
\eta(\gamma_2,\gamma_2)+
\frac{1}{2}\eta(\gamma_2,\gamma_1)
+ \eta(\gamma_1,\gamma_1)
\end{align*}

thus yielding the symmetry.

Using the symmetry property just established, we can write the conclusion of <ref> in a more intrinsic form.
Specifically, set $\gamma:=\gamma_k^{n_k}\cdots \gamma_1^{n_1}$ and then

\begin{align}
\label{eqn:large_parabolic_automorphism}
\begin{split}
\gamma_* \omega
\omega + n_k \left(\alpha_k - \frac{1}{2}\eta(\gamma_k,\gamma_k)\right) + \cdots + n_1\left( \alpha_1 - \frac{1}{2}\eta(\gamma_k,\gamma_k)\right) \\
& + dd^c \left[
\sum_{i=1}^k n_i \phi_i - S_{n_i}\left(\gamma_i, \gamma_k^{n_k}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i\right)
\right]\\
& + \frac{1}{2}\eta(\gamma,\gamma)
\end{split}
\end{align}

In this form, the dominant (quadratic) term of order $-[\xi(\gamma)].[\xi(\gamma)]$ is the last one, whereas the intermediate ones are linear in the size of $\xi(\gamma)$ or bounded.

For any $\gamma\in \Aut^\circ_{\pi}(X)$ and any form $\omega$ on $X$ with $\int_{X_b}\omega=1$, we have that

$$\lim_{n\to+\infty} \frac{\gamma^n_*\omega}{n^2}=\frac{1}{2}\eta(\gamma,\gamma),$$

in the weak topology. In particular we have

\[
\eta(\gamma,\gamma)\geq 0.
\]

In <ref>, take $\omega$ to be a form and express $\gamma$ as a product of generators.
Then any weak limit of $\frac{\gamma^n_*\omega}{n^2}$ is on the one hand a positive current, and on the other hand equal to $\frac{1}{2}\eta(\gamma,\gamma)$.

§.§.§ Extension to real coefficients

Let us note that the current-valued bilinear pairing $\eta$ from <ref> extends from the lattice $[E]^{\perp}/[E]$ to the vector space $([E]^{\perp}/[E])\otimes_{\bZ}\bR=:([E]^{\perp}/[E])_{\bR}$ by taking a set of generators $\gamma_i$ of $\Aut^{\circ}_{\pi}(X)$ and allowing the expression $\sum n_i \gamma_i$ to have real coefficients $n_i$.
We see, in particular, that restricted to the unit sphere, this gives a mapping to positive currents, which is moreover continuous in the $C^0$ topology of potentials.
Indeed, the current for an arbitrary real-valued point is given as a linear combination of a finite set of basis currents, with coefficients determined by the coordinates of the point, and the basis currents have continuous potentials.

§.§.§ Summary

<ref> applies to the case of $[\omega].[E]=1$.
We can reduce to this case by scaling, and the explicit formula is as follows.
Assume that $\gamma_i \omega = \omega + \alpha_i$ and $\phi_i$ is the result of applying <ref> to $\alpha_i$.
Then applying <ref> to $\omega/([\omega].[E])$ and multiplying by $[\omega].[E]$ the result gives (with $\gamma=\gamma_k^{n_k}\cdots \gamma_1^{n_1}$):

\begin{align*}
\gamma_k^{n_k}\cdots \gamma_1^{n_1} \omega
\omega + n_k \left(\alpha_k - \frac{1}{2}\eta(\gamma_k,\gamma_k)\right) + \cdots + n_1\left( \alpha_1 - \frac{1}{2}\eta(\gamma_k,\gamma_k)\right) \\
& + dd^c \left[
\sum_{i=1}^k n_i \phi_i - S_{n_i}\left(\gamma_i, \gamma_k^{n_k}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i\right)
\right]\\
& + \frac{([\omega].[E])}{2}\eta(\gamma,\gamma)
\end{align*}

Note that the middle terms, involving $dd^c$, are cohomologically trivial and also of linear magnitude in the size of $\gamma$.
Note also that the first row of terms has magnitude which is linear in the size of $\gamma$, while the last term is of quadratic magnitude.

For later estimates, we will use the notation

\begin{align}
\label{eqn:norm_gamma}
\norm{\gamma}_{\NS}^2:= -[\xi(\gamma)].[\xi(\gamma)]
\end{align}

which is a natural strictly positive-definite norm on $\Aut^{\circ}_{\pi}(X)$ after it is identified with $[E]^\perp/[E]$ via $\xi$.

Also importantly, let us note that once we fix a positive-definite norm $\norm{\cdot}$ on $\NS(X)$, there exists a constant $C=C(E)$ such that:

\begin{align}
\label{eqn:norm_gamma_operator_square}
\frac 1C \left(1+\norm{\gamma}^{2}_{\NS}\right) \leq \norm{\gamma}_{op} \leq C \left(1+\norm{\gamma}^{2}_{\NS}\right)
\end{align}

where $\norm{\gamma}_{op}$ refers to the operator norm of $\gamma$ acting (as a matrix) on $\NS(X)$.
Note, importantly, that $\norm{\gamma}^2_{\NS}$ appears and not $\norm{\gamma}_{\NS}$.
Indeed, for a unit vector $v$ whose cup product with $[E]$ is bounded below by a constant, the size of $\norm{\gamma v}$ is of the order $\norm{\gamma}_{\NS}^2$, up to some multiplicative constant, because of <ref>.
The estimate in <ref> is compatible (and implies) the fact that the operator norm of $\gamma^k$ grows quadratically in $k$.

§ BOUNDARY CURRENTS, DIRECT CONSTRUCTION

§.§.§ Outline of section

We can now construct the canonical currents on the boundary of the ample cone of a K3 surface.
The main result of this section is <ref> that establishes the existence of a continuous family of boundary currents.

In <ref> we fix smooth representatives for every cohomology class, so that any other current is expressed relative to the fixed basis using a potential function.
In <ref> we establish the existence of the canonical boundary currents and establish their properties.
In <ref> we include a result due to Sibony–Verbitsky that shows the uniqueness of the closed positive currents in irrational boundary classes.
Finally, in <ref> we include some consequences for degenerating Ricci-flat metrics, and discuss further geometric interpretations of our construction.

§.§ Potentials from a basis

§.§.§ Setup

For this section and the next, fix a basis $\omega_0,\omega_1\ldots, \omega_{\rho-1}$ spanning the subspace in cohomology given by the –Severi group $\NS(X)\subseteq H^{1,1}(X)$.
For a closed $(1,1)$-form $\omega$ with $[\omega]\in \NS(X)$, there exists a unique way to write it as

\[
\omega = \sum c_i \omega_i + dd^c \phi(\omega) \qquad \text{ with }\int_X \phi(\omega)\dVol = 0\text{ and }c_i\in \bR.
\]

We will call $\phi(\omega)$ the non-canonical potential of $\omega$.
Given a cohomology class $[\omega]$, we will write $A([\omega])$ for the $(1,1)$-form satisfying

\[
A([\omega]) = \sum c_i \omega_i \qquad \text{ where }[\omega]=\sum c_i[\omega_i].
\]

With this notation, we always have

\begin{align}
\label{eqn:omega_noncanonical_potential}
\omega = A([\omega]) + dd^c \phi(\omega).
\end{align}

We will use the notation $\phi(\omega)$ also for currents which have continuous potentials, such as $\eta(\gamma,\gamma')$ defined in <ref>.
From now on, all constants are allowed to depend on the choice of the initial representative $(1,1)$-forms and this will not be stated additionally.

