label = , label-format = ,) label-width = 12pt MnLargeSymbols’164
MnLargeSymbols’171 section0em1.7em

#  Submultiplicative norms and filtrations on section rings

Siarhei Finski

Dedicated to Xiaonan Ma on the occasion of his 50th birthday

Abstract. We show that submultiplicative norms on section rings of polarised
projective manifolds are asymptotically equivalent to sup-norms associated
with metrics on the polarisation.

As an application, we establish that over canonically polarised manifolds,
convex hull of Narasimhan-Simha pseudonorm over pluricanonical sections is
asymptotically equivalent to the sup-norm associated with the supercanonical
metric of Tsuji, refining a result of Berman-Demailly.

As another application, we deduce that the jumping measures associated with
bounded submultiplicative filtrations on section rings converge to the
spectral measures of their Bergman geodesic rays, generalizing previous
results of Witt Nyström and Hisamoto. We show also that the latter measures
can be described using pluripotential theory. More precisely, we establish
that Bergman geodesic rays are maximal, i.e. they can be constructed through
plurisubharmonic envelopes.

As a final application, for non-continuous metrics on ample line bundles over
projective manifolds, we prove that a weak version of semiclassical
holomorphic extension theorem holds for generic submanifolds. This means that
up to a negligible portion, the totality of holomorphic sections over generic
submanifolds extends in an effective way to the ambient manifold.

As an unexpected byproduct, we show that injective and projective tensor norms
on symmetric algebras of finitely dimensional complex normed vector spaces are
asymptotically equivalent.

Table of contents

######

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Fubini-Study metrics associated to pseudonorms on cohomology
    2. 2.2 Kawamata log terminal divisors and a result of Berman-Demailly
    3. 2.3 Rays of norms and logarithmic relative spectrum
    4. 2.4 Pluripotential theory and quantization of distances
  3. 3 Study of the set of submultiplicative norms
    1. 3.1 Classification of submultiplicative norms and applications
    2. 3.2 Semiclassical holomorphic extension theorem and pluripolar sets
    3. 3.3 Projective geometry and norms on symmetric algebras
    4. 3.4 Homogenization of submultiplicative norms on section rings
  4. 4 Norms on spaces of polynomials
    1. 4.1 Bohnenblust-Hille inequality as ratio of injective and projective norms
    2. 4.2 Projective tensor norms and holomorphic extension theorem
  5. 5 Limiting behavior of jumping measures and geodesic rays
    1. 5.1 From submultiplicative filtrations to rays of submultiplicative norms
    2. 5.2 Maximal geodesic rays from submultiplicative filtrations
    3. 5.3 Filtrations, test configurations and geodesic rays

##  1 Introduction

For a holomorphic line bundle $L$ over a compact complex manifold $X$, we
define the section ring

$R(X,L)\mathrel{\mathop{\ordinarycolon}}=\oplus_{k=1}^{\infty}H^{0}(X,L^{k}).$
(1.1)

A graded norm $N=\sum N_{k}$,
$N_{k}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{k}$, over $R(X,L)$ is
called submultiplicative if for any $k,l\in\mathbb{N}^{*}$, $f\in
H^{0}(X,L^{k})$, $g\in H^{0}(X,L^{l})$, we have

$\|f\cdot g\|_{k+l}\leq\|f\|_{k}\cdot\|g\|_{l}.$ (1.2)

As a basic example, any bounded metric $h^{L}$ on $L$ induces the sequence of
sup-norms
${\rm{Ban}}^{\infty}_{k}(h^{L})\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{L^{\infty}_{k}(X,h^{L})}$
over $H^{0}(X,L^{k})$, defined for $f\in H^{0}(X,L^{k})$ as follows

$\|f\|_{L^{\infty}_{k}(X,h^{L})}=\sup_{x\in X}|f(x)|_{h^{L}}.$ (1.3)

The associated graded norm
${\rm{Ban}}^{\infty}(h^{L})=\sum{\rm{Ban}}^{\infty}_{k}(h^{L})$ is clearly
submultiplicative. The main goal of this article is to prove that,
asymptotically, these are the only possible examples.

More precisely, we say that two graded norms $N=\sum N_{k}$, $N^{\prime}=\sum
N_{k}^{\prime}$ over $R(X,L)$ are equivalent ($N\sim N^{\prime}$) if the
multiplicative gap between the graded pieces, $N_{k}$ and $N_{k}^{\prime}$, is
subexponential. This means that for any $\epsilon>0$, there is
$k_{0}\in\mathbb{N}^{*}$, such that for any $k\geq k_{0}$, we have

$\exp(-\epsilon k)\cdot N_{k}\leq N_{k}^{\prime}\leq\exp(\epsilon k)\cdot
N_{k}.$ (1.4)

Recall that any norm $N_{k}$ on $H^{0}(X,L^{k})$ induces the Fubini-Study
metric $FS(N_{k})$ on $L^{k}$ through the associated Kodaira embedding, see
(2.2). By Fekete’s lemma, for any submultiplicative norm $N$, the sequence of
metrics $FS(N_{k})^{\frac{1}{k}}$ converges, as $k\to\infty$, to a (possibly
only bounded from above and even null) metric on $L$, which we denote by
$FS(N)$, cf. Lemma 3.3.

######  Theorem 1.1.

Assume that a graded norm $N=\sum N_{k}$ over the section ring $R(X,L)$ of an
ample line bundle $L$ is submultiplicative and $FS(N)$ is continuous. Then

$N\sim{\rm{Ban}}^{\infty}(FS(N)).$ (1.5)

######  Remark 1.2.

The continuity of $FS(N)$ without submultiplicativity of $N$ do not determine
the equivalence class of $N$, see Proposition 3.7 or [35, Proposition 4.16]
for examples.

To study submultiplicative norms with non-continuous $FS(N)$, we define a
weaker equivalence relation on the set of graded norms. Let
$\lambda_{1},\cdots,\lambda_{r}$ be the ordered logarithmic relative spectrum
between two norms $N,N^{\prime}$ on a finitely dimensional complex vector
space $V$, $\dim V=r$, see Section 2.3 for the definition. For
$p\in[1,+\infty]$, we let

$d_{p}(N,N^{\prime})\mathrel{\mathop{\ordinarycolon}}=\sqrt[p]{\frac{\sum_{i=1}^{r}|\lambda_{i}|^{p}}{r}}.$
(1.6)

We say that graded norms $N=\sum N_{k}$ and $N^{\prime}=\sum N_{k}^{\prime}$
are $p$-equivalent ($N\sim_{p}N^{\prime}$) if

$\frac{1}{k}d_{p}(N_{k},N_{k}^{\prime})\to 0.$ (1.7)

We show in Section 2.3 that $\sim_{p}$ is an equivalence relation and $\sim$
equals $\sim_{+\infty}$.

A graded norm $N$ on $R(X,L)$ is called bounded if
$N\geq{\rm{Ban}}^{\infty}(h^{L})$ for a certain bounded metric $h^{L}$ on $L$.
For submultiplicative $N$, it is equivalent to the boundness of $FS(N)$, see
(3.7).

######  Theorem 1.3.

Assume that a graded norm $N=\sum N_{k}$ over the section ring $R(X,L)$ of an
ample line bundle $L$ is submultiplicative and bounded. Then for any
$p\in[1,+\infty[$, we have

$N\sim_{p}{\rm{Ban}}^{\infty}(FS(N)).$ (1.8)

######  Remark 1.4.

a) For $p=+\infty$, the analogous statement fails, see Proposition 3.9.

b) In non-Archimedean geometry, Boucksom-Jonsson [18, Theorems D and 2.26] and
Reboulet [57] established analogues of Theorems 1.1 and 1.3. Our approaches
are very different.

Recall that a plurisubharmonic (or psh) metric $h^{L}$ on a holomorphic line
bundle $L$ is a (singular) metric such that for any local holomorphic section
$\sigma$ of $L$, $-\log|\sigma|_{h^{L}}$ is psh. A line bundle is called
pseudoeffective (or psef) if it carries a psh metric.

######  Definition 1.5.

We say that a bounded psh metric $h^{L}$ is regularizable from above if there
is a decreasing sequence of continuous psh metrics $h^{L}_{i}$,
$i\in\mathbb{N}$, converging to $h^{L}$ almost everywhere.

######  Remark 1.6.

a) In [6, Theorem 2] Bedford-Taylor described in a local setting regularizable
from above psh metrics as those having pluripolar discontinuity set.

b) According to Demailly’s regularization theorem, see [28], [29], any psh
metric is regularizable from below, meaning that there is an increasing
sequence of smooth positive metrics $h^{L}_{i}$, $i\in\mathbb{N}$, converging
pointwise to $h^{L}$, cf. [38, Theorem 8.1].

As we explain in Section 2.3, Theorem 1.1 tells us that in $\sim$-equivalence
class of every submultiplicative norm with continuous Fubini-Study metric
there is exactly one sup-norm associated with a continuous psh metric. Theorem
1.3 in its turn tells us that for any $p\in[1,+\infty[$, in
$\sim_{p}$-equivalence class of every bounded submultiplicative norm there is
exactly one sup-norm associated with a regularizable from above psh metric.

The proofs of Theorems 1.1 and 1.3 are based on an interpretation of the
submultiplicativity condition in terms of projective tensor norms, see (3.2).
We then use the techniques from [35] to reduce the proofs to the special case
when $X$ is a projective space and $L$ is the hyperplane bundle. In this
setting, the above statements are essentially equivalent to showing that
injective and projective tensor norms on symmetric algebras of finitely
dimensional complex normed vector spaces are asymptotically equivalent, see
Theorem 3.13. Surprisingly, our proof of the aforementioned functional-
analytic statement uses tools from complex geometry, as Ohsawa-Takegoshi
extension theorem. Remark also that in full tensor algebras the projective and
injective tensor norms are essentially never equivalent by a result of Pisier
[54], see Remark 3.14a).

We now describe some applications of Theorems 1.1 and 1.3. We fix a compact
complex manifold $X$ of dimension $n$ and denote by
$K_{X}\mathrel{\mathop{\ordinarycolon}}=\Lambda^{n}T^{(1,0)*}X$ its canonical
line bundle. Narasimhan-Simha in [47] defined pseudonorms
$\mathcal{NS}_{k}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{\mathcal{NS}}_{k}$,
$k\in\mathbb{N}^{*}$, over the vector space of $k$-th pluricanonical sections,
$f\in H^{0}(X,K_{X}^{k})$, as

$\|f\|^{\mathcal{NS}}_{k}\mathrel{\mathop{\ordinarycolon}}=\Big{(}\int_{X}\big{(}(-\sqrt{-1})^{k(n^{2}+2n)}\cdot
f\wedge\overline{f}\big{)}^{\frac{1}{k}}\Big{)}^{k}.$ (1.9)

By a pseudonorm over a finitely dimensional vector space $V$, we mean a non-
negative absolutely homogeneous continuous function over $V$, which is equal
to $0$ only at $0\in V$. We extend the notion of equivalence, $\sim$,
previously defined for graded norms, to graded pseudonorms.

Remark that the sequence of pseudonorms $\mathcal{NS}_{k}$,
$k\in\mathbb{N}^{*}$, is defined without the use of any fixed metric on
$K_{X}$. In particular, it depends only on the complex structure of $X$. Even
more, it is a birational invariant, as birational equivalence between two
complex manifolds $X$ and $Y$ induces the isometry with respect to
$\mathcal{NS}_{k}$ between $H^{0}(X,K_{X}^{k})$ and $H^{0}(Y,K_{Y}^{k})$ for
any $k\in\mathbb{N}^{*}$, cf. [47].

Assume now more generally that a pair $(X,\Delta)$ of $X$ with a
$\mathbb{Q}$-divisor $\Delta$ is klt, see Section 2.2 for necessary
definitions. We denote the log canonical $\mathbb{Q}$-line bundle by
$K_{X}(\Delta)\mathrel{\mathop{\ordinarycolon}}=K_{X}\otimes\mathscr{O}_{X}(\Delta)$
and let $r\mathrel{\mathop{\ordinarycolon}}=r_{\Delta}\in\mathbb{N}^{*}$ be
the minimal number such that $r\Delta$ is a $\mathbb{Z}$-divisor. Any section
$f\in H^{0}(X,K_{X}(\Delta)^{kr})$, $k\in\mathbb{N}^{*}$, can be then
interpreted as a meromorphic section of $K_{X}^{kr}$. The klt condition
implies, see (2.7) and after, that the integrand in (1.9) is finite. We denote
by
$\mathcal{NS}_{kr}^{\Delta}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{\mathcal{NS},\Delta}_{kr}$
the pseudonorm on $H^{0}(X,K_{X}(\Delta)^{kr})$, given by this integral. Over
log canonical ring, $R(X,K_{X}(\Delta)^{r})$, we define the Narasimhan-Simha
graded pseudonorm

$\mathcal{NS}^{\Delta}\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}\mathcal{NS}_{kr}^{\Delta}.$
(1.10)

For any pseudonorm $N_{V}$ over a finitely dimensional vector space $V$, we
associate the convex hull norm ${\rm{Conv}}{(N_{V})}$ on $V$ in such a way
that the unit ball of ${\rm{Conv}}{(N_{V})}$ is the convex hull of the unit
ball of $N_{V}$. We extend this definition to graded pseudonorms naturally.

Now, any psh metric $h^{K,\Delta}$ on $K_{X}(\Delta)^{r}$ induces a volume
form (with singularities) on $X$, denoted by $dV_{h^{K,\Delta}}$, see (2.8)
for details. If the pair $(X,\Delta)$ is klt, $dV_{h^{K,\Delta}}$ is of finite
volume, see (2.7) and after. Recall that Tsuji in [66] defined the
supercanonical metric $h^{K,\Delta}_{{\rm{can}}}$ on $K_{X}(\Delta)^{r}$ over
klt pairs $(X,\Delta)$ with psef $K_{X}(\Delta)^{r}$ through the following
envelope construction: for $x\in X$, we let

$h^{K,\Delta}_{{\rm{can}}}(x)=\inf\Big{\\{}h^{K,\Delta}(x)\mathrel{\mathop{\ordinarycolon}}\quad
h^{K,\Delta}\text{ is a psh metric on $K_{X}(\Delta)^{r}$, with
}\int_{X}dV_{h^{K,\Delta}}\leq 1\Big{\\}}.$ (1.11)

Recall that a pair $(X,\Delta)$ is called log canonically polarised if
$K_{X}(\Delta)^{r}$ is ample. Berman-Demailly in [10, Theorem 5.5] showed that
$h^{K,\Delta}_{{\rm{can}}}$ is continuous and psh for such pairs. As an easy
application of a result of Berman-Demailly and Theorem 1.1, we obtain the
following statement.

######  Theorem 1.7.

For log canonically polarised klt pairs $(X,\Delta)$, the following
equivalence of graded norms on the log canonical ring $R(X,K_{X}(\Delta)^{r})$
holds

${\rm{Conv}}{(\mathcal{NS}^{\Delta})}\sim{\rm{Ban}}^{\infty}(h^{K,\Delta}_{{\rm{can}}}).$
(1.12)

######  Remark 1.8.

Taking convex hull is necessary. In fact, for any $f\in
H^{0}(X,K_{X}(\Delta)^{kr})$, $k\in\mathbb{N}^{*}$, both the Narasimhan-Simha
pseudonorms and sup-norms behave multiplicatively on the sequence $f^{l}$ for
$l\in\mathbb{N}^{*}$. Hence, if the statement would hold without taking the
convex hull, it would imply that Narasimhan-Simha pseudonorms coincide
identically with the sup-norm, which is false.

Historically, Narasimhan-Simha pseudonorms have been introduced in [47] to
study moduli problems. It is known that for canonically polarised manifolds,
the isomorphism type of pseudonormed vector space
$(H^{0}(X,K_{X}^{k}),\mathcal{NS}_{k})$ for sufficiently big and divisible $k$
determines $X$ up to an isomorphism, see Royden [59, Theorem 1] and Chi [22,
Theorem 1.4]. In family setting, the study of positivity of related
(pseudo)norms is linked to Iitaka conjecture and invariance of plurigenera
problem, see Kawamata [41], Berndtsson-Păun [13], Păun-Takayama [49]. See also
Amini-Nicolussi [1], [2] and Shivaprasad [63], [62] for the study of
Narasimhan-Simha pseudonorms and related objects in singular family setting.

We will now describe an application of Theorem 1.3 related to
submultiplicative filtrations. Recall that a decreasing -filtration
$\mathcal{F}$ of a vector space $V$ is a map from to vector subspaces of $V$,
$t\mapsto\mathcal{F}^{t}V$, verifying
$\mathcal{F}^{t}V\subset\mathcal{F}^{s}V$ for $t>s$, and such that
$\mathcal{F}^{t}V=V$ for sufficiently small $t$ and $\mathcal{F}^{t}V=\\{0\\}$
for sufficiently big $t$. We say that $\mathcal{F}$ is graded if it respects
the grading of $V$. It is left-continuous if for any $t\in\real$, there is
$\epsilon_{0}>0$, such that $\mathcal{F}^{t}V=\mathcal{F}^{t-\epsilon}V$ for
any $0<\epsilon<\epsilon_{0}$. All filtrations in this article are assumed to
be decreasing left-continuous and graded if applicable.

A filtration $\mathcal{F}$ on $R(X,L)$ is called submultiplicative if for any
$t,s\in\real$, $k,l\in\mathbb{N}$, we have

$\mathcal{F}^{t}H^{0}(X,L^{k})\cdot\mathcal{F}^{s}H^{0}(X,L^{l})\subset\mathcal{F}^{t+s}H^{0}(X,L^{k+l}).$
(1.13)

We say that $\mathcal{F}$ is bounded if there is $C>0$ such that for any
$k\in\mathbb{N}^{*}$, $\mathcal{F}^{Ck}H^{0}(X,L^{k})=\\{0\\}$. Studying
asymptotic properties of bounded submultiplicative filtrations is related to
K-stability due to their relation with test configurations, see Section 5.3,
cf. [67], [64], [18].

We define the sequence of jumping measures $\mu_{\mathcal{F},k}$,
$k\in\mathbb{N}^{*}$, on of $\mathcal{F}$ as follows

$\mu_{\mathcal{F},k}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{\dim
H^{0}(X,L^{k})}\sum_{j=1}^{\dim
H^{0}(X,L^{k})}\delta_{k^{-1}e_{\mathcal{F}}(j,k)},$ (1.14)

where $\delta_{x}$ is the Dirac mass at $x\in\real$ and $e(j,k)$ are the
jumping numbers, defined as follows

$e_{\mathcal{F}}(j,k)\mathrel{\mathop{\ordinarycolon}}=\sup\Big{\\{}t\in\real\mathrel{\mathop{\ordinarycolon}}\dim\mathcal{F}^{t}H^{0}(X,L^{k})\geq
j\Big{\\}}.$ (1.15)

Recall now that a ray $h^{L}_{t}$, $t\in[0,+\infty[$, of bounded psh metrics
is called geodesic if the associated rotationally-invariant metric
$\hat{h}^{L}$ on $\pi^{*}L$ over $X\times\mathbb{D}^{*}$, defined as

$\hat{h}^{L}(x,s)=h^{L}_{-\log|s|}(x),$ (1.16)

is psh and provides a solution to the following Monge-Ampère equation

$c_{1}(\pi^{*}L,\hat{h}^{L})^{n+1}=0,$ (1.17)

where the wedge product is interpreted in the Bedford-Taylor [7] sense.

For any bounded submultiplicative filtration $\mathcal{F}$ on $R(X,L)$, we
construct in Section 5.1 a ray of submultiplicative norms interpolating
between a fixed submultiplicative norm and the non-Archimedean
submultiplicative norm associated to the filtration, see (2.18) and (2.30). By
taking the Fubini-Study metric of this ray, we obtain a ray of psh metrics,
emanating from any fixed regularizable from above psh metric on $L$. Building
on Theorem 1.3, we prove in Theorem 5.1 that this ray of metrics is geodesic.
We call it the Fubini-Study geodesic ray. We show in Proposition 5.2 that this
ray, when emanates from a continuous psh metric, coincides with the respective
Bergman geodesic ray, defined by Phong-Sturm [51] and Ross-Witt Nyström in
[58], see (5.4). The major difference between the two constructions (both of
which proceed by quantization) is that the first one is obtained from
submultiplicative norms, hence, it can be analyzed using Theorems 1.1, 1.3.

