label = , label-format = ,) label-width = 12pt MnLargeSymbols’164
MnLargeSymbols’171

#  On the metric structure of section ring

Siarhei Finski

In memory of Jean-Pierre Demailly

Abstract. The main goal of this article is to study for a projective manifold
and an ample line bundle over it the relation between metric and algebraic
structures on the associated section ring.

More precisely, we prove that once the kernel is factored out, the
multiplication operator of the section ring becomes an approximate isometry
(up to a normalization) with respect to the $L^{2}$-norms and the induced
Hermitian tensor product norm. We also show that the analogous result holds
for the $L^{1}$ and $L^{\infty}$-norms if instead of the Hermitian tensor
product norm, we consider the projective and injective tensor norms induced by
$L^{1}$ and $L^{\infty}$-norms respectively.

Then we show that $L^{2}$-norms associated with continuous plurisubharmonic
metrics are actually characterized by the multiplicativity properties of this
type. Using this, we refine the theorem of Phong-Sturm about quantization of
Mabuchi geodesics from the weaker level of Fubini-Study convergence to the
stronger level of norm equivalences.

###### Contents

  1. 1 Introduction
  2. 2 Extension theorem and norms on the section ring
    1. 2.1 Two versions of Ohsawa-Takegoshi extension theorem
    2. 2.2 Bernstein-Markov property and comparison of natural norms
    3. 2.3 Fubini-Study operator and continuous psh metrics
  3. 3 Synergy of metric and algebraic structures on section ring
    1. 3.1 Multiplication map as a restriction morphism, a proof of Theorem 1.1
    2. 3.2 Norms on the product manifold, injective and projective tensor norms
    3. 3.3 Bounds on products of holomorphic sections, a proof of Theorem 3.2
    4. 3.4 Asymptotics of optimal decomposition map, a proof of Theorem 3.4
  4. 4 The set of multiplicatively generated norms
    1. 4.1 Multiplicatively generated norms from plurisubharmonic metrics
    2. 4.2 Metric properties of the Hilbert map and Mabuchi supremum geometry
    3. 4.3 Fubini-Study operator on graded multiplicatively generated norms
    4. 4.4 An example of a graded Hermitian norm on the section ring
    5. 4.5 Inductive construction of the Hilbert map, a proof of Theorem 4.2
    6. 4.6 Mabuchi geodesics and geodesics on the space of norms
  5. A On almost sub-additive sequences of functions
    1. A.1 A version of Fekete’s lemma for almost sub-additive sequences
    2. A.2 Uniform convergence of almost sub-additive sequences of functions
  6. B Normed vector spaces and induced tensor norms

##  1 Introduction

Many questions from algebraic and complex geometry can be restated in terms of
some problems related to the study of the section ring. For example, finite
generation of the canonical ring is of great importance to the minimal model
program, [7]. The K-stability condition, famously known for its connection
with the existence of the constant scalar curvature metrics, is related to the
asymptotic study of certain filtrations on the section ring [42], [54], [10].

In this article, we propose to study further properties of the section ring;
now in connection with the metric structure of the manifold.

More precisely, we fix a complex projective manifold $X$ of dimension $n$ and
an ample line bundle $L$ over $X$. We consider the section ring $R(X,L)$,
defined as follows

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

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

${\rm{Mult}}_{k_{1},\cdots,k_{r}}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k_{1}})\otimes\cdots\otimes
H^{0}(X,L^{k_{r}})\to H^{0}(X,L^{k}),$ (1.2)

as $f_{1}\otimes\cdots\otimes f_{r}\mapsto f_{1}\cdots f_{r}$. Those
multiplication maps endow $R(X,L)$ with the structure of a graded ring. It
follows from the ${\rm{Proj}}$-construction that the (graded) ring structure
of $R(X,L)$ carries the same amount of information as the pair $(X,L)$.

We now fix a positive Hermitian metric $h^{L}$ on $L$. This means that for the
curvature $R^{L}$ of the Chern connection on $(L,h^{L})$, the following closed
real $(1,1)$-differential form is positive

$\omega\mathrel{\mathop{\ordinarycolon}}=c_{1}(L,h^{L})\mathrel{\mathop{\ordinarycolon}}=\frac{\sqrt{-1}}{2\pi}R^{L}.$
(1.3)

For smooth sections $f,f^{\prime}$ of $L^{p}$, $p\in\mathbb{N}$, we define the
$L^{2}$-Hermitian product using the pointwise Hermitian products
$\langle\cdot,\cdot\rangle_{h}$ on $L^{p}$, induced by $h^{L}$, as follows