§.§.§ Mass and normalization

We assume that $\omega_0$, the first element in the basis, is a metric normalized to $[\omega_0]^2=1$ and such that the class $[\omega_0]$ coincides with the reference basepoint from <ref>.
Recall also that we defined

\[
M([\omega]):=[\omega_0].[\omega] \text{ to be the \emph{mass} of }[\omega].
\]

To study the (non-canonical) potentials of forms as they approach the boundary, we will always normalize them to have mass $1$.
We will say that $\omega$, or $[\omega]$, is normalized if we have that

\[
[\omega]^2\geq 0 \text{ and }[\omega].[\omega_0]=1,\text{ i.e. }M([\omega])=1 \text{ and it is in the positive cone.}
\]

§.§.§ Action of automorphisms on potentials

For the action of an automorphism $\gamma\in \Aut(X)$ on forms and potentials, we will use the following expressions.
First we have from the definitions

\[
\gamma_* A([\omega]) = A(\gamma_*[\omega])  + dd^c \phi\Big(\gamma_* A([\omega])\Big)
\]

For a general $(1,1)$-form $\omega$ we have, after acting by $\gamma$ on <ref>:

\begin{align}
\label{eqn:gamma_action_noncanonical_potential}
\begin{split}
\gamma_* \omega & =
\gamma_* A([\omega]) + dd^c \gamma_* \phi(\omega)\\
& =
A\left(\gamma_*[\omega]\right) + dd^c \Big(\phi\big(\gamma_* A([\omega])\big) + \gamma_*\phi(\omega) \Big) \text{ and so:}\\
\phi(\gamma_*\omega) & =
\phi\left(\gamma_{*}A([\omega])\right) + \gamma_*\phi(\omega)
\end{split}
\end{align}

For $\gamma\in \Aut(X)$ fixed, there exists a constant $C(\gamma)$ such that for any $[\omega]$ with $[\omega]^2\geq 0$ and $[\omega].[\omega_0]\geq 0$ we have the bound on potentials:

\[
\norm{\phi\big(\gamma_* A([\omega])\big)}_{C^0} \leq C(\gamma)M([\omega])
\]

Furthermore, for any $[\omega],[\omega']$ which are normalized we have

\[
\norm{\phi\big(\gamma_* A([\omega])\big) - \phi\big(\gamma_* A([\omega'])\big)}_{C^0} \leq 2C(\gamma)
\]

By scaling the first inequality we can normalize to $M([\omega])=1$ and then the bound follows since $[\omega]$ belongs to a compact set and $\phi\big(\gamma_* A([\omega])\big)$ varies continuously (even smoothly) on it.
The second inequality follows from the first by the triangle inequality, and using that $M([\omega])=1$.

§.§.§ Estimates for elliptic fibrations

Suppose now that $X\xrightarrow{\pi}\bP^1$ is an elliptic fibration, with class of fiber $[E]$, with parabolic automorphism group $\Aut^\circ_{\pi}(X)$ as in <ref>.
We will sharpen the estimate in <ref> to include the size of a parabolic automorphism $\gamma$ (see <ref>) and obtain convergence towards a positive current.

Specifically, fix $\gamma_1,\ldots, \gamma_k$ generators of $\Aut_{\pi}^{\circ}(X)$ with $k=\rho-2$ and a constant $C([E])$ depending on the fibration.
We will say that $[\omega]$ is parabolically normalized if, in addition to being normalized (i.e. $[\omega]^2\geq 0$ and $[\omega].[\omega_0]=1$) we also have that $[\omega].[E]\geq \frac{1}{C([E])}$.
Note that this implies that there also exists a constant $C'([E])$ such that $C'([E])\geq [\omega].[E]$ since $[\omega]$ is restricted to a compact set.
Observe that being parabolically normalized is equivalent to being transverse (with a uniform constant) for any element $\gamma\in \Aut_\pi^\circ(X)$, i.e. satisfying the inequality $\norm{\gamma [\omega]}\geq \frac{1}{C''}\norm{\gamma}_{op}\norm{\omega}$ for a constant $C''$ depending only on $[E]$.
Finally, recall that for norms of parabolic automorphisms we have $\norm{\gamma}_{op}\approx \norm{\gamma}_{\NS}^2$, see <ref>.

Consider a parabolic automorphism $\gamma=\gamma_k^{n_k}\cdots \gamma_1^{n_1}$ written as a product of generators and $[\omega]$ a parabolically normalized class.

* There exists a constant $C_1=C_1(E)$ giving the mass estimate:

\[
\norm{
\frac{M(\gamma_*[\omega])}{M\left(\frac{[\omega].[E]}{2}\eta(\gamma,\gamma)\right)} - 1}
\leq
\frac{C_1}{1 + \norm{\gamma}_{\NS}}
\]

There exists a constant $C_2=C_2(E)$ giving the estimate on potentials:

\[
\norm{
\frac{\phi\left(\gamma_*A([\omega])\right)}{M(\gamma_* [\omega])
\frac{\phi\left(\eta(\gamma,\gamma)\right)}{M\left(\eta(\gamma,\gamma)\right)}
\leq
\frac{C_2}{1 + \norm{\gamma}_{\NS}}
\]

We also have the estimate

\[
\norm{
\phi\left(\gamma_* A([\omega])\right)
\leq C_3 \left(1+\norm{\gamma}_{\NS}^2\right)\norm{[\omega]}
\]

which holds for arbitrary $[\omega]$.

The estimates could be rewritten in terms of $\norm{\gamma}_{op}$, keeping in mind the bound in <ref> that gives $\norm{\gamma}_{op}\approx \norm{\gamma}_{\NS}^2$.

We apply the expression in <ref> to $A([\omega])$.
Part (i) then follows immediately since it only involves cohomology classes.
For part (ii), recall that the generators $\gamma_i$ are fixed at the start, so as $[\omega]$ varies in the compact set of parabolically normalized classes, so do the $\alpha_i$ so we have uniform $C^0$ (even $C^k$) bounds on the functions $\phi_i$ appearing in the formula.
We then have the estimates

\begin{align*}
\norm{\phi(A([\omega]))}_{C^0} & = O(1)\\
\norm{\phi\left(\sum n_i (\alpha_i - \tfrac{1}{2}\eta(\gamma_i,\gamma_i))\right)}_{C^0} & = O\left(\norm{\gamma}_{\NS}\right)\\
\norm{\sum {n_i \phi_i} - S_{n_i}(\gamma_i, \gamma_k^{n_k}\cdots \gamma_{i+1}^{n_{i+1}}\phi_i)}_{C^0} & = O\left(\norm{\gamma}_{\NS}\right)
\end{align*}

so the main contribution comes from the term $\frac{[\omega].[E]}{2}\eta(\gamma,\gamma)$ which has mass $\frac{([\omega].[E])}{2}\norm{\gamma}^2_{\NS}$, while the other terms have size $O(\norm{\gamma}_{\NS})$, and hence the quadratic term dominates the rest.
So to get the claimed estimate in part (ii), multiply both sides by $M(\gamma_*[\omega])$ and use the estimates from part (i) that give

\[
\frac{M(\gamma_*[\omega])}{M(\eta(\gamma,\gamma))}=\tfrac12 [\omega].[E] + O\left(\frac{1}{1+\norm{\gamma}_{\NS}}\right)
\]

and then observe that we have

\[
\phi(\gamma_*A([\omega])) = \phi\left(\tfrac{[\omega].[E]}{2}\eta(\gamma,\gamma)\right) + O(\norm{\gamma}_{\NS})
\]

This last estimate also gives the desired bound in (iii) for parabolically normalized classes, keeping in mind that $\eta(\gamma,\gamma)$ scales quadratically in $\gamma$ and is uniformly bounded for $\gamma$ in a bounded set.
To obtain the estimate in (iii) for arbitrary classes, observe that we can fix a basis of parabolically normalized classes, apply the estimate to each of them, and sum up the estimates (using the linearity of the assignments $[\omega]\mapsto A([\omega])$ and $\omega\mapsto \phi(\omega)$).

Suppose that $\{\chi_i\}\in \Aut^{\circ}_{\pi}(X)$ is a sequence of elements with $\norm{\chi_i}\to +\infty$ and such that $\frac{\chi_i}{\norm{\chi_i}_{\NS}}\in \left([E]^\perp/E\right)_\bR$ convergences to an element $\chi$.
Then for any smooth $(1,1)$-form $\omega$ we have the convergence:

\[
\frac{(\chi_i)_* \omega}{M((\chi_i)_*\omega )} \to
\frac{\eta(\chi,\chi)}{M(\eta(\chi,\chi))}
\]

in the $C^0$ topology on potentials.