Now, to any geodesic ray $h^{L}_{t}$, $t\in[0,+\infty[$, emanating from a
smooth positive metric, one can define its spectral measure by
$(-\dot{h}^{L}_{t})_{*}(c_{1}(L,h^{L}_{0})^{n}/\int_{X}c_{1}(L)^{n})$, where
the derivative
$\dot{h}^{L}_{t}\mathrel{\mathop{\ordinarycolon}}=(h^{L}_{t})^{-1}\frac{\partial
h^{L}_{t}}{\partial t}|_{t=0}$ can be defined by convexity, see (2.46),
despite the possible absence of regularity. For this measure, in particular,
the $p$-absolute moments are related with the slopes of the $p$-Finsler
distances between points on the geodesic ray, see (2.47). For the Fubini-Study
geodesic ray constructed from a filtration $\mathcal{F}$, we denote the
associated spectral measure by $\mu_{\mathcal{F}}$. As it follows from the
next theorem (and implicit in the notation), $\mu_{\mathcal{F}}$ is
independent from the initial point of the ray.

######  Theorem 1.9.

For any bounded submultiplicative filtration $\mathcal{F}$ on a section ring
$R(X,L)$ of an ample line bundle $L$, the jumping measures
$\mu_{\mathcal{F},k}$, $k\in\mathbb{N}^{*}$, converge weakly, as $k\to\infty$,
to $\mu_{\mathcal{F}}$.

######  Remark 1.10.

a) The existence of the weak limit was proved by Boucksom-Chen [16] in a more
general setting of big line bundles, refining an earlier work of Chen [20].

b) When the filtration is induced by an ample test configuration, Theorem 1.9
was established for product test configurations by Witt Nyström [67, Theorems
1.1 and 1.4] and for general test configurations by Hisamoto in [40, Theorem
1.1], proving a conjecture [67, after Theorem 1.4].

c) Our method differs from [20], [16], [67] and [40]; it is not algebraic in
nature and instead of Okounkov bodies, we rely on Fubini-Study geodesic rays
and Theorem 1.3.

Our next application says, in a way, that Fubini-Study geodesic rays can be
seen as solutions of the Dirichlet problem for the Monge-Ampère equation
(1.17), where one boundary condition is given by the non-Archimedean Fubini-
Study potential of the filtration, see (5.13). In particular, the Fubini-Study
geodesic ray (and hence the measure $\mu_{\mathcal{F}}$) can be studied using
pluripotential theory.

More precisely, following Berman-Boucksom-Jonsson [8] and Boucksom-Jonsson
[18], we define the maximal geodesic ray, emanating from a fixed bounded psh
metric on $L$, by considering an envelope of psh metrics having singularities
at $t=+\infty$ prescribed by the non-Archimedean potential of the filtration,
see Theorem 5.4 and the paragraph after.

######  Theorem 1.11.

For any bounded submultiplicative filtration $\mathcal{F}$ on a section ring
$R(X,L)$ of an ample line bundle $L$, the Fubini-Study geodesic ray emanating
from a fixed regularizable from above psh metric on $L$ coincides with the
respective maximal geodesic ray.

######  Remark 1.12.

An equivalent result was announced in [8, Example 6.9] without a proof.

To prove Theorem 1.11, we rely on the works of Phong-Sturm [52], Berman-
Boucksom-Jonsson [8] and Boucksom-Jonsson [18], which establish Theorem 1.11
for $\mathcal{F}$ induced by ample test configurations, and on the fact,
remarked by Székelyhidi [64], that any filtration can be approximated by
filtrations induced by ample test configurations. It remains to establish that
both Fubini-Study geodesic rays and maximal geodesic rays behave reasonably
under these approximations. For Fubini-Study geodesic rays, the corresponding
statement is Theorem 5.5, and it follows from Theorem 1.3. For maximal
geodesic rays, the corresponding statement is Theorem 5.7, and it follows from
several results from complex and non-Archimedean pluripotential theory
developed by Berman-Boucksom-Jonsson [8] and Boucksom-Jonsson [18].

Let us, finally, describe an application of Theorem 1.3 to the holomorphic
extension problem. Recall that Ohsawa-Takegoshi in [48] gave a sufficient
condition under which a holomorphic section of a vector bundle on a
submanifold extends to a holomorphic section over an ambient manifold with a
reasonable bound on the $L^{2}$-norm of the extension in terms of the
$L^{2}$-norm of the section. Later in [34], we proved a more precise statement
in the semiclassical limit, i.e. when the vector bundle is given by a
sufficiently high tensor power of a fixed positive line bundle. In particular,
in [34, Theorem 1.1], we established an asymptotically optimal semiclassical
version of Ohsawa-Takegoshi extension theorem and in [34, Theorem 1.10] we
proved its version for sup-norms. See also [36, Theorems 1.1 and 1.3] for a
more general statement about jet extensions.

The regularity of the line bundle and its strict positivity were crucial in
latter developments. Since many constructions in complex geometry (as those
arising from envelopes) yield non-regular metrics with weak positivity, it is
natural to ask to which extent our results remain valid in these
circumstances. As we recall in Theorem 3.11, following Zhang [69], Bost [15],
Randriambololona [56] and [35], for continuous psh metrics a weak analogue of
the semiclassical extension theorem holds. It turns out it doesn’t hold more
generally. As an application of Theorem 1.3, we give in Theorem 3.12 a
characterization of submanifolds for which it, nevertheless, holds. Below we
describe a consequence of it.

Recall that a norm $N_{V}=\|\cdot\|_{V}$ on a finitely dimensional vector
space $V$ naturally induces the norm
$\|\cdot\|_{Q}\mathrel{\mathop{\ordinarycolon}}=[N_{V}]$ on any quotient $Q$,
$\pi\mathrel{\mathop{\ordinarycolon}}V\to Q$ of $V$ as follows

$\|f\|_{Q}\mathrel{\mathop{\ordinarycolon}}=\inf\Big{\\{}\|g\|_{V}\mathrel{\mathop{\ordinarycolon}}\quad
g\in V,\pi(g)=f\Big{\\}},\qquad f\in Q.$ (1.18)

Let $Y$ be a closed submanifold of a compact complex manifold $X$ and $L$ be
an ample line bundle over $X$. It is classical that there is
$k_{0}\in\mathbb{N}$, such that for any $k\geq k_{0}$, the map

${\rm{Res}}_{Y}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k})\to
H^{0}(Y,L|_{Y}^{k}),$ (1.19)

is surjective. Hence, a norm on $H^{0}(X,L^{k})$ induces a norm on
$H^{0}(Y,L|_{Y}^{k})$. In this language, Ohsawa-Takegoshi extension theorem
basically compares the two norms on $H^{0}(Y,L|_{Y}^{k})$: one induced from
the metric on $Y$, another one is the quotient norm induced from the metric on
$X$.

######  Theorem 1.13.

For any regularizable from above psh metrics $h^{L}$ on $L$ and any
$p\in[1,+\infty[$, under (1.19), the following equivalence of norms on
$R(Y,L)$ holds for generic submanifolds $Y$:

$[{\rm{Ban}}^{\infty}_{X}(h^{L})]\sim_{p}{\rm{Ban}}^{\infty}_{Y}(h^{L}).$
(1.20)

Generic here means that it holds for $Y$ not contained in a certain pluripolar
subset of $X$.

######  Remark 1.14.

According to Boucksom-Eriksson [17], cf. Theorem 2.13, to study sup-norms, it
is enough to consider only sup-norms associated to regularizable from above
psh metrics.

This paper is organized as follows. In Section 2, we recall the preliminaries.
In Section 3, we prove Theorems 1.1, 1.3 modulo a certain functional-analytic
statement, to which Section 4 is dedicated. We also deduce Theorem 1.13. In
Section 5, we establish Theorems 1.9, 1.11.

Notation. A sequence of numbers (resp. positive numbers) $a_{k}$,
$k\in\mathbb{N}$, is called subadditive (resp. submultiplicative or
superbadditive) if $a_{k+l}\leq a_{k}+a_{l}$ (resp. $a_{k+l}\leq a_{k}a_{l}$
or $a_{k+l}\geq a_{k}+a_{l}$) for any $k,l\in\mathbb{N}$. We extend these
notions for sequences of functions and for metrics on powers of a line bundle.
A sequence of positive real numbers $a_{k}$ is called subexponential if for
any $\epsilon>0$, $\exp(-\epsilon k)\leq a_{k}\leq\exp(\epsilon k)$ for $k$
big enough.

Over $\mathbb{C}^{l}$, $l\in\mathbb{N}^{*}$, we denote by
$\textit{l}_{1}=\|\cdot\|_{1}$ and $\textit{l}_{\infty}=\|\cdot\|_{\infty}$
the norms, defined for $x=(x_{1},\cdots,x_{l})$ as follows
$\|x\|_{1}=\sum|x_{i}|$, $\|x\|_{\infty}=\max|x_{i}|$. By a multiplicative gap
between the pseudonorms $N_{1}$, $N_{2}$ on a vector space $V$, we mean the
minimal constant $C>0$, such that both inequalities $N_{1}\leq CN_{2}$ and
$N_{2}\leq CN_{1}$ are satisfied. By a seminorm over a finitely dimensional
vector space $V$, we mean a non-negative absolutely homogeneous convex
function over $V$.

We denote by $|\cdot|_{h^{L}}$ the induced pointwise norm on $L$ induced by a
norm $h^{L}$. We sometimes denote ${\rm{Ban}}^{\infty}(h^{L})$ by
${\rm{Ban}}^{\infty}_{X}(h^{L})$ to underline the dependence on the ambient
manifold, $X$. Throughout the whole article $L$ is assumed to be ample and $X$
is a compact complex manifold.

For $0\geq a<b$, we will use the following notation
$\mathbb{D}_{a,b}=\\{z\in\mathbb{C}\mathrel{\mathop{\ordinarycolon}}a<|z|<b\\}$,
$\mathbb{D}_{b}=\\{z\in\mathbb{C}\mathrel{\mathop{\ordinarycolon}}|z|<b\\}$,
$\mathbb{D}\mathrel{\mathop{\ordinarycolon}}=\mathbb{D}_{1}$,
$\mathbb{C}^{*}\mathrel{\mathop{\ordinarycolon}}=\mathbb{C}\setminus\\{0\\}$.
We also denote by $\pi$ the projection
$\pi\mathrel{\mathop{\ordinarycolon}}X\times\mathbb{D}\to\mathbb{D}$ to the
second factor and use the similar notations for all of the above spaces.

For a given function $f$ on a topological space, we denote by $f^{*}$ (resp.
$f_{*}$) the upper (resp. lower) semi-continuous regularization of $f$. The
same notations are used for metrics on line bundles.

Acknowledgement. I would like to thank Hajime Tsuji and Mihai Păun for
discussions related to Narasimhan-Simha pseudonorms during Hayama Symposium
2022 and Oberwolfach workshop # 2236 respectively, as well as the organizers
of these meetings for making the discussions possible. I warmly thank
Sébastien Boucksom for drawing my attention to many relevant works and for all
the stimulating discussions on complex and non-Archimedean pluripotential
theory, leading to a significant improvement of the first version of this
article. I also thank Rémi Reboulet for several useful discussions. It is,
finally, a pleasure to thank Xiaonan Ma for all of his help over the years.

##  2 Preliminaries

This section is organized as follows. In Section 2.1, we recall the definition
of the Fubini-Study operator. In Section 2.2, we recall Kawamata log terminal
condition for $\mathbb{Q}$-divisors and a convergence result of Berman-
Demailly. In Section 2.3, we recall some basic results from the theory of
finitely dimensional normed vector spaces. In Section 2.4, we recall the
basics of pluripotential theory and related quantization results.

###  2.1 Fubini-Study metrics associated to pseudonorms on cohomology

In this section we recall the definition of the Fubini-Study operator and its
positivity properties.

We fix an ample line bundle $L$ over a compact complex manifold $X$. Recall
that for $k$ so that $L^{k}$ is very ample, Fubini-Study operator associates
for any norm $N_{k}=\|\cdot\|_{k}$ on $H^{0}(X,L^{k})$, a continuous metric
$FS(N_{k})$ on $L$, constructed in the following way. Consider the Kodaira
embedding

${\rm{Kod}}_{k}\mathrel{\mathop{\ordinarycolon}}X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{*}).$
(2.1)

The evaluation maps provide the isomorphism
$L^{-k}\to{\rm{Kod}}_{k}^{*}\mathscr{O}(-1),$ where $\mathscr{O}(-1)$ is the
tautological bundle over $\mathbb{P}(H^{0}(X,L^{k})^{*})$. We endow
$H^{0}(X,L^{k})^{*}$ with the dual norm $N_{k}^{*}$ and induce from it a
metric $h^{FS}(N_{k})$ on $\mathscr{O}(-1)$ over
$\mathbb{P}(H^{0}(X,L^{k})^{*})$. We define the metric $FS(N_{k})$ on $L^{k}$
as the only metric verifying under the dual of the above isomorphism the
identity

$FS(N_{k})={\rm{Kod}}_{k}^{*}(h^{FS}(N_{k})^{*}).$ (2.2)

Sometimes, by an abuse of notations, we denote by $FS(N_{k})$ the metric
$h^{FS}(N_{k})^{*}$ on $\mathscr{O}(1)$ over $\mathbb{P}(H^{0}(X,L^{k})^{*})$.
A statement below can be seen as an alternative definition of $FS(N_{k})$.

######  Lemma 2.1.

For any $x\in X$, $l\in L^{k}_{x}$, the following identity takes place

$|l|_{FS(N_{k})}=\inf_{\begin{subarray}{c}s\in H^{0}(X,L^{k})\\\
s(x)=l\end{subarray}}\|s\|_{k}.$ (2.3)

###### Proof.

An easy verification, cf. Ma-Marinescu [45, Theorem 5.1.3]. ∎

The above construction of the Fubini-Study metric works more generally for
pseudonorms $N_{k}$. In this case, since the Fubini-Study operator uses the
dual of the pseudonorm and double dual of a pseudonorm equals to its convex
hull, we clearly have

$FS(N_{k})=FS({\rm{Conv}}{(N_{k})}).$ (2.4)

When the norm $N_{k}$ comes from a Hermitian product on $H^{0}(X,L^{k})$, the
Fubini-Study construction is standard and explicit evaluation shows that in
this case $c_{1}(\mathscr{O}(-1),h^{FS}(N_{k}))$ coincides up to a negative
constant with the Kähler form of the Fubini-Study metric on
$\mathbb{P}(H^{0}(X,L^{k})^{*})$ induced by $N_{k}$. In particular,
$c_{1}(\mathscr{O}(-1),h^{FS}(N_{k}))$ is a negative $(1,1)$-form.

Let us now discuss the positivity properties of the metric $FS(N_{k})$ for
general pseudonorms $N_{k}$. A pseudonorm
$N_{V}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V}$ on a vector space $V$
defines a continuous function
$F_{V}\mathrel{\mathop{\ordinarycolon}}V\to[0,+\infty[$,
$v\mapsto\|v\|_{V}^{2}$. Following Kobayashi’s terminology on Finsler metrics,
[42], we say that $N_{V}$ is pseudoconvex if we have
$\sqrt{-1}\partial\overline{\partial}F_{V}\geq 0$ in the sense of currents.
Clearly, if $N_{V}$ is a norm, the function $F_{V}$ is convex by triangle
inequality and then $N_{V}$ is trivially pseudoconvex.

Pseudonorms $N_{V}$ on $V$ are in one-to-one correspondence with metrics
$h^{N_{V}}$ on the tautological line bundle $\mathscr{O}(-1)$ over
$\mathbb{P}(V)$. According to [42, Lemma on p.160], pseudoconvexity of $N_{V}$
is equivalent to the negativity of the $(1,1)$-current
$c_{1}(\mathscr{O}(-1),h^{N_{V}})$. In particular, for any norm $N_{V}$ on
$V$, we have $c_{1}(\mathscr{O}(-1),h^{N_{V}})\leq 0$ in the sense of
currents. Hence, from (2.2) and (2.4), the (singular) metric $FS(N_{k})$ is
psh for any pseudonorm $N_{k}$ on $H^{0}(X,L^{k})$.

###  2.2 Kawamata log terminal divisors and a result of Berman-Demailly

The main goal of this section is to recall Kawamata log terminal (or klt)
condition for $\mathbb{Q}$-divisors and a convergence result of Berman-
Demailly.

We say that for a normal crossing divisor $\sum D_{i}$ on $X$, the
$\mathbb{Q}$-divisor $\sum d_{i}D_{i}$, $d_{i}\in\mathbb{Q}$, is klt if for
any index $i$, we have $d_{i}<1$. More generally, a $\mathbb{Q}$-divisor $D$
is klt if for a resolution of singularities
$\pi\mathrel{\mathop{\ordinarycolon}}\tilde{X}\to X$ of $|D|$ and the normal
crossing $\mathbb{Q}$-divisor $\tilde{D}$, verifying

$K_{\tilde{X}}+\tilde{D}=\pi^{*}(K_{X}+D),$ (2.5)

the pair $(\tilde{X},\tilde{D})$ is klt. This definition doesn’t depend on the
choice of the resolution $\pi$, cf. [44, Lemma 3.10].

The klt condition can be restated in the following more differential-geometric
way. Let $r\mathrel{\mathop{\ordinarycolon}}=r_{D}\in\mathbb{N}^{*}$ be the
minimal number such that the element $rD$ is a $\mathbb{Z}$-divisor. We denote
by $h^{rD}$ the canonical (singular) metric on the line bundle
$\mathscr{O}_{X}(rD)$, defined as

$|s|_{h^{rD}}(x)=1,\quad\text{for }x\notin|D|,$ (2.6)

where $s$ is the canonical (meromorphic) section of $\mathscr{O}_{X}(rD)$.
According to [44, Proposition 3.20], the klt condition is equivalent to the
fact that

$\Big{(}\frac{h^{rD}}{h^{rD}_{sm}}\Big{)}^{\frac{1}{r}}\quad\text{is
integrable over }X,$ (2.7)

where $h^{rD}_{sm}$ is some (hence, any) smooth metric on
$\mathscr{O}_{X}(rD)$.

Let us now recall that psh metrics on log canonical line bundles of klt pairs
give rise to positive integrable volume forms. More precisely, assume first
that $h^{K}_{0}$ is a smooth metric on $K_{X}$. We define the positive volume
form $dV_{h^{K}_{0}}$ by requiring that for any $x\in X$, we have

$dV_{h^{K}_{0}}(x)=(-\sqrt{-1})^{n^{2}+2n}dz_{1}\wedge\cdots\wedge
dz_{n}\wedge d\overline{z}_{1}\wedge\cdots\wedge d\overline{z}_{n},$ (2.8)

where $|dz_{1}\wedge\cdots\wedge dz_{n}|_{h^{K}_{0}}(x)=1$. This construction
can be extended to psh metrics $h^{K}$ on $K_{X}$ by writing
$h^{K}=e^{-\phi}\cdot h^{K}_{0}$ for $\phi\in L^{1}_{{\rm{loc}}}$ and defining
$dV_{h^{K}}\mathrel{\mathop{\ordinarycolon}}=e^{\phi}\cdot dV_{h^{K}_{0}}$.
Clearly, the result doesn’t depend on the choice of $h^{K}_{0}$. The volume
form $dV_{h^{K}}$ is bounded since any quasi-psh function $\phi$ is bounded.
It might, nevertheless, vanish, as $\phi$ is allowed to take $-\infty$ values.

Now, more generally a psh metric $h^{K,D}$ on
$K_{X}^{r}\otimes\mathscr{O}_{X}(rD)$ defines a singular metric
$h^{K,D}_{sing}=\frac{h^{K,D}}{h^{rD}}$ on $K_{X}^{r}$. Then $h^{K,D}$ defines
a (singular) volume form $dV_{h^{K,D}}$ as $dV_{h^{K,D}}=e^{\phi}\cdot
dV_{h^{K}_{0}}$, where $\phi$ is so that
$h^{K,D}_{sing}=e^{-r\phi}\cdot(h^{K}_{0})^{r}$. For $(X,D)$ klt,
$dV_{h^{K,D}}$ is integrable by (2.7).

This property was of utmost importance in the definition of supercanonical
metric in (1.11). Let us now recall a related result about the convergence of
Fubini-Study metrics associated with Narasimhan-Simha pseudonorms. Below we
use the notations from Section 1.

######  Theorem 2.2 (Berman-Demailly [10, Proposition 5.19 and Remark 5.23] ).

For a log canonically polarised klt pair $(X,\Delta)$, the sequence of metrics
$FS(\mathcal{NS}_{rk}^{\Delta})^{\frac{1}{k}}$ on $K_{X}(\Delta)^{r}$
converges uniformly, as $k\to\infty$, to the supercanonical metric,
$h^{K,\Delta}_{{\rm{can}}}$.

######  Remark 2.3.

In [10], authors assume that $\Delta$ is effective, but this is never used in
the proof.

###  2.3 Rays of norms and logarithmic relative spectrum

The main goal of this section is to recall some basic results from the theory
of finitely dimensional normed vector spaces, emphasizing related metric
structures. We also give two constructions of rays of norms on a finitely
dimensional vector space endowed with a filtration and compare them.