$\langle
f,f^{\prime}\rangle_{L^{2}_{p}(X,h^{L})}\mathrel{\mathop{\ordinarycolon}}=\int_{X}\langle
f(x),f^{\prime}(x)\rangle_{h}dv_{X}(x),$ (1.4)

where $dv_{X}=\frac{1}{n!}\omega^{n}$. We also denote by
$\|\cdot\|_{L^{2}_{p}(X,h^{L})}$ the associated norm. We, finally, denote by
$\|\cdot\|_{L^{1}_{p}(X,h^{L})}$ and $\|\cdot\|_{L^{\infty}_{p}(X,h^{L})}$ the
$L^{1}$ and $L^{\infty}$-norms, defined as follows

$\|f\|_{L^{1}_{p}(X,h^{L})}\mathrel{\mathop{\ordinarycolon}}=\int_{X}|f|_{h}dv_{X}(x),\qquad\|f\|_{L^{\infty}_{p}(X,h^{L})}\mathrel{\mathop{\ordinarycolon}}=\sup_{x\in
X}|f(x)|_{h}.$ (1.5)

When restricted to the vector space of holomorphic sections, $H^{0}(X,L^{p})$,
we sometimes denote the $L^{2}$-norm by ${\rm{Hilb}}_{p}(h^{L})$, and the
$L^{1}$ and $L^{\infty}$-norms by ${\rm{Ban}}_{p}^{1}(h^{L})$,
${\rm{Ban}}_{p}^{\infty}(h^{L})$.

Over $R(X,L)$, for $q=1,\infty$, we define the induced graded norms

${\rm{Hilb}}(h^{L})\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}{\rm{Hilb}}_{k}(h^{L}),\qquad{\rm{Ban}}^{q}(h^{L})\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}{\rm{Ban}}_{k}^{q}(h^{L}).$
(1.6)

The first goal of this paper is to study the metric properties of
$(R(X,L),{\rm{Ban}}^{1}(h^{L}))$, $(R(X,L),{\rm{Hilb}}(h^{L}))$ and
$(R(X,L),{\rm{Ban}}^{\infty}(h^{L}))$ in their relation with the
multiplication map. Roughly, our first result states that one can inductively
recover the approximate metric structure of the section ring from its
multiplicative structure.

To explain this more precisely, let us recall some basic definitions from the
theory of normed vector spaces. Let $V_{1},V_{2}$ be two finitely dimensional
vector spaces endowed with norms
$N_{i}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{i}$, $i=1,2$. There are
several natural constructions of a norm on the tensor product $V_{1}\otimes
V_{2}$.

Recall that the projective norm
$N_{1}\otimes_{\pi}N_{2}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{N_{1}\otimes_{\pi}N_{2}}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{\otimes_{\pi}}$
on $V_{1}\otimes V_{2}$ is defined as

$\mathinner{\\!\left\lVert
f\right\rVert}_{\otimes_{\pi}}=\inf\Big{\\{}\sum\|x_{i}\|_{1}\cdot\|y_{i}\|_{2};\quad
f=\sum x_{i}\otimes y_{i}\Big{\\}},$ (1.7)

where the infimum is taken over different ways of partitioning $f$ into a sum
of decomposable terms.

Recall that the injective norm
$N_{1}\otimes_{\epsilon}N_{2}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{N_{1}\otimes_{\epsilon}N_{2}}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{\otimes_{\epsilon}}$
on $V_{1}\otimes V_{2}$ is defined as

$\mathinner{\\!\left\lVert
f\right\rVert}_{\otimes_{\epsilon}}=\sup\Big{\\{}\big{|}(\phi\otimes\psi)(f)\big{|};\quad\phi\in
V_{1}^{*},\psi\in V_{2}^{*},\|\phi\|_{1}^{*}=\|\psi\|_{2}^{*}=1\Big{\\}}$
(1.8)

where $\|\cdot\|_{i}^{*}$, $i=1,2$, are the dual norms associated to
$\|\cdot\|_{i}$.

When both $\|\cdot\|_{i}$ are induced by Hermitian products (then we say that
$\|\cdot\|_{i}$ are Hermitian norms), we will denote by $N_{1}\otimes
N_{2}=\|\cdot\|_{N_{1}\otimes N_{2}}=\|\cdot\|_{\otimes}$ the Hermitian tensor
product norm induced by the product of the respective Hermitian products. See
Appendix B for relations between the above tensor norms and some of their
basic properties, used sometimes implicitly below.

A norm $N_{V}=\|\cdot\|_{V}$ on a finitely dimensional vector space $V$
naturally induces the norm $\|\cdot\|_{Q}$ on any quotient $Q$,
$\pi\mathrel{\mathop{\ordinarycolon}}V\to Q$ of $V$ through the following
identity

$\|f\|_{Q}\mathrel{\mathop{\ordinarycolon}}=\inf\big{\\{}\|g\|_{V};\quad g\in
V,\pi(g)=f\\},\qquad f\in Q.$ (1.9)

By a slight abuse of notations, we sometimes denote the quotient norm by
$[N_{V}]$, i.e. without the reference to the quotient space. This will not
cause any trouble as the quotient will be explicit.