Given a fixed $\omega$, there is some constant $C(E)$ such that $\omega$ is parabolically normalized.
Then the result follows from part (ii) of <ref>.

§.§ Existence of boundary potentials

§.§.§ Rational boundary points and currents

Recall that the boundary of interest $\partial^\circ \Amp_c(X)$ has a stratification as in <ref>.
It consists of irrational cohomology classes, as well as elements of the form $k[E]\times [\xi]$ where $k$ is a real strictly positive scalar, $[E]$ is the (integral, primitive) class of an elliptic fibration, and $[\xi]\in \bP\left([E]^\perp/[E]\right)$.

For a class $[E]$ associated to a genus one fibration, from now on we denote the current-valued pairing from <ref> by $\eta_{[E]}$, see also <ref>.
This is to distinguish it from pairings of other genus one fibrations, and to distinguish it from the global family of boundary currents that the main result provides.

By our standing assumptions on K3 surfaces in <ref>, and due to Sterk's theorem, the subgroup $\Aut^\circ_\pi(X)$ of parabolic automorphisms gives a lattice in $[E]^\perp/[E]$ using the embedding $\xi_{[E]}$ given by $\xi_{[E]}(\gamma):=\gamma_*[\omega]-[\omega]$ where $[\omega].[E]=1$ is fixed, see <ref>.
By extension of scalars, we will use below the currents $\eta_{[E_0]}(\xi_1,\xi_2)$ where $\xi_i\in [E]^\perp/[E]$ are real vectors.

Here is the main result of this section.
Recall that $\partial^\circ \Amp_c(X)$ is defined in <ref> and $\cZ_{1,1}(X)$ denotes closed $(1,1)$-currents.

There exists a map

\[
\eta_X\colon \partial^\circ \Amp_c(X) \to \cZ_{1,1}(X)
\]

compatible with the map from $\partial^\circ \Amp_c(X)$ to $\partial \Amp(X)\subset \NS(X)$, and from $\cZ_{1,1}(X)$ to $H^{1,1}(X)$.
The map has the following additional properties:

For any $[\alpha]\in \partial^\circ \Amp_c(X)$ the current $\eta_X([\alpha])$ is positive and has continuous potentials.

The map $\eta_X$ is continuous for the $C^0$-topology on potentials.

The map $\eta_X$ is $\Aut(X)$-equivariant, and also equivariant for the scaling action of $\bR_{>0}$.

On the real projective $(\rho-2)$-spaces in $\partial^\circ \Amp_c(X)$ associated to a rational boundary ray $\bR[E]$ corresponding to a genus one fibration, the map $\eta_X$ agrees with the current-valued pairing from <ref> with the assignment

\[
\eta_X(k[E]\times [\xi])=k\cdot \frac{\eta_{[E]}([\xi],[\xi])}{\norm{\xi}_{\NS}^2} \text{ for }[\xi]\in \bP([E]^{\perp}/[E]).
\]

Note that the formula uses, but is independent of, a choice of a lift of $[\xi]$ to $[E]^\perp/[E]$.

* For boundary classes $[\alpha_\pm]$ expanded/contracted by hyperbolic automorphisms of $X$, the currents $\eta_X([\alpha_{\pm}])$ agree with those constructed by Cantat [12].

Furthermore, the combination of either <ref> and <ref>, or the combination of <ref> and <ref>, uniquely determine $\eta_X$.

To each boundary point $[\alpha]\in \partial^\circ \Amp_c(X)$ we have an associated sequence of automorphisms $\gamma_1,\gamma_2,\ldots$ from the list fixed in <ref>.
We will check that the assignment (after normalization)

\[
\frac{[\alpha]}{M([\alpha])} \mapsto \lim_{n\to +\infty} \frac{\gamma_1\cdots \gamma_n \omega_0}{M(\gamma_1\cdots \gamma_n[\omega_0])}
\]

works, i.e. the limit exists and has the required properties.
When we approach a parabolic boundary point, we interpret the limit $\lim_{n\to +\infty}$ as $\lim_{i\to +\infty}$ and we have the sequence $\gamma_1\cdots \gamma_{n-1}\gamma_{n,i}$ instead, where the sequence $\gamma_{n,i}$ belongs to the corresponding parabolic subgroup.

Once we establish <ref> and <ref>, i.e. positivity and continuity in the $C^0$-topology of potentials, the rest follows.
Indeed, uniqueness (and hence equivariance) follows from <ref> below, due to Sibony–Verbitsky, stating that the closed positive currents in the irrational boundary classes are unique.
Since the irrational points are dense in both $\partial \Amp(X)$ and $\partial^\circ \Amp_c(X)$ the uniqueness and equivariance properties of $\eta$ follow from its continuity.
One can alternatively use the dense set of classes expanded by hyperbolic automorphisms, on which uniqueness holds by a simpler, by now standard, argument.
Again continuous dependence of potentials on the boundary class, for the construction of $\eta$ below, implies the uniqueness and equivariance.

To define the map at the set-theoretical level, recall from <ref> that

\[
\partial^\circ\Amp_c(X)=\partial \Amp(X)^{irr}\coprod_{[E]}\left(\bR_{>0}[E]\right)\times \bP\left([E]^{\perp}/[E]\right)
\]

where $\partial \Amp(X)^{irr}$ denotes the irrational points and $\bR_{>0}[E]$ denotes the ray corresponding to the parabolic boundary class $[E]$.
For $[\alpha]\in \partial^\circ\Amp_c(X)$, we divide the discussion into two cases, according to whether it is in the rational or irrational part.

§.§.§ Case $[\alpha]$ is rational

At the rational points of the form $[\alpha]:=[E]\times [\xi]$, where $\xi\in \bP\left([E]^\perp/[E]\right)$ define the current to be

\[
\eta\left([E]\times [\xi]\right) := \frac{\eta_{[E]}(\xi,\xi)}{\norm{\xi}^2_{\NS}}
\]

which is in particular in the cohomology class $[E]$.
Extend the assignment by scaling to the entire ray $\bR_{>0}[E]$.
Positivity follows from <ref>.

For continuity at this point, we argue as follows.
First, by applying a finite sequence of generators $\gamma_1([\alpha]),\ldots,\gamma_{n-1}([\alpha])$ (given by a corresponding geodesic going into that cusp) we can assume that $\left(\gamma_1([\alpha])\cdots\gamma_{n-1}([\alpha])\right)^{-1}[E]$ is one of the fixed representatives for $\Aut(X)$-equivalences of genus one fibrations, say $[E_0]$.
We can thus reduce to this case.
By the construction in <ref> for any $R>0$, there exists a neighborhood $U$ of the point $[\alpha]$ in $\partial^\circ\Amp_c(X)$ such that any geodesic $s$ starting at $[\omega_0]$ and landing in the neighborhood $U$ must have that $\gamma_{1}(s)$ is a parabolic element in $\Gamma_{[E_0]}$, with $\norm{\xi(\gamma_{1}(s))}_{\NS}\geq R$, and furthermore such that the ray determined by $\xi(\gamma_{1}(s))$ is in the prescribed neighborhood of $[\xi]$ in $\bP\left([E]^{\perp}/[E]\right)$.
The estimates in <ref> below establish the required continuity.

§.§.§ Case $[\alpha]$ is irrational.

At the irrational points, the map $\eta$ is defined by the current whose noncanonical potential is provided by <ref>.
The proposition gives us a current normalized to mass $1$, but the map is extended naturally by scaling to the entire ray.
Positivity follows by construction, and continuity follows from the estimate provided by <ref> and the fact that given $n$, there is a neighborhood of the given irrational endpoint such that geodesics $s$ landing in the neighborhood must have the sequence $\gamma_i(s)$ agree with that of the irrational point up to $n$.