Let $N_{i}=\|\cdot\|_{i}$, $i=1,2$, be two norms on a finitely dimensional
vector space $V$. We define the logarithmic relative spectrum of $N_{1}$ with
respect to $N_{2}$ as a non-increasing sequence
$\lambda_{j}\mathrel{\mathop{\ordinarycolon}}=\lambda_{j}(N_{1},N_{2})$,
$j=1,\cdots,\dim V$, defined as follows

$\lambda_{j}\mathrel{\mathop{\ordinarycolon}}=\sup_{\begin{subarray}{c}W\subset
V\\\ \dim W=j\end{subarray}}\inf_{w\in
W\setminus\\{0\\}}\log\frac{\|w\|_{2}}{\|w\|_{1}}.$ (2.9)

Clearly, for any norms $N_{1},N_{2},N_{3}$, $j=1,\cdots,\dim V$, we have

$\lambda_{j}(N_{1},N_{2})+\lambda_{\dim
V}(N_{2},N_{3})\leq\lambda_{j}(N_{1},N_{3})\leq\lambda_{j}(N_{1},N_{2})+\lambda_{1}(N_{2},N_{3}),$
(2.10)

and whenever $N_{1}\leq N_{2}\leq N_{3}$, we have

$\lambda_{j}(N_{1},N_{2})\leq\lambda_{j}(N_{1},N_{3}).$ (2.11)

By [17, Theorem 3.1], the functions $d_{p}$, $p\in[1,+\infty[$, defined in
(1.6), are such that

$d_{p}$ satisfies the triangle inequality over the space of Hermitian norms.
(2.12)

Remark also that John ellipsoid theorem, cf. [55, §3], says that for any
normed vector space $(V,N_{V})$, there is a Hermitian norm $N_{V}^{H}$ on $V$,
verifying

$N_{V}^{H}\leq N_{V}\leq\sqrt{\dim V}\cdot N_{V}^{H}.$ (2.13)

From (2.10), (2.11), (2.12), (2.13) and the fact that $\dim H^{0}(X,L^{k})$
grow polynomially (hence, subexponentially) in $k\in\mathbb{N}$, we see that
$\sim_{p}$ for $p\in[1,+\infty[$ is indeed an equivalence relation.

From (2.9), $\max\\{|\lambda_{1}|,|\lambda_{\dim
V}|\\}=d_{+\infty}(N_{1},N_{2})$ is the minimal constant $C\geq 0$, verifying

$\exp(-C)\cdot N_{2}\leq N_{1}\leq\exp(C)\cdot N_{2}.$ (2.14)

This clearly proves that $\sim_{+\infty}$ coincides with the equivalence
relation $\sim$.

Recall that in a metric space $(\mathcal{M},d)$, a curve $\gamma_{t}$,
$t\in[0,1]$, is called a geodesic if there is $v\geq 0$, verifying
$d(\gamma_{a},\gamma_{b})=v|a-b|$ for any $a,b\in[0,1]$.

It is possible to prove that the space of Hermitian norms on $H^{0}(X,L^{k})$,
endowed with the distance $d_{2}$, is isometric to the space $SL(\dim
H^{0}(X,L^{k}))/SU(\dim H^{0}(X,L^{k}))$, endowed with the distance coming
from the standard $SL(\dim H^{0}(X,L^{k}))$-invariant metric, cf. [17, Theorem
3.7]. The later space is known to be of non-positive sectional curvature, see
[43, Theorem X.8.6], and contractible (by Cartan decomposition). In
particular, by Cartan-Hadamard theorem, it is uniquely geodesic. This goes in
a sharp contrast with the distance $d_{1}$, as we see below.

######  Lemma 2.4.

Assume that Hermitian norms $H_{1},H_{2},H_{3}$ on a vector space $V$ are
ordered as follows $H_{1}\leq H_{2}\leq H_{3}$. Then the following identity
holds

$d_{1}(H_{1},H_{2})+d_{1}(H_{2},H_{3})=d_{1}(H_{1},H_{3}).$ (2.15)

######  Remark 2.5.

Hence, a concatenation of geodesics between $H_{1},H_{2}$ and $H_{2},H_{3}$ is
a geodesic.

###### Proof.

The result follows from (2.12) and Lidskii’s inequality, cf. [27, Theorem
5.1]. It can be, however, proved in a much simpler way. In fact, let $A$ and
$B$ be the transition matrices between $H_{1},H_{2}$ and $H_{1},H_{3}$
respectively. Then $A^{-1}B$ is the transition matrix between $H_{2},H_{3}$.
Since $A$, $B$ and $A^{-1}B$ are transition matrices, the are diagonalizable.
From the ordering of $H_{1},H_{2},H_{3}$, we conclude that the eigenvalues of
$A$, $B$ and $A^{-1}B$ are all bigger than $1$. In particular, we see that
$\exp(d_{1}(H_{1},H_{2}))=(\det A)^{\frac{1}{\dim V}}$,
$\exp(d_{1}(H_{1},H_{3}))=(\det B)^{\frac{1}{\dim V}}$ and
$\exp(d_{1}(H_{2},H_{3}))=(\det A^{-1}B)^{\frac{1}{\dim V}}$. From this,
(2.15) is equivalent to $\det(A)\cdot\det(A^{-1}B)=\det(B)$. ∎

For any $p\in[1,+\infty]$ and graded norms $N=\sum N_{k}$, $N^{\prime}=\sum
N^{\prime}_{k}$ on a section ring $R(X,L)$ of an ample line bundle, we define

$d_{p}(N,N^{\prime})\mathrel{\mathop{\ordinarycolon}}=\limsup_{k\to\infty}\frac{1}{k}d_{p}(N_{k},N^{\prime}_{k}).$
(2.16)

From (2.10), (2.11), (2.12), (2.13) and the fact that $\dim H^{0}(X,L^{k})$
grow polynomially (hence, subexponentially) in $k\in\mathbb{N}$, $d_{p}$ is
symmetric and satisfies the triangle inequality.

We say now that graded norms $N,N^{\prime}$ are in the same asymptotic class
if $d_{1}(N,N^{\prime})<+\infty$ and in (2.16), one can put $\lim$ instead of
$\limsup$ for $p=1$.

######  Lemma 2.6.

Assume that graded norms $N_{1},N_{2},N_{3}$ on $R(X,L)$ are in the same
asymptotic class and they are ordered as follows $N_{1}\leq N_{2}\leq N_{3}$.
Then the following identity holds

$d_{1}(N_{1},N_{2})+d_{1}(N_{2},N_{3})=d_{1}(N_{1},N_{3}).$ (2.17)

###### Proof.

We write $N_{i}=\sum N_{i,k}$ for $i=1,2,3$, and denote by $N_{i,k}^{H}$ the
Hermitian norm on $H^{0}(X,L^{k})$ associated to $N_{i,k}$ as in (2.13). Then
for the graded Hermitian norms
$N_{1}^{H}\mathrel{\mathop{\ordinarycolon}}=\sum\frac{1}{\dim
H^{0}(X,L^{k})}N_{1,k}^{H}$, $N_{2}^{H}\mathrel{\mathop{\ordinarycolon}}=\sum
N_{2,k}^{H}$, $N_{3}^{H}\mathrel{\mathop{\ordinarycolon}}=\sum\dim
H^{0}(X,L^{k})\cdot N_{3,k}^{H}$, by (2.11), (2.13) and the fact that $\dim
H^{0}(X,L^{k})$ grow polynomially (hence, subexponentially) in
$k\in\mathbb{N}$, we have $N_{i}^{H}\sim N_{i}$ for any $i=1,2,3$. By (2.10),
we deduce $d_{1}(N_{i},N_{j})=d_{1}(N_{i}^{H},N_{j}^{H})$ for any $i,j=1,2,3$
and conclude that $N_{1}^{H},N_{2}^{H},N_{3}^{H}$ are in the same asymptotic
class. The proof of Lemma 2.17 follows from this, Lemma 2.4 and the fact that
by (2.13), our norms are ordered as follows $N_{1}^{H}\leq N_{2}^{H}\leq
N_{3}^{H}$. ∎

Let us now discuss two different constructions of rays of norms associated to
a filtration. We fix a finitely-dimensional normed vector space $(V,N_{V})$,
$\|\cdot\|_{V}\mathrel{\mathop{\ordinarycolon}}=N_{V}$ and a filtration
$\mathcal{F}$ of $V$. We construct a ray of norms
$N_{V,\mathcal{F}}^{t}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V,\mathcal{F}}^{t}$,
$t\in[0,+\infty[$, emanating from $N_{V}$, as follows

$\|f\|_{V,\mathcal{F}}^{t}\mathrel{\mathop{\ordinarycolon}}=\inf\Big{\\{}\sum
e^{-t\mu_{i}}\cdot\|f_{i}\|_{V}\,\mathrel{\mathop{\ordinarycolon}}\,f=\sum
f_{i},\,f_{i}\in\mathcal{F}^{\mu_{i}}V\Big{\\}}.$ (2.18)

######  Lemma 2.7.

For any $t\geq s\geq 0$ and $i=1,\ldots,\dim V$, the following estimate holds

$\Big{|}\lambda_{i}(N_{V,\mathcal{F}}^{t},N_{V,\mathcal{F}}^{s})-(t-s)\cdot
e_{\mathcal{F}}(i)\Big{|}\leq\log\dim V,$ (2.19)

where $e_{\mathcal{F}}(i)$ are the jumping numbers of the filtration
$\mathcal{F}$, defined analogously to (1.15).

###### Proof.

First, for any $t\geq s\geq 0$, an easy verification shows for any $f\in V$
the following identity

$\|f\|_{V,\mathcal{F}}^{t}\mathrel{\mathop{\ordinarycolon}}=\inf\Big{\\{}\sum
e^{-(t-s)\mu_{i}}\cdot\|f_{i}\|_{V,\mathcal{F}}^{s}\,\mathrel{\mathop{\ordinarycolon}}\,f=\sum
f_{i},\,f_{i}\in\mathcal{F}^{\mu_{i}}V\Big{\\}}.$ (2.20)

Hence, it is enough to establish Lemma 2.7 for $s=0$.

Let assume first that $N_{V}$ is Hermitian. We then show the following
stronger statement

$\lambda_{i}(N_{V,\mathcal{F}}^{t},N_{V})=t\cdot e_{\mathcal{F}}(i).$ (2.21)

For this, remark that if $x\in V$ is orthogonal (with respect to the scalar
product associated with $N_{V}$) to $\mathcal{F}^{e_{\mathcal{F}}(i-1)}V$,
then $\|x\|_{V,\mathcal{F}}^{t}\geq\|x\|_{V}\cdot\exp(-te_{\mathcal{F}}(i))$.
In particular, since in any $i$-dimensional subspace of $V$, there is an
element orthogonal to the $i-1$-dimensional subspace
$\mathcal{F}^{e_{\mathcal{F}}(i-1)}V$, we have
$\lambda_{i}(N_{V,\mathcal{F}}^{t},N_{V})\leq t\cdot e_{\mathcal{F}}(i)$.
However, by taking a subspace
$W\mathrel{\mathop{\ordinarycolon}}=\mathcal{F}^{e_{\mathcal{F}}(i)}V$ in
(2.9), we obtain $\lambda_{i}(N_{V,\mathcal{F}}^{t},N_{V})\geq t\cdot
e_{\mathcal{F}}(i)$. A combination of the two estimates imply (2.21).

Now, for general norms $N_{V}$, we consider the norm $N_{V}^{H}$ as in (2.13).
Let us denote by $N_{V,\mathcal{F}}^{H,t}$ the ray of norms emanating from
$N_{V}^{H}$ as in (2.18). Clearly, by (2.13), for any $t\in[0,+\infty[$, we
have

$d_{+\infty}(N_{V,\mathcal{F}}^{H,t},N_{V,\mathcal{F}}^{t})\leq\frac{1}{2}\log\dim
V,\qquad d_{+\infty}(N_{V},N_{V}^{H})\leq\frac{1}{2}\log\dim V.$ (2.22)

By (2.10) and (2.21), we conclude the proof of Lemma 2.7. ∎

We now fix a Hermitian norm
$N_{H}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{H}$ on $V$. Let us give an
alternative construction of a ray of norms emanating from $N_{H}$. Consider an
orthonormal basis $s_{1},\ldots,s_{r}$,
$r\mathrel{\mathop{\ordinarycolon}}=\dim V$, of $V$, adapted to the filtration
$\mathcal{F}$, i.e. verifying $s_{i}\in\mathcal{F}^{e_{\mathcal{F}}(i)}V$. We
define the ray of Hermitian norms
$N_{H,\mathcal{F}}^{\perp,t}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{H,\mathcal{F}}^{\perp,t}$,
$t\in[0,+\infty[$, on $V$ by declaring the basis

$(s_{1}^{t},\ldots,s_{r}^{t})\mathrel{\mathop{\ordinarycolon}}=\big{(}e^{te_{\mathcal{F}}(1)}s_{1},\ldots,e^{te_{\mathcal{F}}(r)}s_{r}\big{)},$
(2.23)

to be orthonormal with respect to $N_{H,\mathcal{F}}^{\perp,t}$. The following
result compares the two constructions.

######  Lemma 2.8.

For any (resp. Hermitian) norm $N_{V}$ (resp. $N_{H}$) on $V$ and any
$t\in[0,+\infty[$, we have

$d_{+\infty}(N_{H,\mathcal{F}}^{\perp,t},N_{V,\mathcal{F}}^{t})\leq
d_{+\infty}(N_{H},N_{V})+\log\dim V.$ (2.24)

###### Proof.

Let us denote by $N_{H,\mathcal{F}}^{t}$ the ray of norms emanating from
$N_{H}$ by the construction from (2.18). Let us establish first that for any
$k\in\mathbb{N}$, we have

$\dim V\cdot N_{H,\mathcal{F}}^{t}\geq N_{H,\mathcal{F}}^{\perp,t}.$ (2.25)

By the definition of $N_{H,\mathcal{F}}^{t}$, we conclude that for any
$\lambda\in\real$, $f\in V$, we have

$\|f\|_{H,\mathcal{F}}^{t}\geq e^{-t\lambda}\|Q_{\lambda}(f)\|_{H},$ (2.26)

where $Q_{\lambda}(f)\mathrel{\mathop{\ordinarycolon}}=f-P_{\lambda}(f)$ and
$P_{\lambda}(f)$ is the projection of $f$ to
$\cup_{\epsilon>0}\mathcal{F}^{\lambda+\epsilon}V$ with respect to the norm
$N_{H}$. We take now the decomposition $f=\sum a_{i}s_{1}^{t}$,
$a_{i}\in\mathbb{C}$ of $f\in V$ in basis $(s_{1}^{t},\ldots,s_{r}^{t})$ from
(2.23). Then by the definition of $N_{H,\mathcal{F}}^{\perp,t}$, we have

$\|f\|_{H,\mathcal{F}}^{\perp,t}\mathrel{\mathop{\ordinarycolon}}=\sqrt{\sum|a_{i}|^{2}}.$
(2.27)

By taking sums of (2.26) over all jumping numbers, using (2.27) and the fact
that for any $i=1,\ldots,r$, we have
$\|Q_{e_{\mathcal{F}}(i)}(f)\|_{H}=e^{te_{\mathcal{F}}(i)}\cdot|a_{i}|$, we
deduce (2.25).

Now, directly by the definition of $N_{H,\mathcal{F}}^{t}$, we obtain
$\|f\|_{H,\mathcal{F}}^{t}\leq\sum|a_{i}|$. From this and mean value
inequality, we establish

$N_{H,\mathcal{F}}^{t}\leq\sqrt{\dim V}\cdot N_{H,\mathcal{F}}^{\perp,t}.$
(2.28)

From (2.25) and (2.28), we conclude that
$d_{+\infty}(N_{H,\mathcal{F}}^{\perp,t},N_{H,\mathcal{F}}^{t})\leq\log\dim
V$. To finish the proof, it is only left to use the following trivial bound
$d_{+\infty}(N_{V,\mathcal{F}}^{t},N_{H,\mathcal{F}}^{t})\leq
d_{+\infty}(N_{H},N_{V})$. ∎

Let us remark that the first construction of the ray seems to be more stable
than the second one. In particular, it is monotonic with respect to the
initial norm and the filtration. More importantly, as we shall see in Section
5, when both rays of norms are defined on an algebra (instead of a vector
space), the first ray preserves submultiplicativity. The second one doesn’t
seem to be.

Now, recall that filtrations $\mathcal{F}$ on $V$ are in one-to-one
correspondence with functions
$\chi_{\mathcal{F}}\mathrel{\mathop{\ordinarycolon}}V\to[0,+\infty[$, defined
as

$\chi_{\mathcal{F}}(s)\mathrel{\mathop{\ordinarycolon}}=\exp(-w_{\mathcal{F}}(s)).$
(2.29)

where $w_{\mathcal{F}}(s)$ is the weight associated with the filtration,
defined as
$w_{\mathcal{F}}(s)\mathrel{\mathop{\ordinarycolon}}=\sup\\{\lambda\in\real\mathrel{\mathop{\ordinarycolon}}s\in
F^{\lambda}V\\}$. An easy verification shows that $\chi_{\mathcal{F}}$ is a
non-Archimedean norm on $V$ with respect to the trivial absolute value on
$\mathbb{C}$, i.e. it satisfies the following axioms

  1. 1.

$\chi_{\mathcal{F}}(f)=0$ if and only if $f=0$,

  2. 2.

$\chi_{\mathcal{F}}(\lambda f)=\chi_{\mathcal{F}}(f)$, for any
$\lambda\in\mathbb{C}^{*}$, $k\in\mathbb{N}^{*}$, $f\in V$,

  3. 3.

$\chi_{\mathcal{F}}(f+g)\leq\max\\{\chi_{\mathcal{F}}(f),\chi_{\mathcal{F}}(g)\\}$,
for any $k\in\mathbb{N}^{*}$, $f,g\in V$.

Remark the following relation between the non-Archimedean norm
$\chi_{\mathcal{F}}$ and the rays of submultiplicative norms
$N_{H,\mathcal{F}}^{\perp,t}$, $N_{V,\mathcal{F}}^{t}$ associated to
$\mathcal{F}$ as above: for any $f\in V$, we have

$\log\chi_{\mathcal{F}}(f)=\lim_{t\to\infty}\frac{\log\|f\|_{V,\mathcal{F}}^{t}}{t}=\lim_{t\to\infty}\frac{\log\|f\|_{H,\mathcal{F}}^{\perp,t}}{t}.$
(2.30)

Hence, the rays $N_{H,\mathcal{F}}^{\perp,t}$, $N_{V,\mathcal{F}}^{t}$,
$t\in[1,+\infty[$, should be regarded as interpolations between any given norm
and a non-Archimedean norm associated to $\mathcal{F}$.

Let us, finally, recall some basic constructions of norms on tensor products.
Let $V_{1},V_{2}$ be two finitely dimensional vector spaces endowed with norms
$N_{i}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{i}$, $i=1,2$.

The projective tensor norm
$N_{1}\otimes_{\pi}N_{2}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{\otimes_{\pi}}$
on $V_{1}\otimes V_{2}$ is defined for $f\in V_{1}\otimes V_{2}$ as

$\mathinner{\\!\left\lVert
f\right\rVert}_{\otimes_{\pi}}=\inf\Big{\\{}\sum\|x_{i}\|_{1}\cdot\|y_{i}\|_{2}\mathrel{\mathop{\ordinarycolon}}\quad
f=\sum x_{i}\otimes y_{i}\Big{\\}},$ (2.31)

where the infimum is taken over different ways of partitioning $f$ into a sum
of decomposable terms. The injective tensor norm
$N_{1}\otimes_{\epsilon}N_{2}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{\otimes_{\epsilon}}$
on $V_{1}\otimes V_{2}$ is defined as

$\mathinner{\\!\left\lVert
f\right\rVert}_{\otimes_{\epsilon}}=\sup\Big{\\{}\big{|}(\phi\otimes\psi)(f)\big{|}\mathrel{\mathop{\ordinarycolon}}\quad\phi\in
V_{1}^{*},\psi\in V_{2}^{*},\|\phi\|_{1}^{*}=\|\psi\|_{2}^{*}=1\Big{\\}}$
(2.32)

where $\|\cdot\|_{i}^{*}$, $i=1,2$, are the dual norms associated with
$\|\cdot\|_{i}$. Lemma below compares injective and projective tensor norms,
see [60, Proposition 6.1], [3, Theorem 21] for a proof.