Now, 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. Proposition 3.1. Our first main result shows that the induced
isomorphism quotient map is an approximate isometry (up to a multiplication by
a constant) with respect to the $L^{1},L^{2}$ and $L^{\infty}$-norms once the
tensor product is endowed with an appropriate tensor product norm. More
precisely, in Section 3, we establish our first main result.

######  Theorem 1.1.

There are $C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any $k,l\geq p_{1}$,
for the norms over $H^{0}(X,L^{k+l})$, under (1.2), the following relation
holds

$1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\leq\frac{[{\rm{Hilb}}_{k}(h^{L})\otimes{\rm{Hilb}}_{l}(h^{L})]}{{\rm{Hilb}}_{k+l}(h^{L})}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{\frac{n}{2}}\leq 1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}.$
(1.10)

Similarly, for $L^{1}$ and $L^{\infty}$-norms the analogous results hold.
There, instead of Hermitian tensor product norm, one needs to consider the
projective and injective tensor product norms respectively. More precisely, in
the notations of (1.10), we have

$\displaystyle\frac{1}{4^{n}}\cdot\Big{(}1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\leq$
$\displaystyle\frac{[{\rm{Ban}}_{k}^{1}(h^{L})\otimes_{\pi}{\rm{Ban}}_{l}^{1}(h^{L})]}{{\rm{Ban}}_{k+l}^{1}(h^{L})}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\cdot\frac{1}{2^{n}}$ (1.11)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq
4^{n}\cdot\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)},$
$\displaystyle 1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\leq$
$\displaystyle\frac{[{\rm{Ban}}_{k}^{\infty}(h^{L})\otimes_{\epsilon}{\rm{Ban}}_{l}^{\infty}(h^{L})]}{{\rm{Ban}}_{k+l}^{\infty}(h^{L})}\leq
1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}.$

######  Remark 1.2.

a) According to Pisier [45, Théorème 3.1], cf. also [1, Definition 2 and
Theorem 6], the projective and injective tensor norms differ by a value which
tends to infinity as the dimensions of the vector spaces tends to infinity.
Our result is, hence, sensible to the change of the projective norm by the
injective norm.

b) The constant $4^{n}$ from the first equation of (1.11) is not optimal. It
seems, however, to us that one cannot replace this constant by $1$, see Remark
3.9 for a related discussion.

c) One can guess the appearance of the injective and projective tensor
products in (1.11) from the classical result on the tensor products of $L^{1}$
and $\mathscr{C}^{0}$-spaces, see Lemma B.9.

In the second part of this paper, we show that the $L^{1}$, $L^{2}$ and
$L^{\infty}$-norms are characterized by the multiplicative properties as in
Theorem 1.1. Naturally, for our characterization, we need to consider non-
smooth metrics.

Recall that a continuous metric $h^{L}$ on $L$ is called plurisubharmonic (psh
for short) if the $(1,1)$-current $c_{1}(L,h^{L})$, see (2.17) for a
definition, is positive. We denote by $\mathcal{H}^{L}$ the set of continuous
psh metrics on $L$. Clearly, one can extend the definition of
${\rm{Ban}}^{\infty}(h^{L})$ for any $h^{L}\in\mathcal{H}^{L}$. By using the
Bedford-Taylor definition of Monge-Ampère operator, see [3] or [22,
Proposition I.3.2], cf. also (2.18), one can also extend the definition of the
volume form
$dv_{X}\mathrel{\mathop{\ordinarycolon}}=\frac{1}{n!}c_{1}(L,h^{L})^{n}$ as a
non-negative measure on $X$ for any $h^{L}\in\mathcal{H}^{L}$. Using this
measure, we extend the definition of ${\rm{Ban}}^{1}(h^{L})$ and
${\rm{Hilb}}(h^{L})$ for any $h^{L}\in\mathcal{H}^{L}$. See the discussion
after (2.19) for details.

Over finitely dimensional vector spaces $V$, we will later need to compare
different norms. For this, the Goldman-Iwahori distance (named after [30], cf.
also [9, (1.1)]), $d_{\infty}(N_{1},N_{2})$, between two norms
$N_{1}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{1}$ and
$N_{2}=\mathinner{\\!\left\lVert\cdot\right\rVert}_{2}$ over $V$ is defined as
follows

$d_{\infty}(N_{1},N_{2})=\sup_{v\in
V\setminus\\{0\\}}\big{|}\log\mathinner{\\!\left\lVert
v\right\rVert}_{1}-\log\mathinner{\\!\left\lVert v\right\rVert}_{2}\big{|}.$
(1.12)

Now, we are aiming to classify graded norms on the section ring up to the
equivalence relation ($\sim$) defined as follows. We say that graded norms
$N=\sum_{k=1}^{\infty}N_{k}$ and
$N^{\prime}=\sum_{k=1}^{\infty}N^{\prime}_{k}$ are equivalent if the graded
pieces satisfy the following asymptotics

$\frac{1}{k}d_{\infty}(N_{k},N^{\prime}_{k})\to 0.$ (1.13)

One motivation for considering this equivalence relation is that it
distinguishes elements from the image of the Hilbert map. See Theorem 1.7 for
a more precise statement.