§.§.§ Setup

To ease notation, all actions of group elements $\gamma\in \Aut(X)$ on $(1,1)$-forms and on functions are by pushforward and we write $\gamma \omega$ for $\gamma_*\omega$ and similarly for functions.
Applying <ref> successively, we have the formula

\begin{align}
\label{eqn:gamma1n_omega}
\begin{split}
\phi(\gamma_1\cdots \gamma_n\omega) & =
\phi\big(\gamma_1 A(\gamma_2\cdots \gamma_n[\omega])\big)+ \cdots \\
& + \gamma_1 \cdots \gamma_{i-1}\phi\big(\gamma_i A(\gamma_{i+1}\cdots \gamma_n[\omega])\big)+ \cdots \\
& + \gamma_1 \cdots \gamma_n \phi(\omega)
\end{split}
\end{align}

Note that this is the correct order of multiplication, i.e. in the sequence $\gamma_1\cdots \gamma_n \omega$ the next term is obtained by applying $\gamma_{n+1}$ “based” at the previous element, i.e. we must conjugate $\gamma_{n+1}$ by $\gamma_1\cdots \gamma_n$ and apply it to the point, hence we get $\gamma_1\cdots \gamma_{n+1}\omega$.
We will use the following notation to occasionally shorten expressions:

\[
\omega_{i,j}:=\gamma_i\cdots \gamma_j \omega_0 \qquad
M_{i,j}=M([\omega_{i,j}]) \text{ with }i\leq j.
\]

§.§.§ The necessary bounds

Let us summarize the inequalities that we will use in the proof.
From <ref> and <ref> we have that

\begin{align}
\label{eqn:useful_inequalities}
\begin{split}
e^{\lambda_i + \cdots + \lambda_j}\leqapprox &
\quad M_{i,j}\quad \quad \leqapprox
e^{\lambda_i + \cdots + \lambda_j}\\
e^{\lambda_i + \cdots + \lambda_j}\leqapprox & \norm{\gamma_i\cdots \gamma_j}_{op}\leqapprox
e^{\lambda_i + \cdots + \lambda_j}
\end{split}
\end{align}

for all $j\geq i \geq 1$.
Additionally, we will use that $[\omega_0]$ and $\gamma_{j+1}\cdots \gamma_{j+k}[\omega_0]$ is transverse for $\gamma_i\cdots \gamma_j$ and hence the estimates in <ref> and <ref> hold.
We will also use systematically the bound

\begin{align}
\label{eqn:basic_bound_potential_all_automorphisms}
\norm{\phi\big(\gamma A([\omega])\big)}_{C^0}\leqapprox \norm{\gamma}_{op} \norm{[\omega]}
\end{align}

which is valid when $\gamma$ belongs to the fixed set of generators (since we consider a finite number of such generators), or to one of the finitely many fixed parabolic subgroups.
For elements of parabolic subgroups, this is <ref><ref>, and for a fixed finite list of elements this is <ref>.

Let $s$ be any geodesic starting at $[\omega_0]$ and landing at an irrational point on the boundary, with associated sequence of generators $\gamma_i=\gamma_i(s)$ (see <ref>).
Then the sequence of normalized potentials

\[
\frac{\phi(\gamma_1\cdots\gamma_n \omega_0)}{M(\gamma_1\cdots\gamma_n [\omega_0])}
\]

is bounded by a constant independent of the geodesic.
The same boundedness holds for any fixed $i$ and the sequence $\frac{\phi(\gamma_i\cdots \gamma_n \omega_0)}{M(\gamma_i\cdots \gamma_n[\omega_0])}$, with a uniform constant independent of $i$ or the geodesic.

We estimate the $C^0$ norm of every term in <ref>, taking into account that pushing forward by $\gamma_1\cdots \gamma_{i-1}$ does not affect the $C^0$-norm.
The manipulations are:

\begin{align*}
\norm{\frac{\phi(\gamma_i A(\gamma_{i+1}\cdots \gamma_n[\omega_0]))}{M(\gamma_1\cdots \gamma_n [\omega_0])}}_{C^0} & =
\norm{\frac{\phi(\gamma_i A([\omega_{i+1,n}]))}{M_{1,n}}}_{C^0}\\
& =
\frac{M_{i+1,n}}{M_{1,n}}
\norm{\phi\left(\gamma_i \frac{A([\omega_{i+1,n}])}{M_{i+1,n}}\right)}_{C^0}\\
& \leqapprox
\frac{M_{i+1,n}}{M_{1,n}}\norm{\gamma_i}_{op} \leqapprox
\frac{e^{\lambda_{i+1}+\cdots + \lambda_n}}{e^{\lambda_1 + \cdots + \lambda_n}} \norm{\gamma_i}_{op}\\
& \leqapprox
e^{-(\lambda_1 + \cdots + \lambda_{i-1})}
\end{align*}

Using the estimate from <ref> allows us to sum the geometric series and conclude.

Let $s$ be any geodesic starting at $[\omega_0]$ and landing at an irrational point on the boundary, with associated sequence of generators $\gamma_i=\gamma_i(s)$ (see <ref>).
Then the sequence of normalized potentials

\[
\frac{\phi(\gamma_1\cdots\gamma_n \omega_0)}{M(\gamma_1\cdots\gamma_n [\omega_0])} = \frac{\phi(\omega_{1,n})}{M_{1,n}}
\]

forms a Cauchy sequence and we furthermore have the estimate

\[
\norm{
\frac{\phi(\gamma_1\cdots\gamma_n \omega_0)}{M(\gamma_1\cdots\gamma_n [\omega_0])}
\frac{\phi(\gamma_1\cdots\gamma_{n+m} \omega_0)}{M(\gamma_1\cdots\gamma_{n+m} [\omega_0])}
\leqapprox ne^{-\delta \cdot n}
\]

for some fixed $\delta>0$ (depending only on the geometric constructions in <ref>).

Therefore, there exists a closed positive current with continuous potentials in the boundary class normalized to mass $1$ corresponding to the geodesic $s$.

Since $\omega_0$ is a metric, any weak limit of the sequence is a positive current.
The estimate on the sequence of potentials implies, in particular, that we also have $C^0$ convergence of potentials and the limit has a continuous potential.

To analyze the sequence of potentials, we will use the expansion in <ref> to match up the first $n$ terms and estimate their difference.
For the remaining $m$ terms in $\phi(\omega_{1,n+m})$, we use the same type of estimate as in <ref>, plus the estimates on masses <ref>, to find

\begin{align*}
\frac{\norm{\phi(\gamma_{n+i}A([\omega_{n+i+1,n+m}]))}_{C^0}}{M_{1,n+m}}
& \leqapprox
\frac{\norm{\gamma_{n+i}}_{op} M_{n+i+1,n+m}}{M_{1,n+m}}\\
& \leqapprox e^{-(\lambda_1+\cdots + \lambda_n)}e^{-(\lambda_{n+1} + \cdots + \lambda_{n+i-1})}
\end{align*}

We next apply the estimate from <ref> to sum the above estimate in $i$ and obtain a bound of $\leqapprox e^{-(\lambda_1+\cdots +\lambda_n)}\leqapprox e^{-\delta n}$.

For the remaining $n$ differences, we need to estimate (omitting $\gamma_1\cdots \gamma_{i-1}$ applied to both terms, which does not affect the $C^0$ norm):

\begin{multline*}
\norm{
\frac{\phi(\gamma_iA(\gamma_{i+1}\cdots \gamma_n[\omega]))}{M_{1,n}}
\frac{\phi(\gamma_i A(\gamma_{i+1}\cdots \gamma_{n+m}[\omega]))}{M_{1,n+m}}
\norm{
\frac{\phi(\gamma_i A([\omega_{i+1,n}]))}{M_{1,n}}
\frac{\phi(\gamma_i A([\omega_{i+1,n+m}]))} {M_{1,n+m}}
\end{multline*}

The first step is to rewrite each term in normalized form as

\[
\frac{M_{i+1,n}}{M_{1,n}}{\phi\left(\gamma_i A\left(\left[\frac{\omega_{i+1,n}}{M_{i+1,n}}\right]\right)\right)}
\text{ and }
\frac{M_{i+1,n+m}}{M_{1,n+m}}{\phi\left(\gamma_i A\left(\left[\frac{\omega_{i+1,n+m}}{M_{i+1,n+m}}\right]\right)\right)}
\]