######  Lemma 2.9.

The following inequality between the norms on $V_{1}\otimes V_{2}$ holds

$N_{1}\otimes_{\epsilon}N_{2}\leq N_{1}\otimes_{\pi}N_{2}\leq
N_{1}\otimes_{\epsilon}N_{2}\cdot\min\\{\dim V_{1},\dim V_{2}\\}.$ (2.33)

If, moreover, the norms $N_{1}$ and $N_{2}$ are Hermitian, then

$N_{1}\otimes_{\epsilon}N_{2}\leq N_{1}\otimes N_{2}\leq
N_{1}\otimes_{\pi}N_{2}.$ (2.34)

###  2.4 Pluripotential theory and quantization of distances

The main goal of this section is to recall some basic facts from
pluripotential theory, emphasising metric and quantization aspects of the
theory.

Let us fix a Kähler form $\omega$ on $X$ and consider the space
$\mathcal{H}_{\omega}$ of Kähler potentials, consisting of
$u\in\mathscr{C}^{\infty}(X,\real)$, such that
$\omega_{u}\mathrel{\mathop{\ordinarycolon}}=\omega+\sqrt{-1}\partial\overline{\partial}u$
is strictly positive. We denote by ${\rm{PSH}}(X,\omega)$ the set of
$\omega$-psh potentials; these are upper semi-continuous functions $u\in
L^{1}(X,\real\cup\\{-\infty\\})$, such that $\omega_{u}$ is positive as a
$(1,1)$-current. When the De Rham cohomology $[\omega]$ of $\omega$ satisfies
$[\omega]\in 2\pi H^{2}(X,\mathbb{Z})$, there is a Hermitian line bundle
$(L,h^{L}_{0})$, such that $\omega=2\pi c_{1}(L,h^{L}_{0})$. Hence, upon
fixing $h^{L}_{0}$ (which is uniquely defined up to a multiplication by a
locally constant function), the set $\mathcal{H}_{\omega}$ (resp.
${\rm{PSH}}(X,\omega)$) can be identified with the set of smooth positive
(resp. psh) metrics on $L$ through the correspondence

$u\mapsto h^{L}\mathrel{\mathop{\ordinarycolon}}=e^{-u}\cdot h^{L}_{0}.$
(2.35)

Remark that we then have $\omega_{u}=2\pi c_{1}(L,h^{L})$. This identification
will be implicit later on, and all the constructions (of distances, geodesics,
psh rays, etc.) for elements from $\mathcal{H}_{\omega}$ and
${\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X)$ will be implicitly extended
to the corresponding sets of metrics on the line bundle $L$.

One can introduce on $\mathcal{H}_{\omega}$ a collection of $L^{p}$-type
Finsler metrics. For $u\in\mathcal{H}_{\omega}$, let us first define the
Monge-Ampère operator as
$MA(u)\mathrel{\mathop{\ordinarycolon}}=\frac{\omega_{u}^{n}}{V}$, where
$V=\int\omega^{n}$. If $u\in\mathcal{H}_{\omega}$ and $\xi\in
T_{u}\mathcal{H}_{\omega}\simeq\mathscr{C}^{\infty}(X,\real)$, then the
$L_{p}$-length of $\xi$ is given by the following expression

$\|\xi\|_{p,u}\mathrel{\mathop{\ordinarycolon}}=\sqrt[p]{\int_{X}|\xi|^{p}\cdot
MA(u)}.$ (2.36)

For $p=2$, this was introduced by Mabuchi [46], and for other values of $p$,
by Darvas [24].

Using these Finsler metrics, one can introduce path length metric structures
$(\mathcal{H}_{\omega},d_{p})$. In [24], Darvas studied the completion of
these metric spaces, $(\mathcal{E}_{\omega}^{p},d_{p})$, nowadays called
finite $p$-energy classes, and proved that these completions are geodesic
metric spaces and have a vector space structure. It is also well-known, cf.
Guedj-Zeriahi [39, Exercise 10.2], that

$\cap_{p=1}^{\infty}\mathcal{E}_{\omega}^{p}={\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X).$
(2.37)

Darvas proved in [24, Proposition 4.9] that a monotonic sequence of bounded
psh metrics $h^{L}_{i}$ converges almost everywhere to a bounded psh metric
$h^{L}$, if and only if for any (or for some) $p\in[1,+\infty[$, we have

$\lim_{i\to\infty}d_{p}(h^{L}_{i},h^{L})=0.$ (2.38)

The distance on $\mathcal{E}_{\omega}^{1}$ can be alternatively described in
terms of the Monge-Ampère energy functional $E$. Recall that $E$ is explicitly
given for $u,v\in\mathcal{H}_{\omega}$ by

$E(u)-E(v)=\frac{1}{(n+1)V}\sum_{j=0}^{n}\int_{X}(u-v)w_{u}^{j}\wedge
w_{v}^{n-j}.$ (2.39)

By [39, Proposition 10.14], $E$ is monotonic, i.e. for any $u\leq v$, we have
$E(u)\leq E(v)$. From this and Remark 1.6b), it is reasonable to extend the
domain of the definition of $E$ to ${\rm{PSH}}(X,\omega)$ as

$E(u)\mathrel{\mathop{\ordinarycolon}}=\inf\Big{\\{}E(v)\mathrel{\mathop{\ordinarycolon}}v\in\mathcal{H}_{\omega},u\leq
v\Big{\\}}.$ (2.40)

Darvas proved in [24] that $\mathcal{E}_{\omega}^{1}$ coincides with the set
of $u\in{\rm{PSH}}(X,\omega)$, verifying $E(u)>-\infty$. Moreover, for any
$u,v\in\mathcal{E}_{\omega}^{1}$, verifying $u\leq v$, according to[24,
Corollary 4.14], we have

$d_{1}(u,v)=E(v)-E(u).$ (2.41)

In particular, similarly to Lemma 2.4, $(\mathcal{E}_{\omega}^{1},d_{1})$ is
not a uniquely geodesic space – a fact originally observed by Darvas [25,
comment after Theorem 4.17].

Certain geodesic segments of $(\mathcal{E}_{\omega}^{p},d_{p})$ can be
constructed as upper envelopes of quasi-psh functions. More precisely, we
identify paths $u_{t}\in\mathcal{E}_{\omega}^{p}$, $t\in[0,1]$ with
rotationally-invariant $\hat{u}$ over $X\times\mathbb{D}_{e^{-1},1}$ by
(1.16). We say that a curve $[0,1]\ni t\to v_{t}\in\mathcal{E}_{\omega}^{p}$
is a weak subgeodesic connecting $u_{0},u_{1}\in\mathcal{E}_{\omega}^{p}$ if
$d_{p}(v_{t},u_{i})\to 0$, as $t\to 0$ for $i=0$ and $t\to 1$ for $i=1$, and
we have

$\hat{u}\text{ is $\pi^{*}\omega$-psh on $X\times\mathbb{D}_{e^{-1},1}$.}$
(2.42)

As shown in [24, Theorem 2], a distinguished $d_{p}$-geodesic $[0,1]\ni t\to
u_{t}\in\mathcal{E}_{\omega}^{p}$ connecting $u_{0},u_{1}$ can be then
obtained as the following envelope

$u_{t}\mathrel{\mathop{\ordinarycolon}}=\sup\Big{\\{}v_{t}\,\mathrel{\mathop{\ordinarycolon}}\,t\to
v_{t}\,\text{ is a weak subgeodesic connecting }\,v_{0}\leq u_{0}\text{ and
}v_{1}\leq u_{1}\Big{\\}}.$ (2.43)

When $u_{0},u_{1}\in{\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X)$,
Berndtsson [11, §2.2] in [11, §2.2] proved that $u_{t}$, $t\in[0,1]$, defined
by (2.43), can be described as the only path connecting $u_{0}$ to $u_{1}$, so
that $\hat{u}$ is the solution of the following Monge-Ampère equation

$(\pi^{*}\omega+\sqrt{-1}\partial\overline{\partial}\hat{u})^{n+1}=0,$ (2.44)

where the wedge power is interpreted in Bedford-Taylor sense [7]. For smooth
geodesic segments in $(\mathcal{H}_{\omega},d_{2})$, Semmes [61] and Donaldson
[32] have made similar observations before. The uniqueness of the solution of
(2.44) is assured by [39, Lemma 5.25]. Remark, in particular, that for any
$u_{0},u_{1}\in{\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X)$, the
distinguished weak geodesic connecting them is the same if we regard
$u_{0},u_{1}$ as elements in any of $\mathcal{E}_{\omega}^{p}$,
$p\in[1,+\infty[$.

######  Theorem 2.10 (Darvas-Lu [26, Theorem 2] ).

For any $p\in]1,+\infty[$, $(\mathcal{E}_{\omega}^{p},d_{p})$ is uniquely
geodesic.

Let us now define the spectral measure of a finite geodesic segment. We fix
$u_{0},u_{1}\in{\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X)$ and consider
$u_{t}$, $t\in[0,1]$ as in (2.43). From Berndtsson [11, §2.2], we know that
then $u_{t}\in\mathcal{L}^{\infty}(X)$ and the limits $\lim_{t\to
0}u_{t}=u_{0}$, $\lim_{t\to 1}u_{t}=u_{1}$ hold in the uniform sense. Also,
remark that the condition (2.42) implies that for a fixed $x\in X$, the
function $u_{t}(x)$ is convex in $t\in[0,1]$, see [30, Theorem I.5.13]. Hence,
one-sided derivatives $\dot{u}_{t}^{-}$, $\dot{u}_{t}^{+}$ of $u_{t}$ are
well-defined for $t\in]0,1[$ and they increase in $t$. We denote
$\dot{u}_{0}\mathrel{\mathop{\ordinarycolon}}=\lim_{t\to
0}\dot{u}_{t}^{-}=\lim_{t\to 0}\dot{u}_{t}^{+}$. From [11, §2.2], we know that
$\dot{u}_{0}$ is bounded and by Darvas [25, Theorem 1], we, moreover, have

$\sup|\dot{u}_{0}|\leq\sup|u_{1}-u_{0}|.$ (2.45)

We now assume that $u_{0}\in\mathcal{H}_{\omega}$ and define the spectral
measure $\mu_{u_{0},u_{1}}$ as

$\mu_{u_{0},u_{1}}=(\dot{u}_{0})_{*}(MA(u_{0})),$ (2.46)

where $MA(u_{0})$ was defined in (2.36). Clearly, by (2.35), such definition
coincides with the one from the introduction. Then according to Darvas-Lu-
Rubinstein [27, Lemma 4.5], for any $u_{0}\in\mathcal{H}_{\omega}$,
$u_{1}\in{\rm{PSH}}(X,\omega)\cap\mathcal{L}^{\infty}(X)$, we have

$d_{p}(u_{0},u_{t})=t\cdot\sqrt[p]{\int_{X}|\dot{u}_{0}|^{p}\cdot MA(u_{0})}.$
(2.47)

See also Berndtsson [12] for a more refined result for
$u_{0},u_{1}\in\mathcal{H}_{\omega}$.

Remark that when $u_{1}\geq u_{0}$, from (2.43), the inequality $u_{t}\geq
u_{0}$ holds for any $t\in[0,1]$. In particular, $\dot{u}_{0}\geq 0$ and then
the spectral measure is characterized by the following property

$p$-moments of $\mu_{u_{0},u_{1}}$ coincide with $d_{p}(u_{0},u_{1})^{p}$ for
any $p\in\mathbb{N}^{*}$. (2.48)

Now, for a bounded metric $h^{L}$ on $L$ and a positive volume form $\mu$ of
unit volume on $X$, we denote by
${\rm{Hilb}}_{k}(h^{L},\mu)=\|\cdot\|_{L^{2}_{k}(h^{K},\mu)}$ the $L^{2}$-norm
on $H^{0}(X,L^{k})$, defined for $f\in H^{0}(X,L^{k})$ as

$\|f\|_{L^{2}_{k}(h^{K},\mu)}^{2}=\int_{X}|f(x)|_{h^{L}}^{2}d\mu(x).$ (2.49)

The following result says that the metrics $d_{p}$ on the space of Hermitian
norms on $H^{0}(X,L^{k})$ are quantisations of Darvas metrics $d_{p}$ on the
space of bounded psh metrics on $L$.

######  Theorem 2.11 (Darvas-Lu-Rubinstein [27, Theorem 1.2]).

For any bounded psh metrics $h^{L},h^{L}_{0},h^{L}_{1}$ on an ample line
bundle $L$ and any $p\in[1,+\infty[$, we have

$\displaystyle
d_{p}\Big{(}{\rm{Hilb}}(h^{L}_{0},\mu),{\rm{Hilb}}(h^{L}_{1},\mu)\Big{)}=d_{p}(h^{L}_{0},h^{L}_{1}),$
(2.50) $\displaystyle\lim_{k\to\infty}$ $\displaystyle
d_{p}\Big{(}FS\big{(}{\rm{Hilb}}_{k}(h^{L},\mu)\big{)}^{\frac{1}{k}},h^{L}\Big{)}=0.$

######  Remark 2.12.

a) When $h^{L},h^{L}_{0},h^{L}_{1}$ are smooth and positive, the first result
was established by Chen-Sun [21] for $p=2$ and by Berndtsson [12] for
$p\in[1,+\infty[$; the second result in this more regular setting is a direct
consequence of Tian’s theorem [65]. See also Zelditch [68] and Dai-Liu-Ma [23]
for refinements of the latter statement. These results go in line with the
general philosophy that the geometry of the space of psh metrics on $L$ can be
approximated by the geometry of the space of norms on $H^{0}(X,L^{k})$, as
$k\to\infty$, see Donaldson [32] and Phong-Sturm [50].

b) In particular,
${\rm{Hilb}}(h^{L}_{0},\mu)\not\sim_{p}{\rm{Hilb}}(h^{L}_{1},\mu)$ for any
$p\in[1,+\infty]$ and $h^{L}_{0}\neq h^{L}_{1}$ bounded psh. This goes in
sharp contrast with sup-norms, as we shall see below.

Define now, following Boucksom-Eriksson [17, §7.5], the Fubini-Study envelope
$Q(h^{L})$ of a fixed bounded (not necessarily psh) metric $h^{L}$ as follows

$Q(h^{L})(x)\mathrel{\mathop{\ordinarycolon}}=\inf\Big{\\{}h^{L}_{0}(x)\text{
continous psh on $L$, $h^{L}\leq h^{L}_{0}$}\Big{\\}},\text{ for any }x\in X.$
(2.51)

Remark that by [30, Proposition I.4.24], $Q(h^{L})_{*}$ is psh and
regularizable from above. In particular, $h^{l}=Q(h^{L})_{*}$ if and only if
$h^{L}$ is psh and regularizable from above.

######  Theorem 2.13 ([17, Theorem 7.26 and Corollary 7.27]).

For any bounded metric $h^{L}$ on $L$

$\lim_{k\to\infty}FS\big{(}{\rm{Ban}}^{\infty}_{k}(h^{L})\big{)}^{\frac{1}{k}}=Q(h^{L}),\qquad{\rm{Ban}}^{\infty}(h^{L})={\rm{Ban}}^{\infty}(Q(h^{L})).$
(2.52)

######  Remark 2.14.

From Theorems 2.11, 2.13, a bounded psh metric $h^{L}$ is regularizable from
above if and only if
$\lim_{k\to\infty}d_{p}\big{(}FS\big{(}{\rm{Hilb}}_{k}(h^{L},\mu)\big{)}^{\frac{1}{k}},FS\big{(}{\rm{Ban}}^{\infty}_{k}(h^{L})\big{)}^{\frac{1}{k}}\big{)}=0$
for any (some) $p\in[1,+\infty[$.

######  Proposition 2.15.

For any regularizable from above psh metric $h^{L}$ and $p\in[1,+\infty[$, we
have ${\rm{Hilb}}(h^{L},\mu)\sim_{p}{\rm{Ban}}^{\infty}(h^{L})$. If, moreover,
$h^{L}$ is continuous, then one can take $p=+\infty$.

######  Remark 2.16.

By Theorems 2.11, 2.13, regularizable from above psh metrics is the biggest
subclass of bounded psh metrics for which p-equivalence from Proposition 2.15
holds.

###### Proof.

Since the equivalence relation $\sim_{+\infty}$ equals to $\sim$, the second
part is well-known, cf. [35, Proposition 2.10]. To establish the first part,
take a decreasing sequence of continuous psh metrics $h^{L}_{r}$,
$r\in\mathbb{N}$, as in Definition 1.5. From (2.38), for any $\epsilon>0$,
there is $r_{0}\in\mathbb{N}$, such that

$d_{p}\big{(}h^{L},h^{L}_{r_{0}}\big{)}<\epsilon.$ (2.53)

Remark that we trivially have

${\rm{Hilb}}(h^{L},\mu)\leq{\rm{Ban}}^{\infty}(h^{L})\leq{\rm{Ban}}^{\infty}(h^{L}_{r_{0}}).$
(2.54)

Since $h^{L}_{r}$ are continuous and psh, by the second part of Proposition
2.15, we can find $k_{0}\in\mathbb{N}$, such that for any $k\geq k_{0}$, we
have

${\rm{Ban}}^{\infty}_{k}(h^{L}_{r_{0}})<\exp(\epsilon
k)\cdot{\rm{Hilb}}_{k}(h^{L}_{r_{0}},\mu)$ (2.55)

From Theorem 2.11 and (2.53), we see that

$d_{p}\Big{(}{\rm{Hilb}}(h^{L},\mu),{\rm{Hilb}}(h^{L}_{r_{0}},\mu)\Big{)}<\epsilon.$
(2.56)

From (2.11), (2.54), (2.55) and (2.56), we conclude that

$d_{p}\Big{(}{\rm{Hilb}}(h^{L},\mu),{\rm{Ban}}^{\infty}(h^{L})\Big{)}<2\epsilon.$
(2.57)

Since $\epsilon>0$ was chosen arbitrary, this finishes the proof. ∎

######  Corollary 2.17.

For two distinct regularizable from above psh metrics $h^{L}_{i}$, $i=0,1$,
and any $p\in[1,+\infty]$, we have
${\rm{Ban}}^{\infty}(h^{L}_{0})\not\sim_{p}{\rm{Ban}}^{\infty}(h^{L}_{1})$.

###### Proof.

It follows from Theorem 2.11 and Proposition 2.15. ∎

######  Corollary 2.18.

Assume that a decreasing sequence of continuous psh metrics $h^{L}_{i}$,
$i\in\mathbb{N}$, is uniformly bounded from below. Then for any
$p\in[1,+\infty[$, for $h^{L}\mathrel{\mathop{\ordinarycolon}}=\lim
h^{L}_{i}$,
${\rm{Ban}}^{\infty}(h^{L})\sim_{p}{\rm{Ban}}^{\infty}(h^{L}_{*})$. In
particular, for a bounded metric $h^{L}$ on $L$, we have
${\rm{Ban}}^{\infty}(Q(h^{L}))\sim_{p}{\rm{Ban}}^{\infty}(Q(h^{L})_{*}).$

###### Proof.

It follows directly from the proof of Proposition 2.15 and the fact that
$h^{L}_{*}$ coincides with $h^{L}$ almost everywhere, cf. [30, Proposition
I.4.24]. ∎

##  3 Study of the set of submultiplicative norms

The main goal of this section is to establish a classification of
submultiplicative norms on section rings of ample line bundles. More
precisely, in Section 3.1, we prove Theorems 1.1, 1.3, modulo a certain
statement, which will be interpreted in Section 3.3 in functional-analytic
language. We also deduce Theorem 1.7 from our considerations and previous work
of Berman-Demailly [10]. In Section 3.2, we prove Theorem 1.13. In Section
3.4, as another application of our methods, we give an explicit formula for
the spectral radius seminorm associated with a submultiplicative norm.

###  3.1 Classification of submultiplicative norms and applications

The main goal of this section is to establish Theorems 1.1, 1.3 giving a
characterization of submultiplicative norms in terms of sup-norms, and then to
deduce Theorem 1.7. We conserve the notation from the introduction.

For any $r\in\mathbb{N}^{*}$, $k;k_{1},\ldots,k_{r}\in\mathbb{N}$,
$k_{1}+\cdots+k_{r}=k$, we define the multiplication map

${\rm{Mult}}_{k_{1},\cdots,k_{r}}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k_{1}})\otimes\cdots\otimes
H^{0}(X,L^{k_{r}})\to H^{0}(X,L^{k}),$ (3.1)

as follows $f_{1}\otimes\cdots\otimes f_{r}\mapsto f_{1}\cdots f_{r}$. It is
standard that there is $p_{0}\in\mathbb{N}^{*}$, such that for any
$k_{1},\cdots,k_{r}\geq p_{0}$, the map ${\rm{Mult}}_{k_{1},\cdots,k_{r}}$ is
surjective, cf. [35, Proposition 3.1].