It is well-known that for smooth positive metrics $h^{L}$ on $L$, the
multiplicative gap between ${\rm{Ban}}^{1}_{k}(h^{L})$,
${\rm{Hilb}}_{k}(h^{L})$, ${\rm{Ban}}^{\infty}_{k}(h^{L})$ is at most
polynomial in $k$. From the results of Berman-Boucksom-Witt Nyström [5,
Theorem 1.14], cf. Corollary 2.11, we know that more generally, the graded
norms ${\rm{Ban}}^{1}(h^{L})$, ${\rm{Hilb}}(h^{L})$,
${\rm{Ban}}^{\infty}(h^{L})$ are equivalent for any $h^{L}\in\mathcal{H}^{L}$.
Due to this, we concentrate now only on the study of Hermitian norms.

######  Definition 1.3.

We say that a graded Hermitian norm
$N\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}N_{k}$ on $R(X,L)$ is
multiplicatively generated if there is $p_{0}\in\mathbb{N}$ and a function
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{N}_{\geq p_{0}}\to\real$, verifying
$f(k)=o(k)$, as $k\to\infty$, such that for any $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, under the map (1.2),
we have the following bound

$d_{\infty}\Big{(}N_{k},\big{[}N_{k_{1}}\otimes\cdots\otimes
N_{k_{r}}\big{]}\Big{)}\leq f(k_{1})+\cdots+f(k_{r})+f(k).$ (1.14)

######  Remark 1.4.

The class of graded multiplicatively generated Hermitian norms is closed under
the above equivalence relation.

From Theorem 1.1, ${\rm{Hilb}}(h^{L})$ is multiplicatively generated for any
smooth positive metric $h^{L}$ on $L$, see Remark 4.10. More generally, in
Section 4.1, we establish the following result.

######  Theorem 1.5.

For any $h^{L}\in\mathcal{H}^{L}$, the graded norm ${\rm{Hilb}}(h^{L})$ is
multiplicatively generated.

We denote by $\mathscr{N}^{R(X,L)}$ the space of graded multiplicatively
generated norms on $R(X,L)$, and by $\mathscr{N}^{R(X,L)}_{\sim}$ the set of
their equivalence classes. By an abuse of notation, let

${\rm{Hilb}}\mathrel{\mathop{\ordinarycolon}}\mathcal{H}^{L}\to\mathscr{N}^{R(X,L)}_{\sim}$
(1.15)

denotes the induced map. In Section 4.5, we establish our second main result.

######  Theorem 1.6.

The map ${\rm{Hilb}}$ from (1.15) is a bijection.

We will now refine Theorem 1.6 by proving that the above bijection is actually
an isometry with respect to the natural metric structures on the spaces
$\mathscr{N}^{R(X,L)}_{\sim}$ and $\mathcal{H}^{L}$. For this, we define the
$L^{\infty}$-pseudometric
$d_{\infty}\mathrel{\mathop{\ordinarycolon}}\mathscr{N}^{R(X,L)}\times\mathscr{N}^{R(X,L)}\to[0,+\infty]$
as follows

$d_{\infty}(N,N^{\prime})=\limsup_{k\to\infty}\frac{1}{k}d_{\infty}(N_{k},N^{\prime}_{k}).$
(1.16)

By Remark 4.14, for any $N,N^{\prime}\in\mathscr{N}^{R(X,L)}$, we always have
$d_{\infty}(N,N^{\prime})<+\infty$. Clearly, the Hausdorff quotient of
$(\mathscr{N}^{R(X,L)},d_{\infty})$ (i.e. the set of equivalence classes of
$\mathscr{N}^{R(X,L)}$ with respect to the equivalence relation defined by the
annihilation of $d_{\infty}$) coincides with $\mathscr{N}^{R(X,L)}_{\sim}$,
and $d_{\infty}$, hence, yields the induced $L^{\infty}$-distance function

$d_{\infty}\mathrel{\mathop{\ordinarycolon}}\mathscr{N}^{R(X,L)}_{\sim}\times\mathscr{N}^{R(X,L)}_{\sim}\to[0,+\infty[.$
(1.17)

We also define the $L^{\infty}$-distance
$d_{\infty}\mathrel{\mathop{\ordinarycolon}}\mathcal{H}^{L}\times\mathcal{H}^{L}\to[0,+\infty[$
as follows

$d_{\infty}(h^{L}_{0},h^{L}_{1})=\frac{1}{2}\sup\Big{|}\log\frac{h^{L}_{0}}{h^{L}_{1}}\Big{|}.$
(1.18)

Our next and final result, proved in Section 4.2, is as follows.