To proceed, we will use the estimate from <ref>:

\[
\norm{
\frac{[\omega_{i,j}]}{M_{i,j}} -
\frac{\gamma_i\cdots \gamma_j [\alpha]}{M(\gamma_i\cdots \gamma_j[\alpha])}
\leq
\frac{C}{\norm{\gamma_i\cdots \gamma_j}_{op}}
\]

whenever $[\alpha]$ is transverse to $\gamma_i\cdots \gamma_j$.
Let us apply this estimate to the expressions to which we apply the operator $A$:

\begin{align*}
\frac{[\omega_{i+1,n+m}]}{M_{i+1,n+m}}
& =
\frac{\gamma_{i+1}\cdots \gamma_n [\omega_{n+1,n+m}]}{M(\gamma_{i+1}\cdots \gamma_n [\omega_{n+1,n+m}])}
\frac{[\omega_{i+1,n}]}{M_{i+1,n}} + \frac{O(1)}{\norm{\gamma_{i+1}\cdots \gamma_n}_{op}}
\end{align*}

We now also take $[\alpha]= \frac{[\omega_{n+1,n+m}]}{M_{n+1,n+m}}$ to find

\begin{align*}
\frac{[\omega_{i+1,n}]}{M_{i+1,n}} -
\frac{\gamma_{i+1}\cdots \gamma_n\frac{[\omega_{n+1,n+m}]}{M_{n+1,n+m}}}{M\left(\gamma_{i+1}\cdots \gamma_n\frac{[\omega_{n+1,n+m}]}{M_{n+1,n+m}}\right)}
= \frac{O(1)}{\norm{\gamma_{i+1}\cdots \gamma_n}_{op}}
\end{align*}

So we have the estimate for the masses:

\begin{align*}
\abs{\frac{M_{i+1,n}}{M_{1,n}}
\frac{M_{i+1,n+m}}{M_{1,n+m}}}
& =
\abs{
\frac{1}{M\left(\gamma_1\cdots \gamma_i \frac{\omega_{i+1,n}}{M_{i+1,n}}\right)}
\frac{1}{M\left(\gamma_1\cdots \gamma_i \frac{\omega_{i+1,n+m}}{M_{i+1,n+m}}\right)}
\\
& =
\frac{M\left(\gamma_1\cdots \gamma_i \frac{O(1)}{\norm{\gamma_{i+1}\cdots \gamma_n}_{op}}\right)}
{M\left(\gamma_1\cdots \gamma_i \frac{\omega_{i+1,n}}{M_{i+1,n}}\right)
\cdot
M\left(\gamma_1\cdots \gamma_i \frac{\omega_{i+1,n+m}}{M_{i+1,n+m}}\right)
\\
& \leq
\frac{O(1)}{\norm{\gamma_1\cdots \gamma_i}_{op}{\norm{\gamma_{i+1}\cdots \gamma_n}_{op}}}\\
& \leqapprox e^{-(\lambda_1 + \cdots + \lambda_n)}
\end{align*}

where we use for the last inequality again transversality to say that the mass, and the norm of the operator are the same up to constants.

So the terms we must subtract from each other are (using the above estimates)

\begin{align*}
\left(\frac{M_{i+1,n+m}}{M_{1,n+m}} +
\frac{O(1)}{e^{\lambda_1 + \cdots + \lambda_n}}\right)
& \cdot \phi\left(\gamma_i A \left(\frac{[\omega_{i+1,n}]}{M_{i+1,n}}\right)\right)\\
\intertext{ and }
\left(\frac{M_{i+1,n+m}}{M_{1,n+m}}
\phantom{e^{-(\lambda_1 + \cdots + \lambda_n)}}
\right)
& \cdot \phi\left(\gamma_i A \left(\frac{[\omega_{i+1,n}]}{M_{i+1,n}}
+ \frac{O(1)}{e^{\lambda_{i+1}+\cdots+\lambda_n}}\right)\right)
\end{align*}

Upon subtraction, we are left with the two terms

\begin{align*}
\frac{O(1)}{e^{\lambda_1+\cdots + \lambda_n}}
\cdot \phi\left(\gamma_i A \left(\frac{[\omega_{i+1,n}]}{M_{i+1,n}}\right)\right)\text{ and }
\frac{M_{i+1,n+m}}{M_{1,n+m}}
\cdot \frac{\phi\left(\gamma_i A\left(O(1)\right)\right)}{e^{\lambda_{i+1+\cdots +\lambda_n}}}
\end{align*}

Using that $\norm{\phi(\gamma_iA(O(1)) )}_{C^0}\leq O(1)\norm{\gamma_i}_{op}$ which is valid for both situations, when $\gamma_i$ is parabolic, or when $\gamma_i$ belongs to some fixed set of generators (see <ref>),
we get that each term in the difference is bounded by

\[
\norm{\text{difference}}_{C^0}
\leqapprox e^{-(\lambda_1+\cdots + \lambda_n)}e^{\lambda_i}
\]

We again apply the estimate from <ref> to $\lambda_1+\cdots +\lambda_{i-1}$ and to $\lambda_{i+1}+\cdots + \lambda_n$ to find that the term above is bounded by $O(1)\cdot e^{-n\delta}$, and since we have $n$ terms the result follows.

§.§.§ Estimates for continuity at a parabolic boundary point

We now assemble also the estimates that give continuity at the parabolic boundary points, needed for the proof of <ref>.
Recall that we have a sequence $\gamma_1,\gamma_2,\ldots$ such that $\gamma_1\in \Gamma_{[E_0]}$ and $\norm{\xi(\gamma_1)}_{\NS}\geq R$, and we are allowed to choose $R$ as large as we want.
We need to estimate a potential, which we rewrite using <ref>:

\begin{align*}
\frac{\phi(\gamma_1\cdots \gamma_n \omega_0)}{M(\gamma_1\cdots \gamma_n[\omega_0])} & =
\frac{\phi\left(\gamma_1\cdot A\big( \gamma_2\cdots \gamma_n[\omega_0] \big)\right)}
{M\left(\gamma_1\cdot \big( \gamma_2\cdots \gamma_n[\omega_0] \big)\right)} +
\\
& +
\gamma_1\phi\left(\frac{\gamma_2\cdots \gamma_n[\omega_0]}{M(\gamma_2\cdots \gamma_n[\omega_0])}\right)
\cdot
\frac{M(\gamma_2\cdots \gamma_n[\omega_0])}{M(\gamma_1\cdots \gamma_n[\omega_0])}
\end{align*}

We consider the behavior of this expression as $\norm{\xi(\gamma_1)}_{\NS}\to +\infty$.

The second term goes to zero, since <ref> says that the $C^0$-norm of $\phi\left(\frac{\gamma_2\cdots \gamma_n[\omega_0]}{M(\gamma_2\cdots \gamma_n[\omega_0])}\right)$ is uniformly bounded, it does not change when we apply $\gamma_1$, while the factor $\frac{M(\gamma_2\cdots \gamma_n[\omega_0])}{M(\gamma_1\cdots \gamma_n[\omega_0])}$ goes to zero using the estimates in <ref>.

For the first term, to ease notation let $\xi_1:=\xi(\gamma_1)$.
Recall that as $R\to +\infty$ (i.e. $\norm{\xi_1}_{\NS}\to +\infty$) we have that $\frac{\xi_1}{\norm{\xi_1}_{\NS}}\to \xi$ and thus $\frac{\eta_{[E_0]}(\xi_1,\xi_1)}{\norm{\xi_1}^{2}_{\NS}}\to \eta_{[E_0]}(\xi,\xi)$, in the $C^0$-topology of potentials.
By <ref><ref> we also have that

\[
\frac{\phi\left(\gamma_1\cdot A\big( \gamma_2\cdots \gamma_n[\omega_0] \big)\right)}
{M\left(\gamma_1\cdot \big( \gamma_2\cdots \gamma_n[\omega_0] \big)\right)}
\frac{\phi(\eta_{[E_0]}(\xi_1,\xi_1))}{M(\eta_{[E_0]}(\xi_1,\xi_1))}
\]

The assumption of the proposition are verified since the cohomology class $\gamma_2\cdots \gamma_n[\omega_0]$ is parabolically normalized for the class $[E_0]$ by the geometric interpretation in <ref>.
It remains to observe that $M(\eta_{[E_0]}(\xi_1,\xi_1))=\norm{\xi_1}^2_{\NS}M([E_0])$ and analogously for $\xi$, and from this the convergence to the correctly normalized current follows.
In fact, we get a quantitative estimate depending on the distance between $\frac{\xi_1}{\norm{\xi_1}_{\NS}}$ and $\xi$, which goes to zero as $R\to +\infty$.