Assume now that $k,l\in\mathbb{N}^{*}$ are big enough so that
${\rm{Mult}}_{k,l}$ is surjective. A central idea of our approach to Theorems
1.1, 1.3 is to interpret the submultiplicativity condition in terms of
projective tensor norms, see (2.31). In fact, using notations (1.18), (2.31),
the submultiplicativity condition can be reformulated in terms of inequalities
between the norms on $H^{0}(X,L^{k+l})$ as follows

$N_{k+l}\leq[N_{k}\otimes_{\pi}N_{l}].$ (3.2)

Assume now for simplicity that $L$ is very ample and all the multiplication
maps are surjective. Let $N_{1}$ be a norm on $H^{0}(X,L)$. By the
surjectivity of the multiplication maps, we endow $H^{0}(X,L^{k})$ with the
norms $N_{k}^{\pi}=[N_{1}\otimes_{\pi}\cdots\otimes_{\pi}N_{1}]$ and
$N_{k}^{\epsilon}=[N_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}N_{1}]$,
where the tensor powers are repeated $k$ times. We denote by $N^{\pi}=\sum
N_{k}^{\pi}$ and $N^{\epsilon}=\sum N_{k}^{\epsilon}$ the induced graded norms
on $R(X,L)$. Next result, established in Sections 3.3 and 4, lies in the core
of our approach to the proofs of Theorems 1.1 and 1.3.

######  Theorem 3.1.

The norms $N^{\pi}$, $N^{\epsilon}$ and ${\rm{Ban}}^{\infty}(FS(N_{1}))$ are
equivalent.

######  Remark 3.2.

In [35, Theorem 4.18], we established a similar statement, where we assumed
that $N_{1}$ is Hermitian and projective/injective tensor norms were replaced
by the Hermitian tensor norm. Since according to Lemma 2.9, the Hermitian
tensor norm is pinched between the injective and projective tensor norms,
Theorem 3.1 refines [35, Theorem 4.18]. The Hermitian assumption in [35]
simplified substantially the proof, as it allowed us to do explicit
calculations on the projective space, see [35, the first part of the proof of
Theorem 4.18]. Circumventing these calculations is exactly the content of
Section 4 of this article.

To establish Theorem 1.1, recall the following basic lemma.

######  Lemma 3.3.

The sequence of Fubini-Study metrics $FS(N_{k})$, $k\in\mathbb{N}^{*}$, is
submultiplicative for any submultiplicative graded norm $N=\sum N_{k}$. In
particular, by Fekete’s lemma, the sequence of metrics
$FS(N_{k})^{\frac{1}{k}}$ on $L$ converges, as $k\to\infty$, to a (possibly
only bounded from above and even null) upper semi-continuous metric, which we
denote by $FS(N)$. We, moreover, have

$FS(N)=\inf FS(N_{k})^{\frac{1}{k}}.$ (3.3)

If $N$ is bounded, then $FS(N)_{*}$ is a regularizable from above psh metric.
If $FS(N)$ is lower semi-continuous and everywhere non-null, the convergence
is uniform and $FS(N)$ is psh.

###### Proof.

The first part follows easily from Lemma 2.1. The second part follows from
Lemma 2.1 and some classical results, cf. [30, Proposition I.4.24]. The third
part is a consequence of the well-known subadditive analogue of Dini’s theorem
and a statement asserting that a pointwise limit of subadditive sequence of
continuous functions is upper semi-continuous, cf. [35, Appendix A]. ∎

###### Proof of Theorem 1.1..

Let us fix $\epsilon>0$. By our assumption on the continuity of $FS(N)$ and
Lemma 3.3, there is $k_{0}\in\mathbb{N}$, such that for any $k\geq k_{0}$, we
have

$FS(N_{k})^{\frac{1}{k}}\leq\exp(\epsilon/3)\cdot FS(N).$ (3.4)

We denote by ${\rm{Hilb}}_{k}(h^{L})$ the norm ${\rm{Hilb}}_{k}(h^{L},\mu)$,
defined in (2.49), where $\mu=\frac{1}{n!}c_{1}(L,h^{L})^{n}$ constructed
using the Bedford-Taylor definition of the wedge power, see [35, §2.2] for
details.

Recall that in [35, Theorem 1.5], we proved that for any continuous psh metric
$h^{L}$, the graded norm ${\rm{Hilb}}(h^{L})=\sum{\rm{Hilb}}_{k}(h^{L})$ is
multiplicatively generated in the sense of [35, Definition 1.3]. This means,
in particular, that there is $k_{1}\in\mathbb{N}$, such that for any $k,l\geq
k_{1}$, we have

$\exp(-\epsilon(k+l)/6)\cdot{\rm{Hilb}}_{k+l}(h^{L})\leq[{\rm{Hilb}}_{k}(h^{L})\otimes{\rm{Hilb}}_{l}(h^{L})]\\\
\leq\exp(\epsilon(k+l)/6)\cdot{\rm{Hilb}}_{k+l}(h^{L}).$ (3.5)

Now, from the result of Berman-Boucksom-Witt Nyström [9, Theorem 1.14], we
know that the graded norms ${\rm{Hilb}}(h^{L})$ and
${\rm{Ban}}^{\infty}(h^{L})$ are equivalent. Applying this for
$h^{L}\mathrel{\mathop{\ordinarycolon}}=FS(N)$ with the use of Lemma 2.9, we
see that there is $k_{2}\in\mathbb{N}$, such that for any $k,l\geq k_{2}$, we
have

$\exp(-\epsilon(k+l)/3)\cdot{\rm{Ban}}^{\infty}_{k+l}(FS(N))\leq[{\rm{Ban}}^{\infty}_{k}(FS(N))\otimes_{\pi}{\rm{Ban}}^{\infty}_{l}(FS(N))]\\\
\leq\exp(\epsilon(k+l)/3)\cdot{\rm{Ban}}^{\infty}_{k+l}(FS(N)).$ (3.6)

We fix from now on $k^{\prime}\geq\max\\{k_{0},k_{1},k_{2}\\}$.

Directly from Lemma 2.1, we see that for any $k\in\mathbb{N}^{*}$, we have

$N_{k}\geq{\rm{Ban}}^{\infty}_{k}(FS(N_{k})).$ (3.7)

In conjunction with (3.3), we see that for any $k\in\mathbb{N}^{*}$, we have

$N_{k}\geq{\rm{Ban}}^{\infty}_{k}(FS(N)).$ (3.8)

Now, through iteration of the submultiplicativity condition, (3.2), for any
$l\in\mathbb{N}^{*}$, we have

$N_{k^{\prime}l}\leq[N_{k^{\prime}}\otimes_{\pi}\cdots\otimes_{\pi}N_{k^{\prime}}],$
(3.9)

where the tensor product is repeated $l$ times. By the application of Theorem
3.1, (3.4) and (3.9), we see that there is $l_{0}\in\mathbb{N}^{*}$, such that
for any $l\geq l_{0}$, we have

$N_{k^{\prime}l}\leq\exp(2\epsilon
k^{\prime}l/3)\cdot{\rm{Ban}}^{\infty}_{k^{\prime}l}(FS(N)).$ (3.10)

Remark that since the spaces $H^{0}(X,L^{p})$,
$p=k^{\prime},\ldots,2k^{\prime}-1$, are finitely dimensional, the norms
$N_{p}$ and ${\rm{Ban}}^{\infty}_{p}(FS(N))$ are comparable up to a uniform
constant. From this and (3.6), we deduce that there is $l_{1}\in\mathbb{N}$,
such that for any $0\leq r\leq k^{\prime}-1$, $l\geq l_{1}$, we have

$[{\rm{Ban}}^{\infty}_{k^{\prime}l}(FS(N))\otimes_{\pi}N_{k^{\prime}+r}]\leq\exp(\epsilon
k^{\prime}l/4)\cdot{\rm{Ban}}^{\infty}_{k^{\prime}(l+1)+r}(FS(N)).$ (3.11)

A combination of (3.2), (3.10) and (3.11) yields for $k\geq
2k^{\prime}\max\\{l_{0},l_{1}\\}$ the following estimate

$N_{k}\leq\exp(\epsilon k)\cdot{\rm{Ban}}^{\infty}_{k}(FS(N)).$ (3.12)

The result now follows directly from (3.8) and (3.12). ∎

######  Remark 3.4.

Similarly to [35, Definition 1.3], one can lighten the submultiplicativity
assumption by requiring that there is $p_{0}\in\mathbb{N}$ and
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{N}_{\geq p_{0}}\to\real$, verifying
$f(k)=o(k)$, as $k\to\infty$, such that for any $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, $f_{i}\in
H^{0}(X,L^{k_{i}})$, $i=1,\cdots,r$, we have

$\|f_{1}\cdots
f_{r}\|_{k}\leq\|f_{1}\|_{k_{1}}\cdots\|f_{r}\|_{k_{r}}\cdot\exp\Big{(}f(k_{1})+\cdots+f(k_{r})+f(k)\Big{)}.$
(3.13)

The proof in this case remains the same with only one modification: instead of
the usual Fekete’s lemma for the proof of the convergence of
$FS(N_{k})^{\frac{1}{k}}$, one needs to rely on [35, Appendix A].

Let us now prove Theorem 1.7. For this, we first need the following basic
fact.

######  Lemma 3.5.

For any graded submultiplicative pseudonorm $N$, ${\rm{Conv}}{(N)}$ is
submultiplicative.

###### Proof.

It follows directly from the fact that the convex hull norm
$\|\cdot\|_{V}^{{\rm{conv}}}$ of a pseudonorm $\|\cdot\|_{V}$ can be described
as
$\|v\|_{V}^{{\rm{conv}}}=\inf\big{\\{}\sum\|v_{i}\|_{V}\mathrel{\mathop{\ordinarycolon}}\sum
v_{i}=v\big{\\}}$. ∎

###### Proof of Theorem 1.7..

By Hölder’s inequality, $\mathcal{NS}^{\Delta}$ is submultiplicative. By the
discussion before Theorem 1.7, $h^{K,\Delta}_{{\rm{can}}}$ is continuous and
non-null. Theorem 1.7 now follows from this, Theorems 1.1, 2.2, Lemma 3.5 and
(2.4). ∎

###### Proof of Theorem 1.3.

By our boundness assumption, the fact that $FS(N_{2^{k}})^{\frac{1}{2^{k}}}$,
$k\in\mathbb{N}$, decrease and Lemma 3.3, we conclude from (2.38) that for any
$\epsilon>0$, there is $r\in\mathbb{N}^{*}$, such that

$d_{p}\Big{(}FS(N)_{*},FS(N_{r})^{\frac{1}{r}}\Big{)}\leq\epsilon/2.$ (3.14)

From Theorem 2.11, Proposition 2.15 and (3.14), we conclude that

$d_{p}\Big{(}{\rm{Ban}}^{\infty}(FS(N)_{*}),{\rm{Ban}}^{\infty}(FS(N_{r})^{\frac{1}{r}})\Big{)}\leq\epsilon/2.$
(3.15)

From the proof of Theorem 1.1, there is $k_{0}\in\mathbb{N}$, such that for
any $k\geq k_{0}$, we have

$N_{k}\leq\exp(\epsilon
k/2)\cdot{\rm{Ban}}^{\infty}_{k}(FS(N_{r})^{\frac{1}{r}}).$ (3.16)

From (2.11), (3.8), (3.15) and (3.16), we conclude that
$d_{p}({\rm{Ban}}^{\infty}(FS(N)_{*}),N)\leq\epsilon$. This finishes the proof
by Corollary 2.18, as $\epsilon>0$ was chosen arbitrary. ∎

We define for two bounded metrics $h^{L}_{0}$, $h^{L}_{1}$ on $L$, the
distance $d_{+\infty}(h^{L}_{0},h^{L}_{1})$ as the minimal constant $C\geq 0$
verifying $h^{L}_{0}\leq e^{C}\cdot h^{L}_{1}$ and $h^{L}_{1}\leq e^{C}\cdot
h^{L}_{0}$. The following corollary establishes the relation between distances
of submultiplicative graded norms and their Fubini-Study potentials.

######  Corollary 3.6.

For any bounded submultiplicative graded norms $N,N^{\prime}$ on a section
ring $R(X,L)$, and any $p\in[1,+\infty[$, we have

$d_{p}(N,N^{\prime})=d_{p}\big{(}FS(N)_{*},FS(N^{\prime})_{*}\big{)}.$ (3.17)

If, moreover, $FS(N)$, $FS(N^{\prime})$ are continuous, then one can take
$p=+\infty$ above.

###### Proof.

For $p=[1,+\infty[$, the statement follows directly from Theorems 1.3, 2.11,
Proposition 2.15 and Corollary 2.18. For $p=+\infty$, the statement is a
consequence of Theorem 1.1, Proposition 2.15 and [35, Theorem 1.7]. ∎

One can verify, cf. [35, Lemma 4.12], that for any graded norms
$N,N^{\prime}$, for which the sequences $FS(N_{k})^{\frac{1}{k}}$,
$FS(N^{\prime}_{k})^{\frac{1}{k}}$, $k\in\mathbb{N}$, converge uniformly to
some metrics $FS(N)$, $FS(N^{\prime})$ on $L$, we have
$d_{+\infty}(FS(N),FS(N^{\prime}))\leq d_{+\infty}(N,N^{\prime})$. It is
tempting to think that a similar conclusion holds for $d_{p}$-distances,
$p\in[1,+\infty[$, or even the submultiplicativity assumption in Corollary 3.6
is superfluous. It is not the case. The following example shows that the
$d_{p}$-distances between norms and their Fubini-Study metrics are essentially
unrelated.

######  Proposition 3.7.

There is a bounded graded Hermitian norm $H=\sum H_{k}$ on
$R(\mathbb{P}^{1},\mathscr{O}(1))$, such that $FS(H_{k})^{\frac{1}{k}}$
converge uniformly, as $k\to\infty$, to a (continuous psh) metric $FS(H)$ on
$L$, and there is a continuous psh metric $h^{L}\neq FS(H)$, for which
$H\sim_{p}{\rm{Ban}}^{\infty}(h^{L})$ for any $p\in[1,+\infty[$.

######  Remark 3.8.

In particular, by Corollary 2.18, for any $p\in[1,+\infty[$, we have
$d_{p}(H,{\rm{Ban}}^{\infty}(FS(H)))\neq 0$ and
$d_{p}(H,{\rm{Ban}}^{\infty}(h^{L}))=0$, while $d_{p}(FS(H),h^{L})\neq 0$.

###### Proof.

Our proof is a slight modification of [35, Proposition 4.16]. Let us identify
$\mathbb{P}^{1}$ to $\mathbb{P}(V^{*})$, where $V$ is a vector space generated
by two elements: $x$ and $y$. Let us consider a metric $H$ on $V$, which makes
$x$ and $y$ an orthonormal basis, and denote by $h^{FS}$ the induced Fubini-
Study metric on $\mathscr{O}(1)$. For any $k\in\mathbb{N}^{*}$,
$a,b\in\mathbb{N}$, $a+b=k$, under the isomorphism ${\rm{Sym}}^{k}(V)\to
H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k))$, an easy calculation shows that we
have

$\big{\|}x^{a}\cdot
y^{b}\big{\|}_{{\rm{Hilb}}_{k}(h^{FS})}^{2}=\frac{a!b!}{(k+1)!}.$ (3.18)

Let us consider a Hermitian norm $H_{k}$ on
$H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k))$, for which the basis $x^{a}\cdot
y^{b}$ is orthogonal and in the above notations, we have

$\big{\|}x^{a}\cdot
y^{b}\big{\|}_{H_{k}}^{2}=\begin{cases}\frac{1}{2^{k}+k+1},&\text{if }b=0\\\
\big{\|}x^{a}\cdot
y^{b}\big{\|}_{{\rm{Hilb}}_{k}(h^{FS})}^{2},&\text{otherwise}.\end{cases}$
(3.19)

We will now verify that $H_{k}$ satisfies the assumptions of the proposition.

First of all, from Proposition 2.15, it is trivial to verify that for any
$p\in[1,+\infty[$, we have $H\sim_{p}{\rm{Ban}}^{\infty}(h^{L})$. Remark also
that $H$ is bounded by Proposition 2.15, since we have
$H\geq{\rm{Hilb}}(h^{FS}/3)$. From Lemma 2.1, for any $a,b\in\mathbb{C}$, not
simultaneously equal to zero, we have

$\frac{FS({\rm{Hilb}}_{k}(h^{FS}))^{2}}{FS(H_{k})^{2}}\Big{(}[ax^{*}+by^{*}]\Big{)}=\frac{2^{k}|a|^{k}+(k+1)(|a|+|b|)^{k}}{(k+1)(|a|+|b|)^{k}}.$
(3.20)

In particular, we conclude that

$\lim_{k\to\infty}\bigg{(}\frac{FS(H_{k})}{FS({\rm{Hilb}}_{k}(h^{FS}))}\bigg{)}^{\frac{2}{k}}\Big{(}[ax^{*}+by^{*}]\Big{)}=\frac{\max\\{2|a|,|a|+|b|\\}}{|a|+|b|},$
(3.21)

and the convergence is uniform. This finishes the proof by Tian’s theorem, cf.
Theorem 2.13. ∎

We will now show that one cannot take $p=+\infty$ in Theorem 1.3. Recall that
to any complex normed commutative ring $(A,\|\cdot\|_{A})$, one can associate
the seminorm $\|\cdot\|_{A}^{{\rm{hom}}}$, sometimes called the homogenization
or spectral radius seminorm, defined as follows

$\|f\|_{A}^{{\rm{hom}}}\mathrel{\mathop{\ordinarycolon}}=\lim_{k\to\infty}\|f^{k}\|_{A}^{\frac{1}{k}}.$
(3.22)

The existence of the limit above is assured by Fekete’s lemma.

######  Proposition 3.9.

There is a bounded submultiplicative graded norm $N=\sum N_{k}$ on a section
ring $R(X,L)$, such that $N\not\sim{\rm{Ban}}^{\infty}(h^{L})$ for any bounded
metric $h^{L}$ on $L$.

###### Proof.

We fix a bounded metric $h^{L}$ on $L$, an effective divisor $D\subset X$ and
consider the ray of norms $N_{k}^{t}=\|\cdot\|_{k}^{t}$, $t\in[0,+\infty[$,
constructed by the procedure (2.20) from the norm ${\rm{Ban}}^{\infty}(h^{L})$
and the filtration, for which jumping numbers are given by $ik$, $i=0,1,2$,
and such that
$\mathcal{F}^{ik}H^{0}(X,L^{k})=H^{0}(X,L^{k}\otimes\mathcal{J}_{D}^{i})$,
where $\mathcal{J}_{D}$ is a sheaf of holomorphic germs vanishing along $D$.

Let us verify that the graded norm $N\mathrel{\mathop{\ordinarycolon}}=\sum
N_{k}^{1}$ provides an example for Proposition 3.9. An easy verification shows
that it is submultiplicative. It is also trivially bounded from below by
${\rm{Ban}}^{\infty}(e^{-2}h^{L})$. Let us show that
$N\not\sim{\rm{Ban}}^{\infty}(h^{L})$ for any metric $h^{L}$ on $L$.

Indeed, let us consider a sequence of elements $f_{k}\in
H^{0}(X,L^{k}\otimes\mathcal{J}_{D})\setminus
H^{0}(X,L^{k}\otimes\mathcal{J}_{D}^{2})$. The existence of such $f_{k}$ for
$k$ large enough is assured by ampleness of $L$ and effectivity of $D$. We
denote $g_{k}\mathrel{\mathop{\ordinarycolon}}=f_{k}-P_{k}(f_{k})$, where
$P_{k}(f_{k})$ is a projection (with respect to the norm
${\rm{Ban}}^{\infty}_{k}(h^{L})$) of $f_{k}$ to
$H^{0}(X,L^{k}\otimes\mathcal{J}_{D}^{2})$. We then see that
$\|g_{k}\|_{k}^{1}=e^{-k}\cdot\|g_{k}\|_{L^{\infty}_{k}(X,h^{L})}$. However,
in the notations of Section 3.4, we obviously have
$\|g_{k}\|_{k}^{1,\rm{hom}}\leq\sqrt{\|g_{k}^{2}\|^{1}_{2k}}$. Since
$g_{k}^{2}\in H^{0}(X,L^{2k}\otimes\mathcal{J}_{D}^{2})$, we deduce that
$\|g_{k}\|_{k}^{1,\rm{hom}}\leq
e^{-2k}\cdot\|g_{k}\|_{L^{\infty}_{k}(X,h^{L})}$. Hence, $N\not\sim
N^{{\rm{hom}}}$, which implies that $N\not\sim{\rm{Ban}}^{\infty}(h^{L})$ for
any bounded metric $h^{L}$ on $L$. ∎

######  Remark 3.10.

Our example is given by a ray of submultiplicative norms constructed using a
non-Archimedean submultiplicative norm from Boucksom-Jonsson [18, Example
2.25] through a general procedure outlined in (2.18). As it follows from
Theorems 1.1 and 5.1, the reason why we have
$N\not\sim{\rm{Ban}}^{\infty}(FS(N))$ is that the Fubini-Study geodesic ray
associated with our filtration turns out to be non-continuous, see Section 5.1
for details.