######  Theorem 1.7.

The map ${\rm{Hilb}}$ from (1.15) preserves distances (1.17) and (1.18).

We will now describe an application of Theorem 1.6. We fix
$h^{L}_{0},h^{L}_{1}\in\mathcal{H}^{L}$. Consider the weak Mabuchi geodesic
$h^{L}_{t}\in\mathcal{H}^{L}$, $t\in[0,1]$, connecting $h^{L}_{0}$ and
$h^{L}_{1}$, see Section 4.2 for necessary definitions. Now, for any
$k\in\mathbb{N}^{*}$, consider a geodesic $H_{k,t}$, $t\in[0,1]$, connecting
${\rm{Hilb}}_{k}(h^{L}_{0})$ and ${\rm{Hilb}}_{k}(h^{L}_{1})$ in the space of
Hermitian norms on $H^{0}(X,L^{k})$. Explicitly, if
$A_{k}\in\text{End}{(H^{0}(X,L^{k}))}$ is the self-adjoint map, relating the
scalar products on $H^{0}(X,L^{k})$ associated to ${\rm{Hilb}}_{k}(h^{L}_{0})$
and ${\rm{Hilb}}_{k}(h^{L}_{1})$ as
$\langle\cdot,\cdot\rangle_{L^{2}_{k}(X,h^{L}_{1})}=\langle
A_{k}\cdot,\cdot\rangle_{L^{2}_{k}(X,h^{L}_{0})},$ then $H_{k,t}$ is the
Hermitian norm associated to the scalar product $\langle
A^{t}_{k}\cdot,\cdot\rangle_{L^{2}_{k}(X,h^{L}_{0})}$. Consider the associated
graded norm $H_{t}\mathrel{\mathop{\ordinarycolon}}=\sum H_{k,t}$ on $R(X,L)$.

######  Theorem 1.8.

For any $t\in[0,1]$, the graded Hermitian norms $H_{t}$ and
${\rm{Hilb}}(h^{L}_{t})$ are equivalent.

Let us now explain briefly the ideas of the proofs of the above results and
their places in the context of previous works.

The central idea of our approach in the whole article is to give various
interpretations of the multiplication map on the section ring in terms of the
restriction morphisms. This is useful because the latter morphism can be
studied by the methods developed previously by author, [27], [28], [29].

Theorem 1.1 relies on the interpretation of the multiplication map as a
restriction map from the product manifold $X\times X$ to the diagonal $\Delta$
inside of it, see (3.4), and on a combination of several statements. First,
through Künneth theorem, one can view the space of holomorphic sections over
$X\times X$ as the tensor product of the spaces of holomorphic sections over
$X$, see (3.3). In Theorem 3.6, we then express using this isomorphism the
$L^{1}$, $L^{2}$ and $L^{\infty}$-norms on $X\times X$ in terms of projective,
Hermitian and injective tensor product norms respectively induced by the
$L^{1}$, $L^{2}$ and $L^{\infty}$-norms on $X$. Second, in Theorem 3.2, we
establish the lower bounds from Theorem 1.1 by relying on some ideas from the
proof of the semiclassical trace theorem from [27, §4.3] applied for the pair
of the product manifold $X\times X$ and the diagonal $\Delta$. The upper
bounds are then established in Theorem 3.4 using some tools developed
previously in the proof of the semiclassical Ohsawa-Takegoshi extension
theorem in [27], [28], [29], applied here again for the pair $(X\times
X,\Delta)$.

Theorem 1.5 is established using the result of Berman-Boucksom-Witt Nyström
[5, Theorem 1.14], cf. Theorem 2.9, about the Bernstein-Markov property of the
Monge-Ampère operator and the Ohsawa-Takegoshi extension theorem with the
uniform constant, cf. Theorem 2.4, applied to the product manifold and the
diagonal in it.

To show Theorem 1.6, we construct explicitly an inverse of the map
${\rm{Hilb}}$. This is done using the Fubini-Study operator which associates
through the Kodaira map for any norm $N_{k}$ on $H^{0}(X,L^{k})$ (for $k$ big
enough) a continuous psh metric over $L$, which we denote by
$FS(N_{k})^{\frac{1}{k}}$, see (2.30) for details. We study in Theorem 4.1 the
convergence of the Fubini-Study metrics on $L$ associated to a
multiplicatively generated norm $N$. The main content of Theorem 1.6 is to
show that when the graded Hermitian metric is multiplicatively generated, its
equivalence class is determined by its Fubini-Study metric. Note that the
existence of the limit of the Fubini-Study metric alone doesn’t determine the
equivalence class of a general graded Hermitian metric on $R(X,L)$, see
Section 4.4.