§.§.§ Laminarity

It follows from the work of Dujardin [27] (see also de Thélin <cit.>) that the currents constructed in <ref> are also weakly laminar, in the sense of <cit.>.
Indeed, all the currents can also be obtained as weak limits of $\frac{1}{m_n}(\gamma_n)_{*} T_C$, where $C$ is a fixed smooth ample curve, and $T_C$ is the current of integration on it, $\gamma_n$ is a sequence of automorphisms, and $m_n\to +\infty$ is a sequence of normalization factors.
By the results the aforementioned papers, it suffices to provide a bound on the genus and singularities of the curves $(\gamma_n)_* T_C$ in terms of $m_n$, but since the $\gamma_n$ is an automorphism the genus doesn't change and no singularities are introduced.

Specifically, fix an ample linear divisor $A$ on $X$ and let $\pi_{A}\colon X\dashrightarrow \bP^1$ be a rational projection determined by a $2$-dimensional linear subsystem.
We can select the $2$-dimensional subspace in a Baire-generic way such that the locus of indeterminacy of $\pi_A$ doesn't contain any point on any of the curves $\gamma_n(C)$.
By construction it follows that $\deg \pi_A\vert_{\gamma_n(C)}=[A].[\gamma_n(C)]=:d_n$.
By Riemann–Hurwitz the number of ramification points of $\pi_A\vert_{\gamma_n(C)}$ is equal to $2\left(g(C)-1+d_n\right)$, since the genus of $C$ and $C_n$ agree.
Now the required bounds in the proofs of Dujardin <cit.> hold.
The proof as written in loc. cit. is for $\bP^2$, but for generalizing to the case of K3s it suffices to bound the number of ramification points by $O(d_n)$, as just established.

§.§ Uniqueness results, after Sibony–Verbitsky

We quote here some results established by Sibony and Verbitsky.
While these results are not needed for the construction of currents in <ref>, they do show that the resulting currents are unique and hence canonical.

Let $X$ satisfy the standing assumptions in <ref>.
Suppose that $[\alpha]\in \partial \Amp(X)$ is an irrational class, i.e. no nonzero rescaling of $[\alpha]$ is integral.
Then the closed positive current in the class $[\alpha]$ is unique.

For completeness, we include the proof but we emphasize that the result is due to Sibony–Verbitsky.

Let $\cP_{[\alpha]}$ denote the closed positive currents in the class $[\alpha]$.
It is a metric space equipped with the $L^1$ metric:

\[
\dist(\alpha,\alpha'):=\int_X |\phi|\dVol \text{ where } \alpha=\alpha'+dd^c \phi \text{ and } \int_X \phi\dVol =0.
\]

Let $d([\alpha]):=\diam \cP_{[\alpha]}$ denote the diameter of this metric space.
This is an upper semicontinuous function, since if $\alpha_i\to \alpha$ and $\alpha_i'\to \alpha'$ weakly in the sense of currents, and $[\alpha_i]=[\alpha_i']$ then $\dist([\alpha],[\alpha'])=\lim_i \dist([\alpha_i],[\alpha_i'])$ since we have $L^1$ convergence of the normalized difference of potentials (see e.g. <cit.> for the local version of the statement).
The diameter function also satisfies $d(\gamma[\alpha]) = d([\alpha])$ for any $\gamma\in \Aut(X)$ and is homogeneous of degree $1$, i.e. $d\left(e^{\lambda}[\alpha]\right)=e^{\lambda} d([\alpha])$ for any $\lambda\in \bR$.

Take now a hyperbolic automorphism $\gamma$ and boundary class $[\alpha_+]$ such that $\gamma[\alpha_+]=e^\lambda [\alpha_+]$ with $\lambda>0$.
From the relation

\[
= e^{\lambda}d([\alpha_+])
\]

it follows that $d([\alpha_+])=0$.

Take now an arbitrary irrational class $[\alpha]\in \partial \Amp(X)$.
By <ref> below, its orbit under $\Aut(X)$ is dense in $\partial \Amp(X)$.
So there exists a sequence of automorphisms $\gamma_i$ such that $\gamma_i[\alpha]\to [\alpha_+]$ and by upper semicontinuity of the diameter function, combined with the trivial bound $d([\alpha])\geq 0$, it follows that $d([\alpha])=0$.

We think it is instructive to visualize the diameter function in the simplest situation, when $\Aut(X)$ is a lattice in $\SO_{1,2}(\bR)$.
The action of $\Aut(X)$ on $\partial \Amp(X)$ in this case is isomorphic to the action of a lattice in $\SL_2(\bR)$ on $\bR^2\setminus 0$.
There is a finite set of nonzero values corresponding to the diameters for the classes $[E]$ of elliptic fibrations (which fall finitely many $\Aut(X)$ equivalence classes).

In the simplest case when the lattice is commensurable to $\SL_2(\bZ)$, the function $d\colon \bR^2\setminus 0\to \bR_{\geq 0}$ is equal to one of those constants at the primitive integral vectors and extended by homogeneity to the each rational ray, and equals zero on irrational rays.
The function is upper semicontinuous since there are only finitely many primitive vectors in a disc of given radius.
Note also that restricting to say a vertical line $x=1$, the diameter function becomes at rational points of the form $\frac{p}{q}\mapsto \frac{c(p/q)}{q}$ where $c(p/q)$ takes finitely many values, and the function vanishes at irrational points.

§.§ Boundary currents and Ricci-flat metrics

§.§.§ Setup

For each class $[\omega]$ on a K3 surface, Yau's solution of the Calabi conjecture [77] ensures that there exists a unique Ricci-flat representative $[\omega]$.
The degeneration of these metrics, as the cohomology class $[\omega]$ approaches the boundary, has been the focus of intense study, see e.g. the second-named author's recent survey [72] and references therein.
We now interpret our earlier constructions and deduce some consequences for these degenerations.

Fix, for definiteness, a class in $\NS(X)_{\bR}$ with a Ricci-flat metric $\omega_0$ inside.

§.§.§ Irrational boundary classes

Suppose that $[\alpha]\in \partial \Amp(X)$ is an irrational class.
Let also $\gamma_1,\gamma_2,\ldots$ be a sequence of generators as in <ref>, such that $\frac{1}{m_n}(\gamma_1)_*\cdots (\gamma_n)_{*}[\omega_0]\to [\alpha]$, where $m_n$ is an appropriate sequence of scalars with $m_n\to +\infty$.
From the arguments in the proof of <ref>, it follows that the currents $\frac{1}{m_n}(\gamma_1\cdots \gamma_n)_*\omega_0$ converge in the $C^0$ sense of potentials to the unique closed positive current in the class $[\alpha]$.

At the same time, the metrics $(\gamma_1\cdots \gamma_n)\omega_0$ and $\omega_0$ are isometric, since $\gamma_i$ are automorphisms.
Therefore, in the Gromov–Hausdorff sense, the metrics $\frac{1}{m_n}(\gamma_1\cdots \gamma_n)_*\omega_0$ converge to a point, since $m_n\to +\infty$. This confirms <cit.> for projective K3 surfaces with Picard number at least $3$ and no $(-2)$-curves, for classes on the boundary of the ample cone.

We thus see that for irrational boundary points, the Gromov–Hausdorff limit and weak limit as currents (or even stronger $C^0$-potentials limit) disagree.