###  3.2 Semiclassical holomorphic extension theorem and pluripolar sets

The main goal of this section is to deduce from Theorem 1.3 a characterization
of submanifolds for which a weak version of semiclassical Ohsawa-Takegoshi
extension theorem holds and to deduce Theorem 1.13 from this characterization.
We also recall the semiclassical holomorphic extension theorem for continuous
metrics – a central statement for many proofs in this article.

We follow below the notations set up after (1.19). A semiclassical version of
holomorphic extension theorem for continuous metrics goes as follows.

######  Theorem 3.11.

For any continuous psh metric $h^{L}$ on an ample line bundle $L$ over a
compact complex manifold $X$, under surjection (1.19), the following
equivalence of norms on $R(Y,L)$ holds
$[{\rm{Ban}}^{\infty}_{X}(h^{L})]\sim{\rm{Ban}}^{\infty}_{Y}(h^{L})$.

###### Proof.

This result was proved by Bost [15, Theorem A.1] with stronger assumption of
strict positivity on the curvature of $(L,h^{L})$, refining previous result of
Zhang [69]. See also Randriambololona [56] for a statement which requires
laxer assumptions on the manifolds $X$ and $Y$. The proof of exactly this
version of the theorem can be found in [35, Corollary 2.12]. ∎

We will now show that for non-continuous metrics $h^{L}$, things become far
more complicated. Recall that a subset $E\subset X$ is called pluripolar if it
is a subset of a complete pluripolar set, where the latter is defined as
$\\{x\in X\mathrel{\mathop{\ordinarycolon}}u(x)=-\infty\\}$ for a certain
$u\in{\rm{PSH}}(X,\omega)$ and a Kähler form $\omega$. These definitions do
not depend on the choice of the form $\omega$, cf. [38, Proposition 2.3].

Recall that Bedford-Taylor in [7, Theorem 7.1] proved in local setting that
for a uniformly bounded sequence $\phi_{i}$, $i\in\mathbb{N}$ of psh
functions, we have $\sup\phi_{i}=(\sup\phi_{i})^{*}$ away from a pluripolar
subset. Since on a Kähler manifold $X$, any locally pluripolar subset is
pluripolar, by a theorem of Josefson, cf. [38, Theorem 7.2], the same
conclusion holds for functions from ${\rm{PSH}}(X,\omega)$.

Now, from Theorem 2.13, we see that to study sup-norms, it is enough to
consider regularizable from above psh metrics $h^{L}$. For such $h^{L}$, we
define the contact subset $E(h^{L})\subset X$ as follows

$E(h^{L})\mathrel{\mathop{\ordinarycolon}}=\big{\\{}x\in
X\mathrel{\mathop{\ordinarycolon}}h^{L}_{x}\neq Q(h^{L})_{x}\big{\\}},$ (3.23)

where $Q(h^{L})$ is the Fubini-Study envelope of $h^{L}$, defined in (2.51).
From the above results of Bedford-Taylor and Josefson, the set $E(h^{L})$ is
pluripolar. The following result gives a criteria for a weaker version of
semiclassical Ohsawa-Takegoshi extension theorem. It classifies submanifolds
for which the asymptotic contribution of holomorphic sections over the
submanifold, which cannot be effectively extended to the ambient manifold, is
“negligible”.

######  Theorem 3.12.

For any regularizable from above psh metrics $h^{L}$ on an ample line bundle
over a compact complex manifold $X$, the following conditions are equivalent.

1) A submanifold $Y\subset X$ intersects $E(h^{L})$ over a pluripolar subset
(of $Y$).

2) For any $p\in[1,+\infty[$, under (1.19), the following equivalence of norms
on $R(Y,L)$ holds

$[{\rm{Ban}}^{\infty}_{X}(h^{L})]\sim_{p}{\rm{Ban}}^{\infty}_{Y}(h^{L}).$
(3.24)

Moreover, (3.24) holds for $p=+\infty$ if $Y\cap E(h^{L})=\emptyset$.

###### Proof.

The norm $N\mathrel{\mathop{\ordinarycolon}}=[{\rm{Ban}}^{\infty}_{X}(h^{L})]$
on $R(Y,L)$ is submultiplicative as a quotient of a submultiplicative norm.
From Lemma 3.3, we have $FS(N)=FS({\rm{Ban}}^{\infty}_{X}(h^{L}))|_{Y}$. From
Theorem 2.13, this yields $FS(N)=Q(h^{L})|_{Y}$. Hence, by Theorem 1.3 and
Corollaries 2.17, 2.18, we conclude that (3.24) holds if and only if
$(Q(h^{L})|_{Y})_{*}=h^{L}|_{Y}$. Remark, however, that by the already
mentioned result [7, Theorem 7.1] of Bedford-Taylor, this happens if and only
if $Q(h^{L})|_{Y}=h^{L}|_{Y}$ away from a pluripolar subset (of $Y$). This
concludes the proof of the first part of Theorem 3.12 by the definition of the
subset $E(h^{L})$.

Now, points of discontinuity of $h^{L}$ are contained in $E(h^{L})$. This is
due to the fact that $h^{L}$ is lower semi-continuous, $Q(h^{L})$ is upper
semi-continuous and $Q(h^{L})\geq h^{L}$. In particular, $h^{L}|_{Y}$ is
continuous if $Y\cap E(h^{L})=\emptyset$. By the above, we also have
$FS(N)=h^{L}|_{Y}$ under the assumption $Y\cap E(h^{L})=\emptyset$. The second
part of Theorem 3.12 now follows from Theorem 1.1. ∎

###### Proof of Theorem 1.13.

Let $E^{*}(h^{L})$ be the pluripolar hull of $E(h^{L})$, i.e. the intersection
of all complete pluripolar subsets in $X$ containing $E(h^{L})$. The subset
$E^{*}(h^{L})$ is clearly pluripolar, and when $Y$ is not contained in
$E^{*}(h^{L})$, it intersects $E^{*}(h^{L})$ over a pluripolar subset (of
$Y$). Hence, Theorem 1.13 follows from Theorem 3.12. ∎

###  3.3 Projective geometry and norms on symmetric algebras

In this section, we reduce the proof of Theorem 3.1 to a functional-analytic
statement about the norms on the symmetric algebra of complex vector spaces.
We also show that the aforementioned statement can be seen as a special case
of Theorem 3.1, applied for the projective space.

We fix a finitely dimensional complex vector space $V$ with a norm
$N_{V}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V}$. Recall that for any
$k\in\mathbb{N}^{*}$, we have the polarisation map
${\rm{Pol}}\mathrel{\mathop{\ordinarycolon}}{\rm{Sym}}^{k}(V)\to V^{\otimes
k}$ and the symmetrization map
${\rm{Sym}}\mathrel{\mathop{\ordinarycolon}}V^{\otimes
k}\to{\rm{Sym}}^{k}(V)$. Consider two norms
${\rm{Sym}}^{k}_{\epsilon}(N_{V})\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{{\rm{Sym}},\epsilon}_{N_{V},k}$
and
${\rm{Sym}}^{k}_{\pi}(N_{V})\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{{\rm{Sym}},\pi}_{N_{V},k}$
on symmetric tensors ${\rm{Sym}}^{k}(V)$, induced by the polarisation map, and
the norms $N_{V}\otimes_{\epsilon}\cdots\otimes_{\epsilon}N_{V}$,
$N_{V}\otimes_{\pi}\cdots\otimes_{\pi}N_{V}$ on $V^{\otimes k}$. Define the
norm
${\rm{Sym}}^{k}_{{\rm{ev}}}(N_{V})\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{{\rm{ev}}}_{N_{V},k}$
on ${\rm{Sym}}^{k}(V)$ as

$\|P\|^{{\rm{ev}}}_{N_{V},k}\mathrel{\mathop{\ordinarycolon}}=\sup_{\begin{subarray}{c}v\in
V^{*}\\\ \|v\|_{V}^{*}\leq 1\end{subarray}}|P(v)|,\qquad
P\in{\rm{Sym}}^{k}(V).$ (3.25)

We construct from these norms the graded norms
${\rm{Sym}}_{{\rm{ev}}}(N_{V})$, ${\rm{Sym}}_{\epsilon}(N_{V})$ and
${\rm{Sym}}_{\pi}(N_{V})$ on the symmetric algebra ${\rm{Sym}}(V)$. Similarly
to (1.4), we define the equivalence relation on the set of graded norms over
${\rm{Sym}}(V)$. The following result will be established in Sections 4.1 and
4.2.

######  Theorem 3.13.

The norms ${\rm{Sym}}_{\pi}(N_{V})$, ${\rm{Sym}}_{\epsilon}(N_{V})$ and
${\rm{Sym}}_{{\rm{ev}}}(N_{V})$ are equivalent.

######  Remark 3.14.

a) Restriction to symmetric tensors is absolutely necessary for this
statement. In fact, as it follows from the work of Pisier [54, Théorème 3.1],
see also more recent result of Aubrun-Müller-Hermes [4, Theorem 1.1], in the
full tensor algebra
$T(V)\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}V^{\otimes i}$, the gap
between injective and projective tensor norms on the graded pieces is
exponential for any normed vector space $(V,N_{V})$ of dimension bigger than
$1$.

b) Surprisingly, the corresponding statement for real vector spaces is false.
In fact, if we consider a polynomial $P(x,y)=xy(x^{2}-y^{2})$ and view it as a
polynomial on $(\mathbb{R}^{2},\textit{l}_{1})$, then an easy calculation
shows that for any $k\in\mathbb{N}^{*}$, we have
$\|P^{k}\|^{{\rm{ev}}}_{\textit{l}_{1},4k}=\sup_{-1\leq x,y\leq
1}|P^{k}(x,y)|=\big{(}\frac{2\sqrt{3}}{9}\big{)}^{k}$, cf. [19, proof of
Theorem 4.2]. But from the proof of Theorem 3.13, we know that
${\rm{Sym}}_{\pi}(\textit{l}_{1})$ corresponds to the sum of the absolute
values of the coefficients. Hence, we have
$\|P^{k}\|^{{\rm{Sym}},\pi}_{\textit{l}_{1},4k}=2^{k}$.

We now explain that Theorem 3.13 is in fact a special case of Theorem 3.1. For
this, we give geometric interpretations for some of the above norms. First of
all, directly from (2.32), we have

$\|P\|^{{\rm{Sym}},\epsilon}_{k}\mathrel{\mathop{\ordinarycolon}}=\sup_{\begin{subarray}{c}v_{1},\cdots,v_{k}\in
V^{*}\\\ \|v_{i}\|_{V}^{*}\leq
1\end{subarray}}\Big{|}{\rm{Pol}}(P)\big{(}v_{1},\cdots,v_{k}\big{)}\Big{|},\qquad
P\in{\rm{Sym}}^{k}(V).$ (3.26)

Hence, from Lemma 2.9, (3.25) and (3.26), the following chain of inequalities
holds

${\rm{Sym}}_{{\rm{ev}}}(N_{V})\leq{\rm{Sym}}_{\epsilon}(N_{V})\leq{\rm{Sym}}_{\pi}(N_{V}).$
(3.27)

In particular, we see that for the proof of Theorem 3.13, it is enough to
establish the equivalence of the norms ${\rm{Sym}}_{{\rm{ev}}}(N_{V})$ and
${\rm{Sym}}_{\pi}(N_{V})$.

######  Remark 3.15.

By (3.26), the equivalence of ${\rm{Sym}}_{{\rm{ev}}}(N_{V})$ and
${\rm{Sym}}_{\epsilon}(N_{V})$ from Theorem 3.13 means exactly that the
polarisation constant, cf. [31, (4)] for a definition, for finitely
dimensional complex normed vector spaces is equal to $1$. This fact was
recently established by Dimant-Galicer-Rodríguez [31, Theorem 1.1] through
different methods.

Let us now give an interpretation of the norm ${\rm{Sym}}_{{\rm{ev}}}(N_{V})$
through projective spaces. We view the symmetric algebra ${\rm{Sym}}(V)$ as
the section ring $R(\mathbb{P}(V^{*}),\mathscr{O}(1))$ through the
identification

${\rm{Sym}}^{k}(V)=H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k)).$ (3.28)

Under this isomorphisms, we have the following identification of norms

${\rm{Sym}}_{{\rm{ev}}}(N_{V})={\rm{Ban}}^{\infty}_{\mathbb{P}(V^{*})}(FS(N_{V})).$
(3.29)

We now consider the norms ${\rm{Sym}}^{k}_{\epsilon,0}(N_{V})$ and
${\rm{Sym}}^{k}_{\pi,0}(N_{V})$ on ${\rm{Sym}}^{k}(V)$, given by the quotients
of $N_{V}\otimes_{\epsilon}\cdots\otimes_{\epsilon}N_{V}$,
$N_{V}\otimes_{\pi}\cdots\otimes_{\pi}N_{V}$ on $V^{\otimes k}$ through the
symmetrization map, ${\rm{Sym}}$.

######  Lemma 3.16.

The norms ${\rm{Sym}}^{k}_{\epsilon,0}(N_{V})$ (resp.
${\rm{Sym}}^{k}_{\pi,0}(N_{V})$) and ${\rm{Sym}}^{k}_{\epsilon}(N_{V})$ (resp.
${\rm{Sym}}^{k}_{\pi}(N_{V})$) over ${\rm{Sym}}^{k}(V)$ coincide.

###### Proof.

It follows directly from the fact that permutations of coordinates are
isometries for both norms
$N_{V}\otimes_{\epsilon}\cdots\otimes_{\epsilon}N_{V}$,
$N_{V}\otimes_{\pi}\cdots\otimes_{\pi}N_{V}$. ∎

Remark that symmetrization and multiplication maps (3.1) can be put under the
isomorphisms (3.28) into the following commutative diagram

$\begin{CD}H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(1))^{\otimes
k}@>{{\rm{Mult}}_{1,\cdots,1}}>{}>H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k))\\\
\Big{\|}{}\Big{\|}{}\\\ V^{\otimes
k}@>{{\rm{Sym}}}>{}>{\rm{Sym}}^{k}(V).\end{CD}$ (3.30)

Lemma 3.16, (3.29) and (3.30) imply that Theorem 3.13 is a specialisation of
Theorem 3.1 to $X=\mathbb{P}(V^{*})$, $L=\mathscr{O}(1)$ and
$N_{1}\mathrel{\mathop{\ordinarycolon}}=N_{V}$. Remark, however, that our
proof proceeds in another direction: we first establish Theorem 3.13 and then
prove Theorem 3.1.

###### Proof of Theorem 3.1 assuming Theorem 3.13..

Let us first prove the following inequalities

${\rm{Ban}}^{\infty}(FS(N_{1}))\leq N^{\epsilon}\leq N^{\pi}.$ (3.31)

The first inequality is a direct consequence of (3.7) and [35, Lemma 4.3]. The
second inequality follows directly from Lemma 2.9.

From (3.31), it is enough to establish that the norm $N^{\pi}$ can be bounded
from above by ${\rm{Ban}}^{\infty}(FS(N_{1}))$, considered up to a
subexponential factor. The proof of this result is essentially a word-to-word
repetition of the proof of the second part of [35, Theorem 4.18]. We only need
to replace the use of the first part of the proof of [35, Theorem 4.18] by
Theorem 3.13. For the convenience of the reader, we reproduce the argument
below.

Let us consider the Kodaira embedding ${\rm{Kod}}_{1}$ from (2.1). We denote
by
${\rm{Res}}_{{\rm{Kod}}}\mathrel{\mathop{\ordinarycolon}}R(\mathbb{P}(H^{0}(X,L)^{*}),\mathscr{O}(1))\to
R(X,L)$ the associated restriction operator, and by
${\rm{Res}}_{{\rm{Kod}},k}$, $k\in\mathbb{N}^{*}$, the restriction operators
on the associated graded pieces. The multiplication operator
${\rm{Mult}}_{1,\cdots,1}$ from (3.1) factorizes under the identification
(3.28) through symmetrization and restriction as

${H^{0}(X,L)^{\otimes
k}}$${{\rm{Sym}}^{k}(H^{0}(X,L))}$${H^{0}(\mathbb{P}(H^{0}(X,L)^{*}),\mathscr{O}(k))}$${H^{0}(X,L^{k}).}$$\scriptstyle{{\rm{Mult}}_{1,\cdots,1}}$$\scriptstyle{{\rm{Sym}}}$$\scriptstyle{{\rm{Res}}_{{\rm{Kod}},k}}$
(3.32)

Now, from (3.32), it is sufficient to show that by a subsequent quotient of
the projective tensor norm induced by $N_{1}$ through the symmetrization map
${\rm{Sym}}$ and the map ${\rm{Res}}_{{\rm{Kod}}}$, we get the norm
${\rm{Ban}}^{\infty}_{X}(FS(N_{1}))$ on $H^{0}(X,L^{k})$. From Theorem 3.13,
this quotient norm on ${\rm{Sym}}(H^{0}(X,L))$ is equivalent to
${\rm{Sym}}_{{\rm{ev}}}(N_{V})$, which by (3.29) coincides with
${\rm{Ban}}^{\infty}_{\mathbb{P}(H^{0}(X,L)^{*})}(FS(N_{1}))$ under the
identification (3.28). But by Theorem 3.11, the quotient of the norm
${\rm{Ban}}^{\infty}_{\mathbb{P}(H^{0}(X,L)^{*})}(FS(N_{1}))$ under the map
${\rm{Res}}_{{\rm{Kod}}}$ is equivalent to
${\rm{Ban}}^{\infty}_{X}(FS(N_{1}))$. This finishes the proof. ∎

###  3.4 Homogenization of submultiplicative norms on section rings

The main goal of this section is to give an explicit formula for the spectral
radius seminorm, defined in (3.22), associated with a submultiplicative graded
norm on a section ring.

Let us first recall why the spectral radius seminorm is a seminorm. We assume
that $A$ has a unit (otherwise one can formally add it). Gelfand’s spectral
radius formula says, cf. [37, Theorems I.4.4, I.4.6], that
$\|\cdot\|_{A}^{{\rm{hom}}}$ can be alternatively described as follows

$\|f\|_{A}^{{\rm{hom}}}=\max\big{\\{}|\lambda|\mathrel{\mathop{\ordinarycolon}}\lambda\in{\rm{spec}}(f)\big{\\}},$
(3.33)

where ${\rm{spec}}(f)$ is the spectrum of $f$, given by $\lambda\in\mathbb{C}$
such that $\lambda-f$ is not invertible in the completion of
$(A,\|\cdot\|_{A})$. The formula (3.33) implies that
$\|\cdot\|_{A}^{{\rm{hom}}}$ is a seminorm.

######  Theorem 3.17.

Assume that a graded norm $N=\sum N_{k}$ over the section ring $R(X,L)$ of an
ample line bundle $L$ is submultiplicative. Then
$N^{{\rm{hom}}}={\rm{Ban}}^{\infty}(FS(N))$.

######  Remark 3.18.

In non-Archimedean geometry, an analogue of Theorem 3.17 was established by
Boucksom-Jonsson [18, Theorem 2.16].

###### Proof.

From (3.3), (3.7) and (3.16), we conclude that for any $l\in\mathbb{N}^{*}$,
we have

${\rm{Ban}}^{\infty}(FS(N))\leq
N^{{\rm{hom}}}\leq{\rm{Ban}}^{\infty}(FS(N_{l})^{\frac{1}{l}})$ (3.34)

It is, hence, enough to establish that
${\rm{Ban}}^{\infty}(FS(N))=\inf_{l\in\mathbb{N}}{\rm{Ban}}^{\infty}(FS(N_{l})^{\frac{1}{l}})$.