In realms of non-Archimedean geometry, a result, similar to Theorem 1.6, was
established by Boucksom-Jonsson [10, Theorem 6.1] and Reboulet [48, Theorem
A]. Remark that in the non-Archimedean setting, this statement is obtained as
a consequence of certain algebraic results, see [10, proof of Theorem 2.3].
One of them is the non-Archimedean GAGA principle, another is the study of
spectral radius of norms over reduced k-affinoid algebras. Our methods, on the
contrary, are purely analytic in nature and the only nontrivial ingredient in
our proof is the semiclassical version of the Ohsawa-Takegoshi extension
theorem from [27]. Interestingly enough, our proof of Theorem 1.6 is built
once again on an interpretation of the multiplication map as a version of the
restriction map, see (4.62). But now the restriction is done with respect to
the Kodaira embedding and not with respect to the diagonal in the product
manifold.

We prove Theorem 1.7 by standard techniques, essentially relying on the
Ohsawa-Takegoshi extension theorem. As we will show in Section 4.2 after
explaining the necessary definitions, this result can be restated in a form
which relates the limit of Goldman-Iwahori distance on the space of norms on
$H^{0}(X,L^{k})$, as $k\to\infty$, and the $L^{\infty}$-distance of the
Mabuchi weak geodesic in $\mathcal{H}^{L}$. Hence, the former distance should
be regarded as a quantization of the latter one.

This goes in line with the general philosophy that the geometry of
$\mathcal{H}^{L}$ can be approximated by the geometry on the space of norms on
$H^{0}(X,L^{k})$, as $k\to\infty$, see Donaldson [25] and Section 4.2 for a
more detailed review of related results.

The proof of Theorem 1.8, which goes further in this philosophy, is based on
the results of Phong-Sturm [44, Theorem 1] and Berndtsson [6, Theorem 1.2]
about the convergence of the Fubini-Study metric associated to $H_{t}$, on the
proof of Theorem 1.6 and on some elementary statements about the monotonicity
of geodesics in the space of Hermitian norms, which we establish by
interpolation theory from functional analysis. Theorem 1.8 refines the
mentioned result of Phong-Sturm and Berndtsson by Theorem 2.15, Lemma 4.12 and
Proposition 4.15.

This article is organized as follows. In Section 2, we introduce a set of
techniques which will be later used to compare various norms on the section
ring. In Section 3, we study the relation between metric and algebraic
structure of the section ring and establish Theorem 1.1. In Section 4, we
prove Theorem 1.5, give a classification of multiplicatively generated norms
on the section ring and study the metric properties of this set. In
particular, we establish Theorems 1.6, 1.7, 1.8. In Appendix A, we develop the
theory of almost sub-additive sequences, essential for Section 4. In Appendix
B, we recall some classical results about normed vector spaces and the induced
norms on the tensor products, which is of utmost importance to the study of
$L^{1}$ and $L^{\infty}$-norms.

Notation. For vector bundles $E,F$ over $X$, the vector bundle $E\boxtimes F$
over $X\times X$ is defined as the tensor product
$\pi_{1}^{*}E\otimes\pi_{2}^{*}F$, where $\pi_{1},\pi_{2}$ are the projections
$X\times X\to X$ on the first and the second components.

Let $\mu$ be an arbitrary measure on $X$. For a fixed Hermitian metric $h^{L}$
on $L$, we define the $L^{2}$-norm, which we denote by
$\|\cdot\|_{L^{2}_{k}(X,h^{L},\mu)}$ on $H^{0}(X,L^{k})$ in the same way as in
(1.4), but with the measure $dv_{X}$ replaced by $\mu$. When the metric $\mu$
is given by the symplectic volume form of a Kähler metric $\omega$, we denote
the above norm by $\|\cdot\|_{L^{2}_{k}(X,h^{L},\omega)}$. Similarly, we let
${\rm{Hilb}}_{k}(h^{L},\mu)$ and ${\rm{Hilb}}_{k}(h^{L},\omega)$ be the
restrictions of the above norms to $H^{0}(X,L^{k})$. Sometimes, to strengthen
the dependence on the manifold $X$, we denote the above metrics by
${\rm{Hilb}}^{X}_{k}(h^{L})$, etc. Similarly, we use the notations
${\rm{Ban}}^{1,X}(h^{L})$ and ${\rm{Ban}}^{\infty,X}(h^{L})$. When
$h^{L}\in\mathcal{H}^{L}$, and $\mu=\frac{1}{n!}c_{1}(F,h^{F})^{n}$ is
constructed using Bedford-Taylor extension of the Monge-Ampère operator, cf.
(2.18), we will omit $\mu$ from the notation and borrow the notation for psh
metrics from the smooth situation $\|\cdot\|_{L^{2}_{k}(X,h^{L})}$. Similarly,
we extend the $L^{1}$-norm and use the analogous notation.