§.§.§ Rational boundary points

Suppose now that $[E]$ is the class of the fiber in a genus one fibration $X\xrightarrow{\pi}B$.
Then it is a famous result of Gross-Wilson [42] that, under the assumption that all singular fibers of $\pi$ are of type $I_1$, the Ricci-flat metrics in the class $t[\omega_0]+[E]$ converge as $t\to 0$ in the Gromov–Hausdorff sense to the metric completion of $(B^\circ,c\cdot \omega_B)$, where $B^\circ$ denotes the complement of the image of the singular fibers of $\pi$, $\omega_B:=\pi_*\dVol$ is the $(1,1)$-form on $B$ obtained by pushing forward the volume on $X$, and $c>0$ is a normalizing constant.
Furthermore, these Ricci-flat metrics converge locally smoothly on $X^\circ=\pi^{-1}(B^\circ)$ to $\pi^*\omega_B$, and the above metric completion is homeomorphic to $B$, has singularities at finitely many points, and on $B^\circ$ there is an affine structure that makes $\omega_B$ a “Monge-Ampère metric” in the sense of Cheng-Yau [19]. Later a new proof of all these results was given in [39, 40], which also allows for arbitrary singular fibers, and this was extended to higher-dimensional hyperkähler manifolds in [73].

It is therefore natural to compare $\omega_B$, which is canonically associated to the class $[E]$, and the currents coming from <ref> via the dynamics of parabolic automorphisms.
With the same normalization, there is a $\bR\bP^{\rho-3}$ of currents coming from dynamics, and these are in general distinct as soon as $\rho\geq 4$, as follows from <cit.>.
So it is not possible for the currents coming from dynamics to be equal to the current obtained by degenerating the Ricci-flat metric.
We suspect that even in the case $\rho=3$, the unique dynamically defined current does not, in general, equal the one coming from degenerations of Ricci-flat metrics.
See also <ref> for an illustration of why the currents are, in general, different.

§.§.§ Conical vs tangential approach to the boundary

To put some of the discussion in this section into context, let us recall the notion of “conical convergence” to a boundary point $[\alpha]\in \partial\Amp^1(X)$, viewed as a point on the boundary of hyperbolic space.
Namely, $x_i\to[\alpha]$ conically, if there exists a geodesic ray $s$ starting at some point $p_0$ in the interior of hyperbolic space and going to $[\alpha]$, and a constant $C$, such that $\dist(x_i,s)\leq C$ and $\dist(p_0,x_i)\to+\infty$.
The construction in <ref> uses a sequence of points converging conically to an irrational boundary point.
On the other hand, the currents associated to rational boundary points via <ref>, are obtained from points in hyperbolic space that converge tangentially to the rational boundary point, since they all lie on some fixed horosphere.
By contrast, the Gromov–Hausdorff collapse for elliptic fibrations happens along conical approaches to the rational boundary point.

§.§.§ An application

We now give another application of our main theorem <ref> to degenerations of Ricci-flat Kähler metrics. The setup is the following: $X$ is a K3 surface to which Theorem <ref> applies (so $X$ projective with Picard number at least $3$ and no $(-2)$-curves), with a fixed unit-volume Ricci-flat Kähler metric $\omega$ and $\alpha$ a closed real $(1,1)$-form whose class $[\alpha]\in \NS_{\mathbb{R}}$ is on the boundary of the ample cone and satisfies $\int_X\alpha^2=0$. As in <ref>, for $t>0$ we let $\omega_t$ be the Ricci-flat Kähler metric on $X$ cohomologous to $\alpha+t\omega$, so that we can write $\omega_t=\alpha+t\omega+i\partial\overline{\partial}\vp_t$ where the smooth potentials $\vp_t$ are normalized by $\int_X\vp_t\omega^2=0$ and satisfy the complex Monge-Ampère equation

\begin{equation}\label{ma_eqn}
\frac{\omega_t^2}{\int_X(\alpha+t\omega)^2}=\omega^2.
\end{equation}

Under these assumptions, we have the uniform $L^\infty$ estimate

\begin{equation}\label{goal}
\sup_X|\vp_t|\leq C,
\end{equation}

for all small $t>0$.

Note that while $C^0$ convergence of $\vp_t$ also seems plausible, the geometric arguments using $\Aut(X)$ can only establish it only along subsequences of $t$ which project to a compact set modulo $\Aut(X)$.

This result follows from <ref> by applying <cit.>, which uses sophisticated pluripotential theory (and holds in all dimensions). In our setting we can give a simple proof using a Moser iteration type argument as in the classical work of Yau [77], as follows.

First, observe that

\begin{equation}\label{volbd}
\int_X (\alpha+t\omega)^2\leq Ct,
\end{equation}

as $t\to 0$. Thanks to <ref> we can find a closed positive current $\eta=\alpha+i\partial\overline{\partial} \vp_0\geq 0$ with $\vp_0\in C^0(X)$ normalized by $\int_X\vp_0\omega^2=0$. From $\int_X\vp_t\omega^2=0$ we see that $\sup_X\vp_t\geq 0$, and since

$$0<\omega_t=\alpha+t\omega+i\partial\overline{\partial}\vp_t\leq C_0\omega+i\partial\overline{\partial}\vp_t,$$

for a uniform $C_0$ independent of $t$ (small), and therefore

$$\Delta_{\omega}\vp_t\geq -2C_0.$$

It then follows from the Green's formula for $\omega$ that $\sup_X\vp_t\leq C$ for $C$ also independent of $t$. Indeed, picking a point $x\in X$ where $\vp_t(x)=\sup_X\vp_t$, we have

$$\vp_t(x)=-\int_X \Delta_{\omega}\vp_t(y) G(x,y) \omega^2\leq 2C_0\int_XG(x,y) \omega^2\leq C,$$

where $G(x,y)$ is the nonnegative Green's function for $\omega$, which is in $L^1(X)$. The upshot is that

\begin{equation}\label{stupid}
|\sup_X\vp_t|\leq C,
\end{equation}

uniformly as $t\to 0$.

Next applying the regularization theorems in <cit.> or [22] to $\vp_0$ (basically given by convolution with a mollifier in local charts, plus a patching argument, see also [8]), we can find smooth functions $\psi_t,t>0$ with $\int_X\psi_t\omega^2=0$ such that $\psi_t\to\vp_0$ uniformly on $X$ as $t\to 0$ and

$$\alpha+i\partial\overline{\partial}\psi_t\geq -\frac{t}{2}\omega,$$

on $X$.
We then define

\begin{equation}\label{lowbd}
\hat{\omega}_t=\alpha+t\omega+i\partial\overline{\partial}\psi_t\geq \frac{t}{2}\omega,
\end{equation}

so these are Kähler forms cohomologous to $\omega_t$, and letting

we can then write

and $\sup_X\hat{\vp}_t=-1.$ Since $\psi_t$ converges uniformly to $\vp_0$, its $L^\infty$ norm is uniformly bounded as $t\to 0$, so to prove (<ref>) it suffices to obtain a uniform $L^\infty$ bound for $\vp_t-\psi_t$, and recalling (<ref>) we can equivalently derive a uniform $L^\infty$ bound for $\hat{\vp}_t$ instead.

For this we employ the Moser iteration method. For $p\geq 2$ compute using (<ref>), (<ref>) and $\hat{\vp}_t\leq -1$

$$\int_X (-\hat{\vp}_t)^{p-1}(\omega_t^2-\hat{\omega}_t^2)\leq \int_X (-\hat{\vp}_t)^{p-1}\omega_t^2\leq Ct \int_X (-\hat{\vp}_t)^{p-1}\omega^2\leq Ct \int_X (-\hat{\vp}_t)^{p}\omega^2,$$

while also

\[\begin{split}
\int_X (-\hat{\vp}_t)^{p-1}(\omega_t^2-\hat{\omega}_t^2)&=\int_X (-\hat{\vp}_t)^{p-1}i\partial\overline{\partial}\hat{\vp}_t\wedge(\omega_t+\hat{\omega}_t)\\
&\geq (p-1)\int_X(-\hat{\vp}_t)^{p-2}i\partial\hat{\vp}_t\wedge\overline{\partial}\hat{\vp}_t\wedge \hat{\omega}_t\\
&\geq \frac{(p-1)t}{2}\int_X(-\hat{\vp}_t)^{p-2}i\partial\hat{\vp}_t\wedge\overline{\partial}\hat{\vp}_t\wedge\omega\\
\end{split}\]

using (<ref>), and so

$$\int_X\left|\partial\left((-\hat{\vp}_t)^{\frac{p}{2}}\right)\right|^2_{g}\omega^2\leq Cp\int_X (-\hat{\vp}_t)^{p}\omega^2,$$