Clearly, by considering a subsequence $l=2^{k}$, $k\in\mathbb{N}$, from (3.3),
it is enough to prove that for a decreasing sequence of upper semi-continuous
functions $\phi_{i}$ on a compact manifold $X$, the following identity holds
$\lim_{i\to\infty}\sup_{x\in X}\phi_{i}(x)=\sup_{x\in
X}\lim_{i\to\infty}\phi_{i}(x)$. It is trivial that
$\lim_{i\to\infty}\sup_{x\in X}\phi_{i}(x)\geq\sup_{x\in
X}\lim_{i\to\infty}\phi_{i}(x)$. To argue in another direction, we mimic the
proof of Dini’s theorem. Let us denote $M=\sup_{x\in
X}\lim_{i\to\infty}\phi_{i}(x)$. Then for any $\epsilon>0$, the sets
$U_{i}=\\{x\in X\mathrel{\mathop{\ordinarycolon}}\phi_{i}(x)<M+\epsilon\\}$
provide a cover of $X$. By our assumption on upper semi-continuity of
$\phi_{i}$, this cover is open. By compactness of $X$, there is a finite
subcover. But since $\phi_{i}$ decrease, the sets $U_{i}$ are nested, and
hence there is $i_{0}\in\mathbb{N}$, such that $U_{i_{0}}=X$. Hence,
$\sup_{x\in X}\phi_{i_{0}}(x)<M+\epsilon$. Since $\epsilon>0$ was chosen
arbitrary and $\phi_{i}$ decrease, we deduce the inverse direction
$\lim_{i\to\infty}\sup_{x\in X}\phi_{i}(x)\leq M$, which finishes the proof. ∎

######  Remark 3.19.

We learned from Sébastien Boucksom that one can alternatively prove Theorem
3.17 relying only on the Gelfand’s spectral radius formula and the
interpretation of the spectrum of a section ring as an affine cone, very much
in spirit of Fang [33, Proposition 3.15].

In particular, directly from Theorems 1.3 and 3.17, we deduce the following
result.

######  Corollary 3.20.

For any bounded submultiplicative graded norm $N$ over the section ring
$R(X,L)$ of an ample line bundle $L$ and any $p\in[1,+\infty[$, we have
$N\sim_{p}N^{{\rm{hom}}}$.

##  4 Norms on spaces of polynomials

The main goal of this self-contained section is to establish Theorem 3.13. In
Section 4.1, we establish Theorem 3.13 in the special case $V=\mathbb{C}^{r}$,
$r\in\mathbb{N}^{*}$, and
$N_{V}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V}\mathrel{\mathop{\ordinarycolon}}=\textit{l}_{1}$,
and in Section 4.2, we prove Theorem 3.13 in its full generality by relying on
some tools from complex geometry.

###  4.1 Bohnenblust-Hille inequality as ratio of injective and projective
norms

The main goal of this section is to establish Theorem 3.13 in the special case
$V=\mathbb{C}^{r}$, $r\in\mathbb{N}^{*}$, and
$N_{V}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V}\mathrel{\mathop{\ordinarycolon}}=\textit{l}_{1}$.
To establish this, we rely on a recent result about the optimal estimate in
Bohnenblust-Hille inequality, which we now recall. Consider a vector space
$V_{r,k}$ of homogeneous complex polynomials of degree $k$ in $r$ variables.
We represent an element $P\in V_{r,k}$ as

$P(x_{1},\cdots,x_{r})=\sum_{|\alpha|=k}a_{\alpha}x^{\alpha}.$ (4.1)

Since $\dim V_{r,k}=\binom{r+k}{r}<+\infty$, any two norms on $V_{r,k}$ are
equivalent. In particular, for any $\beta\geq 1$, there is a constant
$B_{r,k}^{\beta}>0$, such that for any $P\in V_{r,k}$ as in (4.1), we have

$\Big{(}\sum_{|\alpha|=k}|a_{\alpha}|^{\beta}\Big{)}^{\frac{1}{\beta}}\leq
B_{r,k}^{\beta}\cdot\|P\|,$ (4.2)

where the sup-norm $\|P\|$ is defined as follows

$\|P\|\mathrel{\mathop{\ordinarycolon}}=\sup_{\begin{subarray}{c}x_{i}\in\mathbb{C}\\\
|x_{i}|\leq 1\end{subarray}}\big{|}P(x_{1},\cdots,x_{r})\big{|}.$ (4.3)

We assume that the constants $B_{r,k}^{\beta}$ for $r,k\in\mathbb{N}^{*}$,
$\beta\geq 1$, are the minimal constants verifying the inequality (4.2). The
main result of this section goes as follows.

######  Proposition 4.1.

For any fixed $r\in\mathbb{N}^{*}$, the sequence $B_{r,k}^{1}$,
$k\in\mathbb{N}$, grows subexponentially in $k$.

Recall that Bohnenblust–Hille in [14] showed that for
$\beta\mathrel{\mathop{\ordinarycolon}}=\frac{2k}{k+1}$, the constant

$B_{k}\mathrel{\mathop{\ordinarycolon}}=\sup_{r\in\mathbb{N}}B_{r,k}^{\beta}$
(4.4)

is finite. In other words, for this choice of $\beta$, the bound like (4.2)
can be made uniform in the number of variables. We need the following recent
result about the asymptotics of $B_{k}$.

######  Theorem 4.2 (Bayart-Pellegrino-Seoane-Sepúlveda [5, Corollary 5.3] ).

The constants $B_{k}$ grow subexponentially in $k$.

###### Proof of Proposition 4.1..

By the generalized mean inequality and (4.2), we have

$\sum_{|\alpha|=k}|a_{\alpha}|\leq
B_{k}\cdot\binom{r+k}{r}^{1-\frac{k+1}{2k}}\cdot\|P\|,$ (4.5)

in the notations (4.1). In particular, since the binomial coefficients
$\binom{r+k}{r}$ are polynomials in $k$ for fixed $r$ (and, hence,
subexponential in $k$), we deduce Proposition 4.1 from Theorem 4.2. ∎

###### Proof of Theorem 3.13 in the special case when $V=\mathbb{C}^{r}$ and
$N_{V}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{V}\mathrel{\mathop{\ordinarycolon}}=\textit{l}_{1}$..

From (3.27), it is sufficient to show that ${\rm{Sym}}_{\pi}(N_{V})$,
considered up to a subexponential constant, is bounded from above by
${\rm{Sym}}_{{\rm{ev}}}(N_{V})$.

Let us denote by $x_{1},\ldots,x_{r}$ the coordinate vectors in
$\mathbb{C}^{r}$. We use the notation (4.1) for $P\in{\rm{Sym}}^{k}(V)$,
$k\in\mathbb{N}^{*}$. Since the dual of the $\textit{l}_{1}$-norm is given by
the $\textit{l}_{\infty}$-norm on $\mathbb{C}^{r}$, (3.25) gives us

$\|P\|^{{\rm{ev}}}_{\textit{l}_{1},k}=\|P\|.$ (4.6)

On another hand, since projective tensor norms behave multiplicatively on
$\textit{l}_{1}$-spaces, i.e.
$(\mathbb{C}^{n},\textit{l}_{1})\otimes_{\pi}(\mathbb{C}^{n},\textit{l}_{1})=(\mathbb{C}^{nm},\textit{l}_{1})$,
cf. [60, Exercise 2.8], the norm ${\rm{Sym}}_{\pi}(\textit{l}_{1})$
corresponds to the sum of absolute values of the coefficients occurring in the
representation (4.1), i.e. we have

$\|P\|^{{\rm{Sym}},\pi}_{\textit{l}_{1},k}=\sum_{|\alpha|=k}|a_{\alpha}|.$
(4.7)

We conclude by Proposition 4.1 and (4.6), (4.7) that
${\rm{Sym}}_{\pi}(N_{V})$, considered up to a subexponential constant, can be
bounded from above by ${\rm{Ban}}^{\infty}(FS(N_{V}))$. ∎

###  4.2 Projective tensor norms and holomorphic extension theorem

The main goal of this section is to prove Theorem 3.13 in its full generality.
Surprisingly, our main technical tool in the proof of this purely functional-
analytic statement comes from complex geometry. We also use the following
classical result.

######  Lemma 4.3 (cf. [31, Lemma 2.2]).

For any finitely dimensional complex normed vector space $(V,\|\cdot\|_{V})$,
and any $\epsilon>0$, there is $l\in\mathbb{N}^{*}$ and a surjective map
$\pi\mathrel{\mathop{\ordinarycolon}}\mathbb{C}^{l}\to V$, such that
$\|\cdot\|_{V}$ is related to the quotient norm associated with the
$\textit{l}_{1}$-norm on $\mathbb{C}^{l}$ as follows

$\exp(-\epsilon)\cdot[\textit{l}_{1}]\leq\|\cdot\|_{V}\leq[\textit{l}_{1}].$
(4.8)

###### Proof of Theorem 3.13..

Since Theorem 3.13 holds for $\textit{l}_{1}$-norms by the result from Section
4.1, we deduce by Lemma 4.3 that it is enough to show that the validity of
Theorem 3.13 is stable under taking quotients, i.e. if Theorem 3.13 holds for
a normed vector space $(U,N_{U})$, then it holds for any normed quotient
$(V,N_{V})$, $\pi\mathrel{\mathop{\ordinarycolon}}U\to V$. As we shall see
below, this is a consequence of the semiclassical version of Ohsawa-Takegoshi
extension theorem. We consider the embedding

${\rm{Im}}_{\pi}\mathrel{\mathop{\ordinarycolon}}\mathbb{P}(V^{*})\to\mathbb{P}(U^{*}).$
(4.9)

Clearly, under this embedding, the associated restriction operator, which we
denote by ${\rm{Res}}_{\pi,k}$, and the projection map to the symmetric
tensors induced by $\pi$, which we denote by ${\rm{Sym}}^{k}\pi$, can be put
with the identifications (3.28) into the following commutative diagram

$\begin{CD}H^{0}(\mathbb{P}(U^{*}),\mathscr{O}(k))@>{{\rm{Res}}_{\pi,k}}>{}>H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k))\\\
\Big{\|}{}\Big{\|}{}\\\
{\rm{Sym}}^{k}(U)@>{{\rm{Sym}}^{k}\pi}>{}>{\rm{Sym}}^{k}(V).\end{CD}$ (4.10)

Since $(V,N_{V})$ is a quotient of $(U,N_{U})$, we also have

$FS(N_{V})=FS(N_{U})|_{\mathbb{P}(V^{*})}.$ (4.11)

From Theorem 3.11, (3.29), (4.10) and (4.11), we conclude that for any
$\epsilon>0$, there is $k_{0}\in\mathbb{N}^{*}$, such that for any $k\geq
k_{0}$, $f\in{\rm{Sym}}^{k}(V)$, there is $g\in{\rm{Sym}}^{k}(U)$, such that
${\rm{Sym}}^{k}\pi(g)=f$, and

$\|f\|^{{\rm{ev}}}_{N_{V},k}\geq\exp(-\epsilon
k)\cdot\|g\|^{{\rm{ev}}}_{N_{U},k}$ (4.12)

Now, since Theorem 3.13 holds for $(U,N_{U})$, we deduce that there is
$k_{1}\in\mathbb{N}^{*}$, such that for any $k\geq k_{1}$,
$g\in{\rm{Sym}}^{k}(U)$, we have

$\|g\|^{{\rm{ev}}}_{N_{U},k}\geq\exp(-\epsilon
k)\cdot\|g\|^{{\rm{Sym}},\pi}_{N_{U},k}.$ (4.13)

Since $(V,N_{V})$ is a quotient of $(U,N_{U})$, for any $x\in U$, we have

$\|x\|_{U}\geq\|\pi(x)\|_{V}.$ (4.14)

From this, the definition of the projective tensor norm and Lemma 3.16, we
deduce that for any $k\in\mathbb{N}^{*}$, $f\in{\rm{Sym}}^{k}(V)$ and
$g\in{\rm{Sym}}^{k}(U)$, verifying ${\rm{Sym}}^{k}\pi(g)=f$, we have

$\|g\|^{{\rm{Sym}},\pi}_{N_{U},k}\geq\|f\|^{{\rm{Sym}},\pi}_{N_{V},k}.$ (4.15)

From (4.12), (4.13) and (4.15), we see that for any
$k\geq\max\\{k_{0},k_{1}\\}$, $f\in{\rm{Sym}}^{k}(V)$, we have

$\|f\|^{{\rm{ev}}}_{N_{V},k}\geq\exp(-2\epsilon
k)\cdot\|f\|^{{\rm{Sym}},\pi}_{N_{V},k}.$ (4.16)

As $\epsilon>0$ is arbitrary, from (3.27) and (4.16), we conclude that
${\rm{Sym}}_{{\rm{ev}}}(N_{V})$ and ${\rm{Sym}}_{\pi}(N_{V})$ are
asymptotically equivalent. As described after (3.27), this finishes the proof.
∎

##  5 Limiting behavior of jumping measures and geodesic rays

The main goal of this section is to study the limiting behavior of jumping
measures of submultiplicative filtrations and to establish equivalence of
several definitions of geodesic rays. More precisely, in Section 5.1, we
establish Theorem 1.9 and in Section 5.2, we establish Theorem 1.11 modulo a
certain statement, to which Section 5.3 is dedicated.

###  5.1 From submultiplicative filtrations to rays of submultiplicative norms

The main goal of this section is to prove Theorem 1.9. For this, on an
arbitrary section ring, we associate with any submultiplicative filtration
$\mathcal{F}$ a ray of submultiplicative norms. When this ray emanates from
${\rm{Ban}}^{\infty}(h^{L}_{0})$ for a certain regularizable from above psh
metric $h^{L}_{0}$ on $L$ and bounded $\mathcal{F}$, we obtain a geodesic ray
of metrics on the line bundle through the Fubini-Study construction. We then
compare this construction of a geodesic ray with the Bergman geodesic ray of
Phong-Sturm [51] and Ross-Witt Nyström [58].

Now, let us associate with any given submultiplicative filtration a ray of
submultiplicative graded norms. More precisely, we fix a submultiplicative
graded norm $N=\sum N_{k}$,
$N_{k}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|_{k}$, over the section ring
$R(X,L)$. For a fixed submultiplicative filtration $\mathcal{F}$ on $R(X,L)$,
we define by the procedure (2.18) for any $t\in[0,+\infty[$,
$k\in\mathbb{N}^{*}$, the ray of norms
$N^{t}_{\mathcal{F},k}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{t}_{\mathcal{F},k}$,
over $H^{0}(X,L^{k})$, emanating from $N_{k}$. An easy verification shows that
by submultiplicativity of $N$ and $\mathcal{F}$, for any $t\in[0,+\infty[$,
the graded norm $N^{t}_{\mathcal{F}}=\sum N^{t}_{\mathcal{F},k}$ is
submultiplicative.

We will now specify this to $N={\rm{Ban}}^{\infty}(h^{L}_{0})$ for a certain
regularizable from above psh metric $h^{L}_{0}$ on $L$ and a bounded
submultiplicative filtration $\mathcal{F}$. From the boundness of
$\mathcal{F}$ (and of $h^{L}_{0}$), we deduce that the norm
$N^{t}_{\mathcal{F}}$ is bounded for any $t\in[0,+\infty[$. Hence, by Theorem
1.3 and Corollary 2.18, we conclude that for any $p\in[1,+\infty[$,
$t\in[0,+\infty[$, the following equivalence holds

$N^{t}_{\mathcal{F}}\sim_{p}{\rm{Ban}}^{\infty}(FS(N^{t}_{\mathcal{F}})_{*}).$
(5.1)

Recall that in (1.17), we defined the notion of geodesic ray.

######  Theorem 5.1.

For any regularizable from above psh metric $h^{L}_{0}$ on $L$ and any bounded
submultiplicative filtration $\mathcal{F}$ on $R(X,L)$,
$FS(N^{t}_{\mathcal{F}})_{*}$, $t\in[0,+\infty[$ is a geodesic ray emanating
from $h^{L}_{0}$.

###### Proof.

From Lemma 2.7 and Minkowski inequality, for any $t\geq s\geq 0$,
$p\in[1,+\infty[$, we deduce

$d_{p}(N_{\mathcal{F}}^{t},N_{\mathcal{F}}^{s})=(t-s)\cdot
d_{p}(N_{\mathcal{F}}^{1},N_{\mathcal{F}}^{0})$ (5.2)

Remark, however, that by Corollary 3.6, we have

$d_{p}(N_{\mathcal{F}}^{t},N_{\mathcal{F}}^{s})=d_{p}\big{(}FS(N^{t}_{\mathcal{F}})_{*},FS(N^{s}_{\mathcal{F}})_{*}\big{)}.$
(5.3)

By (5.2) and (5.3), we conclude that the curve $FS(N^{t}_{\mathcal{F}})_{*}$,
$t\in[0,+\infty[$, is geodesic in any of the metric spaces
$(\mathcal{E}_{\omega}^{p},d_{p})$, $p\in[1,+\infty[$. By Theorem 2.10, we
yield that $FS(N^{t}_{\mathcal{F}})_{*}$, $t\in[0,+\infty[$, is a geodesic
ray. As we assumed that $h^{L}_{0}$ is regularizable from above,
$FS(N^{t}_{\mathcal{F}})_{*}$ emanates from $h^{L}_{0}$ by Theorem 2.13. ∎

We call the above ray the Fubini-Study geodesic ray and denote it by
$h^{L,FS}_{\mathcal{F},t}\mathrel{\mathop{\ordinarycolon}}=FS(N^{t}_{\mathcal{F}})_{*}$,
$t\in[0,+\infty[$. Let us now compare the Fubini-Study geodesic ray with the
Bergman geodesic ray, defined by Phong-Sturm [51] and Ross-Witt Nyström [58].
To recall the definition of the latter one, we fix a continuous psh metric
$h^{L}_{0}$ on $L$, a volume form $\mu$ of unit volume on $X$, and consider
the $L^{2}$-norm ${\rm{Hilb}}(h^{L},\mu)$ on $R(X,L)$, defined as in (2.49).
Now, for a fixed bounded submultiplicative filtration $\mathcal{F}$ on
$R(X,L)$, and any $k\in\mathbb{N}^{*}$, we define the ray of Hermitian norms
$H^{t}_{\mathcal{F},k}\mathrel{\mathop{\ordinarycolon}}=\|\cdot\|^{t,H}_{\mathcal{F},k}$
on $H^{0}(X,L^{k})$, emanating from ${\rm{Hilb}}(h^{L},\mu)$, as in (2.23).
Let $H^{t}_{\mathcal{F}}=\sum H^{t}_{\mathcal{F},k}$ be the induced graded
norm on $R(X,L)$. We define the Bergman geodesic ray
$h^{L,\mathcal{B}}_{\mathcal{F},t}$, $t\in[0,+\infty[$ as

$h^{L,\mathcal{B}}_{\mathcal{F},t}\mathrel{\mathop{\ordinarycolon}}=\big{(}\lim_{k\to\infty}\inf_{l\geq
k}FS(H^{t}_{\mathcal{F},l})^{\frac{1}{l}}\big{)}_{*}.$ (5.4)

From Lemma 2.1, we see that this definition is equivalent to the one of Ross-
Witt Nyström [58, Definition 9.1] up a change of variables $t\mapsto 2t$. The
following result ties the two constructions.

######  Proposition 5.2.

For any $t\in[0,+\infty[$, $H^{t}_{\mathcal{F}}\sim N^{t}_{\mathcal{F}}$. In
particular, Fubini-Study geodesic ray emanating from a fixed continuous psh
metric coincides with the respective Bergman geodesic ray.

###### Proof.

It follows directly from Lemma 2.8 and Proposition 2.15. ∎

######  Remark 5.3.

Phong-Sturm in [51, Theorem 1] established that Bergman geodesic ray emanating
from a smooth positive initial point is indeed a geodesic ray for any
filtration arising from a test configuration, see Section 5.3 for a recap of
the relation between the two. An alternative proof was given by Ross-Witt
Nyström in [58, §9]. From Proposition 5.2, Theorem 5.1 gives a new proof of
this result more general initial points of the ray.

###### Proof of Theorem 1.9.

Let us first remark that it is enough to establish Theorem 1.9 in the special
case when the filtration $\mathcal{F}$ satisfies the additional assumption

$\mathcal{F}^{0}R(X,L)=\\{0\\}.$ (5.5)

To see this, remark that since $\mathcal{F}$ is bounded, there is $C>0$,
verifying $\mathcal{F}^{Ck}H^{0}(X,L^{k})=\\{0\\}$. Consider now another
filtration $\mathcal{F}_{0}$ on $R(X,L)$, defined for any $k\in\mathbb{N}$,
$\lambda\in\real$, as follows
$\mathcal{F}_{0}^{\lambda}H^{0}(X,L^{k})=\mathcal{F}^{\lambda+Ck}H^{0}(X,L^{k})$.
Clearly, $\mathcal{F}_{0}$ is submultiplicative and bounded whenever
$\mathcal{F}$ is submultiplicative and bounded. An easy verification shows
that establishing Theorem 1.9 for $\mathcal{F}_{0}$ and $\mathcal{F}$ is
equivalent. We, hence, assume from now on that $\mathcal{F}$ satisfies (5.5).