A Hermitian metric $h^{F}$ on a vector bundle $F$ is called bounded if it can
be bounded from above and below by some smooth metrics. For a complex manifold
$X$, let $K_{X}$ be the canonical line bundle over it, given by
$\wedge^{n}(T^{(1,0)*}X)$. For a submanifold $Y$ in $X$, we denote by
${\rm{Res}}_{Y}$ the restriction operator from $X$ to $Y$.

Acknowledgement. We thank wholeheartedly Sébastien Boucksom for many
stimulating and fruitful discussions, which helped us to significantly improve
this article. We thank Rémi Reboulet for several discussions we had after the
first version of this paper was available. We also thank Yoshinori Hashimoto
for an interesting conversation about Section 4.4 and László Lempert for
allowing us to reproduce his unpublished example from Proposition 4.16. We,
finally, thank the colleagues and staff of École Polytechnique, CMLS and CNRS
for their support and wonderful working conditions.

Dedication. This article is dedicated to the memory of Jean-Pierre Demailly,
whose mathematics, teaching and personality have profoundly influenced me.

##  2 Extension theorem and norms on the section ring

The main goal of this section is to introduce a set of techniques to compare
various norms on the section ring. This will be intimately related with
Ohsawa-Takegoshi extension theorem, two versions of which we recall in Section
2.1, and to Bernstein-Markov property, which we recall in Section 2.2. Those
preliminaries will be used in Section 2.3. There we will recall the definition
of the Fubini-Study operator, which will be our main technical tool in Section
4, and recall that any continuous psh metric can be obtained through
approximation using this map.

###  2.1 Two versions of Ohsawa-Takegoshi extension theorem

The main goal of this section is to recall a version of Ohsawa-Takegoshi
extension theorem with uniform constant due to Demailly [23, Theorem 2.8] and
Ohsawa [43] and the semiclassical version due to the author [27].

In the first part of this section, we follow closely the presentation from
[23] and [21]. Only Theorem 2.5 seems to be new.

Let $Y$ be a complex submanifold of a compact complex manifold $X$. Let $F$ be
a holomorphic vector bundle $X$. Ohsawa-Takegoshi theorem addresses the
following extension problem. Given a holomorphic section $f$ of $F$ on $Y$, we
would like to find a holomorphic extension $\tilde{f}$ of $f$ to $X$, together
with a good estimate on the $L^{2}$-norm of $\tilde{f}$ in terms of the
$L^{2}$-norm of $f$. It turns out that it is always possible modulo some
positivity condition on the curvature of $F$. To state this result precisely,
we need to fix some notations first.

######  Definition 2.1.

We call a function
$\psi\mathrel{\mathop{\ordinarycolon}}X\to[-\infty,+\infty[$ on a complex
manifold $X$ quasi-plurisubharmonic (quasi-psh) if $\psi$ is locally the sum
of a psh function and of a smooth function (or equivalently, if
$\sqrt{-1}\partial\overline{\partial}\psi$ is locally bounded from below). In
addition, we say that $\psi$ has neat analytic singularities if every point
$x\in X$ possesses an open neighborhood $U$ on which $\psi$ can be written as

$\psi(z)=c\cdot\log\sum_{1\leq j\leq N}\big{|}g_{k}(z)\big{|}^{2}+w(z),$ (2.1)

where $c\geq 0$, $g_{k}$ are holomorphic over $U$ and $w$ is smooth.

######  Definition 2.2.

If $\psi$ is a quasi-psh function on a complex manifold $X$, the multiplier
ideal sheaf $\mathcal{J}(\psi)$ is the coherent analytic subsheaf of
$\mathscr{O}_{X}$, cf. [21, Proposition 5.7], defined by

$\mathcal{J}(\psi)\mathrel{\mathop{\ordinarycolon}}=\Big{\\{}f\in\mathscr{O}_{X,x};\quad\text{there
exists }U\ni x,\int_{U}|f|^{2}e^{-\psi}d\lambda<+\infty\Big{\\}}.$ (2.2)

where $U$ is an open coordinate neighborhood of $x$, and $d\lambda$ the
standard Lebesgue measure in the corresponding open chart of $\mathbb{C}^{n}$.
We say that the singularities of $\psi$ are log canonical singularities along
the zero variety $Y=V(\mathcal{J}(\psi))$ if
$\mathcal{J}(\psi)=\mathcal{J}_{Y}$, where $\mathcal{J}_{Y}$ is the sheaf of
holomorphic sections vanishing along $Y$.