for all $p\geq 2$ (and $C$ independent of $p$) and using the Sobolev inequality this gives

$$\left(\int_X (-\hat{\vp}_t)^{2p}\omega^2\right)^{\frac{1}{2}}\leq C\left(\int_X\left|\partial\left((-\hat{\vp}_t)^{\frac{p}{2}}\right)\right|^2_{g}\omega^2+\int_X (-\hat{\vp}_t)^{p}\omega^2\right)
\leq Cp\int_X (-\hat{\vp}_t)^{p}\omega^2,$$

$$\|\hat{\vp}_t\|_{L^{2p}(X)}\leq C^{\frac{1}{p}}p^{\frac{1}{p}} \|\hat{\vp}_t\|_{L^p(X)},$$

and substituting $p\mapsto 2p$ and iterating this inequality, a standard Moser iteration argument (taking in the end $p=2$) gives

$$\sup_X (-\hat{\vp}_t)\leq C\left(\int_X (-\hat{\vp}_t)^{2}\omega^2\right)^{\frac{1}{2}}\leq C(\sup_X (-\hat{\vp}_t))^\frac{1}{2}\left(\int_X (-\hat{\vp}_t)\omega^2\right)^{\frac{1}{2}},$$

and so

$$\sup_X (-\hat{\vp}_t)\leq C\int_X (-\hat{\vp}_t)\omega^2=\sup_X(\vp_t-\psi_t)+1\leq C,$$

by (<ref>) (and again the uniform boundedness of $\psi_t$). This proves the uniform $L^\infty$ bound for $\hat{\vp}_t$, and therefore also for $\vp_t$.

§.§.§ An example with elliptic fibrations

Let $F_0,F_1\in \bC[X_0,X_1,X_2]$ be two polynomials of homogeneous degree $3$, determining two distinct smooth cubics in $\bP^2$ intersecting at $9$ points.
They also a determine a pencil of cubics $C_t:=\{aF_0+bF_1=0\}$ and we assume additionally that the cubics are in general position so that the singular elements of the pencil are nodal.
Let $\wtilde{\bP}^2$ be the blowup of $\bP^2$ at the base locus of the pencil (the $9$ points) so we have a genus one fibration

\[
\wtilde{\bP}^2 \xrightarrow{\pi'} \bP^1
\]

which in addition has $9$ sections $\sigma_i'\colon \bP^1\to \wtilde{\bP}^2$ corresponding to the exceptional divisors of the blowup.
By selecting one of them as a basepoint, we obtain an abelian group of parabolic automorphisms of rank $8$.
The discussion in <ref> applies (since only nodal singular fibers occur) and associates to each parabolic automorphism a corresponding current on $\bP^1$.

In order to build a K3 surface, let us choose two distinct points $t_1,t_2$ in the pencil, with corresponding smooth cubics $C_{t_1},C_{t_2}$.
The double cover $X_{t_1,t_2}\to \bP^2$ ramified over $C_{t_1}\cup C_{t_2}$ is a K3 surface, after we blow up the $9$ intersection points of the cubics.
In fact we have the morphisms

\begin{equation}
\label{eqn_cd:elliptic_fibration_double_plane}
\begin{tikzcd}
\arrow[r, "c'"]
\arrow[d, "\pi"]
\wtilde{\bP}^2
\arrow[d, "\pi'"]
\\
\bP^1
\arrow[r, "2:1"', "c"]
\bP^1
\end{tikzcd}
\end{equation}

such that $\pi$ is the genus one fibration, and the map $\bP^1\xrightarrow{c}\bP^1 $ is a two-to-one cover ramified over $t_1,t_2$.
The genus one fibration[This example does not fit our standing assumptions on K3s, since the fibrations have sections and hence the K3 has $(-2)$ curves. But the formalism in <ref> does apply.] $X\xrightarrow{\pi}\bP^1$ also has (at least) $9$ sections $\sigma_i'$, one for each intersection point of the cubics.
There is correspondingly a rank $8$ abelian group of parabolic automorphisms of $X$, and the diagram in <ref> commutes for the action of $\bZ^8$ on $X$ and $\wtilde{\bP}^2$.

The dynamical currents on $\bP^1$ associated to parabolic automorphisms on $X$ are therefore pulled back from those on the right side of the diagram via $c^*$, and consequently if we push them back via $c_*$ we get twice the original currents.
In particular, these pushed-forward currents are independent of the parameters $t_1,t_2$ that we used to make the ramified double cover.

On the other hand, we can compute $c_*\pi_*\dVol_X$ by viewing it alternatively as $\pi'_*c'_*\dVol_X$.
To compute this explicitly, let us work in an affine chart where the degree $3$ polynomials are $f_0(x_1,x_2)$ and $f_1(x_1,x_2)$ and $X$ is given by $y^2=(f_0+af_1)(f_0+bf_1)$.
Then the holomorphic $2$-form on $X$ is given by $\Omega:=\frac{dx_1\wedge dx_2}{y}$ and the volume form is $\dVol_X=\frac{|dx_1\wedge dx_2|}{|y|^2}$.
This is invariant by the involution $y\mapsto -y$ and pushes down to $\bA^2$ to become

\[
\dVol_{\wtilde{\bP}^2}(a,b) = \frac{|dx_1\wedge dx_2|^2}{|(f_0 + af_1)(f_0+bf_1)|}
\]

where $(a,b)$ on the left are introduced to note that the volume form does depend on the choice of these parameters.
We now observe that this volume form, as well as its pushforward to $\bP^1$ via the map $(x_1,x_2)\mapsto\frac{f_0(x_1,x_2)}{f_1(x_1,x_t)}=:t$, do depend on the parameters.
Indeed, we have

\begin{align*}
\frac{\dVol_{\wtilde{\bP}^2}(a,b)}{\dVol_{\wtilde{\bP}^2}(a',b')}
& =
\frac{|(f_0 + a'f_1)(f_0+b'f_1)|}{|(f_0 + af_1)(f_0+bf_1)|}\\
& = \frac{\left|\left(\tfrac{f_0}{f_1} + a'\right) \left(\tfrac{f_0}{f_1} + b'\right) \right|}{\left| (\tfrac{f_0}{f_1} + a) (\tfrac{f_0}{f_1} + b)\right|}\\
& = \frac{|t+a'||t+b'|}{|t+a||t+b|}
\end{align*}

So the proportionality factor is a nontrivial function pulled back from $\bP^1$, therefore distinct choices of cubics for ramification lead to distinct pushed-forward volume forms.

§ ELLIPTIC FIBRATIONS AND HEIGHTS

§.§.§ Outline of section

In this section we establish some preliminary properties of heights on surfaces with elliptic fibrations.
The main result of this section is <ref> that establishes the existence of preferred height functions on an elliptic surface, associated to its parabolic automorphisms.
Our basic conventions are spelled out in <ref>.

In <ref> we recall classical results of Silverman and Tate and formulate them in our setting of genus one fibrations without a section.
The dynamical constructions in <ref> are then used in <ref> to construct the preferred height functions.

§.§ The pairing from the variation of canonical height

§.§.§ Setup

Fix a number field $k$ and an algebraic closure $\ov{k}\supset k$.
For all constructions, we allow implicitly to pass to a finite extension of $k$, so that various geometric objects are always defined over $k$.
In particular, all morphisms are over $k$.
Throughout $X$ and $B$ will be a smooth projective surface, respectively curve, defined over the number field $k$.
The notation follows closely that in <ref>.
The morphism $X\xrightarrow{\pi}B$, also defined over $k$, is a fibration in genus one curves and we set $X^\circ\xrightarrow{\pi}B^{\circ}$ to be the morphism restricted to the open locus where the fibration is smooth.
We denote by $k(B)$ the function field of $B$ and by $X_{k(B)}$ the elliptic curve over $k(B)$ obtained from $X$ (i.e. the restriction of $X$ to the generic point of $B$).

§.§.§ Standing assumptions

For this section only, we relax our earlier assumptions that all singular fibers are reduced and irreducible, and consider a slightly more general case.