By Lemma 2.7, (5.5) and Minkowski inequality, the jumping measures
$\mu_{\mathcal{F},k}$ of $\mathcal{F}$ are related to the ray of
submultiplicative norms $N_{\mathcal{F},k}^{t}$, $t\in[0,+\infty[$,
constructed in (2.18), by

$\Big{|}d_{p}(N_{\mathcal{F},k}^{1},N_{\mathcal{F},k}^{0})-k\cdot\sqrt[p]{\int(-x)^{p}\mu_{\mathcal{F},k}(x)}\Big{|}\leq\log\dim
H^{0}(X,L^{k}).$ (5.6)

Since $N_{\mathcal{F}}^{t}\geq N_{\mathcal{F}}^{0}$, we have
$h^{L,FS}_{\mathcal{F},1}\geq h^{L,FS}_{\mathcal{F},0}$ and Theorem 1.9
follows from (2.48), (5.3) and (5.6). ∎

###  5.2 Maximal geodesic rays from submultiplicative filtrations

The main goal of this section is to prove that Fubini-Study geodesic rays are
maximal geodesic rays. More precisely, we establish Theorem 1.11.

We first recall that the definition of maximal geodesic rays requires fixing
initial point of the geodesic ray and its singularities at $+\infty$. Maximal
geodesic ray is then the supremum over all geodesic rays, verifying these
“boundary conditions”. The correct description of this requires some basic
notions from non-Archimedean geometry.

Denote by $X^{an}$ the Berkovich analytification of the projective manifold
$X$ with respect to the trivial absolute value on the ground field
$\mathbb{C}$. We view $X^{an}$ as a topological space, whose points can be
understood as semivaluations on $X$, i.e. valuations
$v\mathrel{\mathop{\ordinarycolon}}\mathbb{C}(Y)^{*}\to\real$ on the function
field $\mathbb{C}(Y)$ of subvarieties $Y$ of $X$, trivial on $\mathbb{C}$. In
particular, $X^{an}$ contains the set $X^{div}$ of divisorial valuations on
$\mathbb{C}(X)$, i.e. valuations
$v\mathrel{\mathop{\ordinarycolon}}\mathbb{C}(X)^{*}\to\real$ of the form
$v=c\cdot{\rm{ord}}_{E}$, where $c\in\mathbb{Q}_{>0}$, $E$ is a prime divisor
on some normal variety $Y$ mapping birationally to $X$ and ${\rm{ord}}_{E}$
corresponds to the valuation calculating the order of vanishing along $E$.
Remark, in particular, that ${\rm{ord}}_{\Sigma}$ is well-defined for any
submanifold $\Sigma\subset X$ through the divisorial valuation of the
exceptional divisor in the blow-up of $X$ along $\Sigma$. The space $X^{an}$
can be seen as a compactification of $X^{div}$, endowed with the topology of
pointwise convergence, see [8, §6.1].

Now, the projection $\pi\mathrel{\mathop{\ordinarycolon}}X\times\mathbb{D}\to
X$ induces a map $(X\times\mathbb{D})^{div}\to X^{div}$; this has a canonical
section
$\sigma\mathrel{\mathop{\ordinarycolon}}X^{div}\to(X\times\mathbb{D})^{div}$,
the Gauss extension, defined by

$\sigma(v)\Big{(}\sum f_{i}\tau^{i}\Big{)}=\min_{i}\\{v(f_{i})+i\\}.$ (5.7)

The rationale behind this is that we have
$\sigma({\rm{ord}}_{\Sigma})={\rm{ord}}_{\Sigma\times\\{0\\}}$ for any
submanifold $\Sigma\subset X$.

Recall that a psh ray is a map
$U\mathrel{\mathop{\ordinarycolon}}]0,+\infty[\to{\rm{PSH}}(X,\omega)$, such
that in the notations (1.16), (2.42), the function $\hat{U}$ is
$\pi^{*}\omega$-psh on $X\times\mathbb{D}^{*}$. Of particular importance are
psh rays
$U\mathrel{\mathop{\ordinarycolon}}]0,+\infty[\to\mathcal{E}_{\omega}^{1}$
called geodesic rays; the restriction of such $U$ for each
$[a,b]\in]0,+\infty[$ coincides (up to affine reparametrization) with the
distinguished psh geodesic joining $U_{a}$ to $U_{b}$, see (2.43). We see from
(2.44) that for bounded psh rays, this definition coincides with (1.17) modulo
identification (2.35). Given now a psh ray
$U\mathrel{\mathop{\ordinarycolon}}]0,+\infty[\to{\rm{PSH}}(X,\omega)$ of
linear growth (i.e. such that there is $a>0$, for which $U(t)-at$ is bounded
from above, as $t\to+\infty$), define the $S^{1}$-invariant
$\pi^{*}\omega$-psh function $V$ on $X\times\mathbb{D}^{*}$, in the notations
of (1.16) by

$V(x,\tau)\mathrel{\mathop{\ordinarycolon}}=\hat{U}+a\log|\tau|.$ (5.8)

Then $V$ is bounded above near $X\times{0}$, hence, it uniquely extends to a
$\pi^{*}\omega$-psh function on $X\times\mathbb{D}$, cf. [30, Theorem I.5.23].
For each divisorial valuation $w$ on $X\times\mathbb{D}$, we then can make
sense of $w(V)\geq 0$ as a generic Lelong number on a suitable blowup, see [8,
§B.6]. We set $w(U)\mathrel{\mathop{\ordinarycolon}}=w(V)-aw(\tau)$. This is
independent of the choice of the constant $a$ by the additivity of Lelong
numbers. We construct the function
$U_{NA}\mathrel{\mathop{\ordinarycolon}}X^{div}\to\real$, decoding the
singularities of $U$ at $+\infty$, by

$U_{NA}(v)=-\sigma(v)(U).$ (5.9)

Following [8, Definition 6.5], we say that a psh geodesic ray
$U\mathrel{\mathop{\ordinarycolon}}[0,+\infty[\to\mathcal{E}_{\omega}^{1}$ is
maximal if for any psh ray
$V\mathrel{\mathop{\ordinarycolon}}]0,+\infty[\to\mathcal{E}_{\omega}^{1}$ of
linear growth with $\lim_{t\to 0}V_{t}\leq U_{0}$ and $V_{NA}\leq U_{NA}$, we
have $V\leq U$. A maximal geodesic ray is thus uniquely determined by $U_{0}$
and $U_{NA}$. Remark the analogy between this and (2.43). However, not every
geodesic ray is maximal, see [8, Example 6.10].

Now, as in (2.29), we denote by
$\chi_{\mathcal{F},k}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k})\to[0,+\infty[$,
$k\in\mathbb{N}^{*}$ the non-Archimedean norm on $H^{0}(X,L^{k})$ associated
with a restriction to $H^{0}(X,L^{k})$ of a graded submultiplicative
filtration $\mathcal{F}$ on $R(X,L)$. The associated graded norm
$\chi_{\mathcal{F}}=\max\chi_{\mathcal{F},k}$ on $R(X,L)$ is
submultiplicative, i.e. for any $f\in H^{0}(X,L^{k})$, $g\in H^{0}(X,L^{l})$,
$k,l\in\mathbb{N}^{*}$, we have

$\chi_{\mathcal{F},k+l}(f\cdot
g)\leq\chi_{\mathcal{F},k}(f)\cdot\chi_{\mathcal{F},l}(g).$ (5.10)

A graded (non-Archimedean) norm $\chi=\max\chi_{k}$ on $R(X,L)$ is called
bounded if there is $C>0$, such that for any $k\in\mathbb{N}^{*}$, the
following inequality is satisfied $\chi_{k}\geq\exp(-Ck)$. Remark that since
$R(X,L)$ is finitely generated, the existence of $C>0$ such that for any
$k\in\mathbb{N}^{*}$, $\chi_{k}\leq\exp(Ck)$ is automatic for
submultiplicative norms by (5.10). Clearly, if $\mathcal{F}$ is a bounded
filtration on $R(X,L)$, $\chi_{\mathcal{F}}$ is bounded, and bounded
submultiplicative filtrations on $R(X,L)$ are in one-to-one correspondence
with bounded submultiplicative non-Archimedean norms on $R(X,L)$.

In this perspective, the construction of Boucksom-Jonsson [18] of a non-
Archimedean potential on $X^{an}$ from $\mathcal{F}$ realizes in the non-
Archimedean context the complex-geometric philosophy we recalled in Section
2.1 and Lemma 3.3 that associates to any bounded submultiplicative norm $N$ on
$R(X,L)$ the Fubini-Study metric $FS(N)$ on $L$ through the associated Kodaira
embedding.

To describe it precisely, recall that a semivaluation $v\in X^{an}$ can be
naturally evaluated on a section $s\in H^{0}(X,M)$ of any line bundle $M$ on
$X$, by defining $v(s)\in[0,+\infty]$ as the value of $v$ on the local
function corresponding to $s$ in any local trivialization of $M$ at the center
of the valuation, see [18, p. 15] for details. For any $s\in H^{0}(X,M)$, we
then can define $|s|\mathrel{\mathop{\ordinarycolon}}X^{an}\to[0,1]$ by
setting

$|s|(v)\mathrel{\mathop{\ordinarycolon}}=\exp(-v(s)).$ (5.11)

Now, Boucksom-Jonsson in [18, (4.3)] associated with any non-Archimedean norm
$\chi_{k}$ on $H^{0}(X,L^{k})$ the Fubini-Study potential,
$FS(\chi_{k})\mathrel{\mathop{\ordinarycolon}}X^{an}\to\real$, defined as
follows

$FS(\chi_{k})\mathrel{\mathop{\ordinarycolon}}=\sup_{s\in
H^{0}(X,L^{k})\setminus\\{0\\}}\big{\\{}\log|s|-\log\chi_{k}(s)\big{\\}}.$
(5.12)

Whenever $\chi_{k}$ is associated with a graded filtration $\mathcal{F}$ on
$R(X,L)$, we denote the associated Fubini-Study potentials by
$FS(\mathcal{F})_{k}$. For bounded $\mathcal{F}$, the resulting sequence of
potentials $FS(\mathcal{F})_{k}$, $k\in\mathbb{N}^{*}$, is uniformly bounded
from above. If the filtration $\mathcal{F}$ is, moreover, submultiplicative,
then the sequence of potentials $FS(\mathcal{F})_{k}$ is superadditive and the
Fubini-Study potential of a bounded submultiplicative filtration is now
defined by Fekete’s lemma as

$FS(\mathcal{F})=\sup_{k\in\mathbb{N}^{*}}\Big{\\{}\frac{1}{k}FS(\mathcal{F})_{k}\Big{\\}}.$
(5.13)

To justify the existence of maximal geodesic rays, let us recall some basics
from non-Archimedean pluripotential theory. Following [18, §1.8], we say that
a function $f$ on $X^{an}$ is a Fubini-Study function if there is
$m\in\mathbb{N}^{*}$, base point free $s_{1},\ldots,s_{r}\in H^{0}(X,L^{m})$,
and some $\lambda_{1},\ldots,\lambda_{r}\in\real$, such that

$f=\frac{1}{m}\max_{j=1,\ldots,r}\\{\log|s_{j}|+\lambda_{j}\\},$ (5.14)

where $|s_{j}|$ were defined in (5.11). Following [8, §6.2, 6.3], we say that
a function $\phi\mathrel{\mathop{\ordinarycolon}}X^{an}\to[-\infty,+\infty[$
is in $PSH^{NA}(X)$ if it can be obtained as the pointwise limit of a
decreasing net of Fubini-Study functions, excluding $\phi=-\infty$. See the
analogy with Remark 1.6b).

Analogously to the complex situation, see (2.39), using the non-Archimedean
mixed Monge-Ampère operator, one can define an energy functional $E$ on the
space
$CPSH^{NA}(X)\mathrel{\mathop{\ordinarycolon}}=PSH^{NA}(X)\cap\mathscr{C}^{0}(X^{an})$,
see [18, (4.3)]. This energy functional satisfies similar monotonicity
properties as its complex analogue. Using this, it is then possible to extend
$E$ to $PSH^{NA}(X)$ through the same procedure as in (2.40). We denote by
$\mathcal{E}^{1,NA}$ the subset of $PSH^{NA}(X)$ with finite (i.e. not equal
to $-\infty$) energy.

######  Theorem 5.4 ([8, Theorems 6.2, 6.4 and 6.6] and [18, Lemma 4.3]).

For any psh ray of linear growth
$U\mathrel{\mathop{\ordinarycolon}}]0,+\infty[\to\mathcal{E}_{\omega}^{1}$,
the function $U_{NA}\mathrel{\mathop{\ordinarycolon}}X^{div}\to\real$ extends
uniquely to $U_{NA}\in PSH^{NA}(X)$, and we, moreover, have
$U_{NA}\in\mathcal{E}^{1,NA}$. Similarly, for any bounded submultiplicative
filtration $\mathcal{F}$ on $R(X,L)$, we have
$FS(\mathcal{F})\in\mathcal{E}^{1,NA}$. For any $u\in\mathcal{E}_{\omega}^{1}$
and $\phi\in\mathcal{E}^{1,NA}$, there exists a unique maximal geodesic ray
$U\mathrel{\mathop{\ordinarycolon}}[0,+\infty[\to\mathcal{E}_{\omega}^{1}$
emanating from $u$ such that $U_{NA}=\phi$.

From Theorems 5.4, we see, in particular, that for any bounded psh metric
$h^{L}_{0}$ and any bounded submultiplicative filtration $\mathcal{F}$, there
is the maximal geodesic ray $h^{L,\max}_{\mathcal{F},t}$, emanating from
$h^{L}_{0}$, corresponding on the potential level, see (2.35), to the ray $U$,
verifying $U_{NA}=FS(\mathcal{F})$.

Define now, following Székelyhidi [64], for any graded filtration
$\mathcal{F}$ on $R(X,L)$ the sequence of canonical approximations
$\mathcal{F}_{(k)}$ of $\mathcal{F}$, as the filtrations induced by
$\mathcal{F}H^{0}(X,L^{k})$ on $R(X,L^{k})$, for $k\in\mathbb{N}^{*}$ big
enough so that $H^{0}(X,L^{k})$ generates the algebra $R(X,L^{k})$. Taking
into account the identification of submultiplicative filtrations and
submultiplicative non-Archimedean norms, see (2.29), remark that
$\mathcal{F}_{(k)}$ are the formal analogues of the norm $N^{\pi}$ from
Theorem 3.1.

The proof of Theorem 1.11 decomposes into several statements. The first step
is to establish that Fubini-Study geodesic rays behave reasonably under these
approximations.

######  Theorem 5.5.

For any bounded submultiplicative filtration $\mathcal{F}$ on a section ring
$R(X,L)$ of an ample line bundle $L$ and any regularizable from above psh
metric $h^{L}_{0}$, Fubini-Study geodesic rays emanating from $h^{L}_{0}$
behave continuously in the topology of $\mathcal{E}_{\omega}^{1}$ with respect
to canonical approximations $\mathcal{F}_{(k)}$ of $\mathcal{F}$. In other
words, in $\mathcal{E}_{\omega}^{1}$, we have

$\lim_{k\to\infty}\big{(}h^{L^{k},FS}_{\mathcal{F}_{(k)},t}\big{)}^{\frac{1}{k}}=h^{L,FS}_{\mathcal{F},t},$
(5.15)

where the limit is taken over multiplicative sequence $k\in\mathbb{N}^{*}$, as
for example $k=2^{l}$, $l\in\mathbb{N}$.

The proof of Theorem 5.5 is based on Theorem 1.3 and the study of the volume
functional for canonical approximations. Recall that the volume of a bounded
submultiplicative filtration $\mathcal{F}$ is defined as follows

${\rm{vol}}(\mathcal{F})\mathrel{\mathop{\ordinarycolon}}=\lim_{k\to\infty}\int
x\mu_{\mathcal{F},k},$ (5.16)

where $\mu_{\mathcal{F},k}$, $k\in\mathbb{N}$ are jumping measures (1.14) of
the filtration. The existence of the aforementioned limit is a consequence of
[20], [16] or of Theorem 1.9.

######  Theorem 5.6 (Boucksom-Jonsson [18, Theorem 3.18 and (3.14)]).

Volumes depend continuously under canonical approximations, i.e. for any
bounded submultiplicative filtration $\mathcal{F}$, we have

$\lim_{k\to\infty}{\rm{vol}}(\mathcal{F}_{(k)})={\rm{vol}}(\mathcal{F}),$
(5.17)

where the limit is taken over multiplicative sequence $k\in\mathbb{N}^{*}$, as
for example $k=2^{l}$, $l\in\mathbb{N}$.

Let us give an alternative proof of Theorem 5.6, relying solely on Theorem 1.3
instead of Okounkov bodies, used in [18]. In fact, as we shall see later,
Theorem 5.5 would follow rather easily from this new proof of Theorem 5.6.

###### Proof.

First of all, from considerations, similar to the ones from the proof of
Theorem 1.9, we can assume that the filtration $\mathcal{F}$ satisfies the
additional assumption (5.5).

From Corollary 3.6, we see that for any $k\in\mathbb{N}^{*}$,
$t\in[0,+\infty[$, we have

$d_{1}\big{(}h^{L,FS}_{\mathcal{F},t},(h^{L^{k},FS}_{\mathcal{F}_{(k)},t})^{\frac{1}{k}}\big{)}=\frac{1}{k}d_{1}\big{(}N^{t}_{\mathcal{F}}|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}.$
(5.18)

Remark, however, that since $\mathcal{F}$ satisfies (5.5), and since the
weight of any element of $R(X,L^{k})$ with respect to $\mathcal{F}$ is at
least as big as the weight with respect to $\mathcal{F}_{(k)}$, we have

$N^{t}_{\mathcal{F}_{(k)}}\geq N^{t}_{\mathcal{F}}|_{R(X,L^{k})}\geq
N|_{R(X,L^{k})}.$ (5.19)

In particular, by Lemma 2.6, we conclude that

$d_{1}\big{(}N^{t}_{\mathcal{F}}|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}=d_{1}\big{(}N|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}-d_{1}\big{(}N|_{R(X,L^{k})},N^{t}_{\mathcal{F}}|_{R(X,L^{k})}\big{)}.$
(5.20)

However, by Lemma 2.7 and the fact that $\mathcal{F}$ satisfies (5.5), we have

$\displaystyle
d_{1}\big{(}N|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}=-t\cdot
k\cdot{\rm{vol}}(\mathcal{F}_{(k)}),$ (5.21) $\displaystyle
d_{1}\big{(}N|_{R(X,L^{k})},N^{t}_{\mathcal{F}}|_{R(X,L^{k})}\big{)}=-t\cdot
k\cdot{\rm{vol}}(\mathcal{F}).$

By Lemma 3.3, the trivial fact that
$N^{t}_{\mathcal{F}_{(k)},1}=N^{t}_{\mathcal{F},k}$ and (5.19), we deduce

$FS(N^{t}_{\mathcal{F}})^{k}\leq FS(N^{t}_{\mathcal{F}_{(k)}})\leq
FS(N^{t}_{\mathcal{F},k}).$ (5.22)

From this, (2.11) and (5.18), we conclude that

$\frac{1}{k}d_{1}\big{(}N^{t}_{\mathcal{F}}|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}\leq
d_{1}\big{(}FS(N^{t}_{\mathcal{F}})_{*},FS(N^{t}_{\mathcal{F},k})_{*}^{\frac{1}{k}}\big{)}.$
(5.23)

From Lemma 3.3, (2.38), (5.23) and the fact that the sequence of metrics
$FS(N^{t}_{\mathcal{F},k})^{\frac{1}{k}}$ is decreasing over multiplicative
sequence $k\in\mathbb{N}^{*}$, we deduce the following convergence

$\lim_{k\to\infty}\frac{1}{k}d_{1}\big{(}N^{t}_{\mathcal{F}}|_{R(X,L^{k})},N^{t}_{\mathcal{F}_{(k)}}\big{)}=0,$
(5.24)

where $k$ runs over a multiplicative sequence. We deduce (5.17) from (5.20),
(5.21) and (5.24). ∎

###### Proof of Theorem 5.5..

It follows directly from Theorem 5.6, (2.38), (5.18), (5.20) and (5.21). ∎

The second step of the proof of Theorem 1.11 consists in the following result.

######  Theorem 5.7.

An analogue of Theorem 5.5 holds for maximal geodesic rays emanating from
bounded psh metrics. In other words, under the assumptions of Theorem 5.5, in
$\mathcal{E}_{\omega}^{1}$, we have

$\lim_{k\to\infty}\big{(}h^{L^{k},\max}_{\mathcal{F}_{(k)},t}\big{)}^{\frac{1}{k}}=h^{L,\max}_{\mathcal{F},t},$
(5.25)

where the limit is taken over multiplicative sequence $k\in\mathbb{N}^{*}$, as
for example $k=2^{l}$, $l\in\mathbb{N}$.

###### Proof.

First, remark that maximal geodesic rays are monotonic with respect to the
data. In other words, for $u_{0},u_{1}\in\mathcal{E}_{\omega}^{1}$ and