Let us define a bump function
$\rho\mathrel{\mathop{\ordinarycolon}}\real\to[0,1]$ as follows

$\rho(x)=\begin{cases}1,\quad\text{for $|x|<\frac{1}{4}$},\\\ 0,\quad\text{for
$|x|>\frac{1}{2}$}.\end{cases}$ (2.3)

######  Lemma 2.3.

Let $X$ be a compact Kähler manifold of dimension $n$, and $\omega$ be a
Kähler form on $X$. Let $Y$ be a closed submanifold of $X$ of dimension $m$.
Then for $r>0$ small enough, the function
$\delta_{Y}\mathrel{\mathop{\ordinarycolon}}X\setminus Y\to\real$, defined as

$\delta_{Y}(x)\mathrel{\mathop{\ordinarycolon}}=2(n-m)\log\big{(}{\rm{dist}}_{X}(x,Y)\big{)}\cdot\rho\Big{(}\frac{{\rm{dist}}_{X}(x,Y)}{r}\Big{)},$
(2.4)

is quasi-psh. Moreover, it has neat analytic log canonical singularities along
$Y$.

###### Proof.

Remark first that only the statement about the quasi-plurisubharmonicity is
non-trivial. For the proof of a similar statement, which works for general
analytic submanifolds, see Demailly [20, Proposition 1.4]. For exactly this
statement, proved in a more general setting of manifolds of bounded geometry,
a reader may consult [27, Theorem 2.31]. ∎

Let us now consider quasi-psh function $\psi$ with log canonical singularities
on $X$ such that $Y\mathrel{\mathop{\ordinarycolon}}=V(\mathcal{J}(\psi))$ is
a submanifold. Let $\mu_{X}$ be a fixed measure on $X$ with smooth positive
density with respect to the Lebesgue measure on $X$. Let us associate in a
canonical way to it a measure on $Y$. We fix $g\in\mathscr{C}^{\infty}_{c}(Y)$
and let $\tilde{g}$ be a compactly supported continuous extension of $g$ on
$X$. We define the measure $\mu_{Y}[\psi]$ on $Y$ through the following limit

$\int_{Y}gd\mu_{Y}[\psi]\mathrel{\mathop{\ordinarycolon}}=\limsup_{t\to-\infty}\int_{U_{t}}\tilde{g}e^{-\psi}d\mu_{X},$
(2.5)

where $U_{t}\subset X$, $t\in\real$, is defined as
$U_{t}\mathrel{\mathop{\ordinarycolon}}=\\{x\in X;t<\psi(x)<t+1\\}$. It is
possible to see that the limit does not depend on the continuous extension
$\tilde{g}$, and that one gets in this way a measure with smooth positive
density with respect to the Lebesgue measure on $Y$, cf. [23, Proposition
4.5].

In the special case when $\mu$ corresponds to the Riemannian volume form
$dV_{X,\omega}$ of Kähler manifold $(X,\omega)$ and $\psi$ equals to
$\delta_{Y}$ from (2.4), an easy verification, cf. [23, (2.6)], shows that the
volume form $\mu_{Y}[\psi]$ is related to the Riemannian volume form
$dV_{Y,\omega|_{Y}}$ induced by $\omega|_{Y}$ as follows

$\mu_{Y}[\psi]\mathrel{\mathop{\ordinarycolon}}=\frac{2^{n-m+1}\cdot\pi^{n-m}}{(n-m-1)!}dV_{Y,\omega|_{Y}}.$
(2.6)

Let us now introduce the function
$\gamma\mathrel{\mathop{\ordinarycolon}}\real\to{}_{+}$ by the following
formula

$\gamma(x)=\begin{cases}\exp(-x),&\text{if }\,x>0,\\\
\frac{1}{1+4x^{2}}&\text{if }\,x\leq 0.\end{cases}$ (2.7)

######  Theorem 2.4 ( [23, Theorem 2.8, Remark 2.9b)] ).

Let $X$ be a compact Kähler manifold, and $\omega$ be a Kähler form on $X$.
Let $(L,h^{L})$ be a holomorphic line bundle equipped with a continuous
Hermitian metric $h^{L}$ on $X$, and let
$\psi\mathrel{\mathop{\ordinarycolon}}X\to[-\infty,+\infty[$ be a quasi-psh
function on $X$ with neat analytic singularities. Assume that the submanifold
$Y$ of $X$ defined by $Y=V(\mathcal{J}(\psi))$ is smooth and $\psi$ has log
canonical singularities. Finally, assume that the $(1,1)$-current

$c_{1}(L,h^{L})+\frac{\sqrt{-1}\alpha}{2\pi}\partial\overline{\partial}\psi,$
(2.8)

is non-negative for any $\alpha\in[1,2]$. Then for any section $f\in
H^{0}(Y,(L\otimes K_{X})|_{Y})$, there is a holomorphic extension
$\tilde{f}\in H^{0}(X,L\otimes K_{X})$ of $f$, such that the following
$L^{2}$-bound is satisfied

$\int_{X}\gamma(\psi)|\tilde{f}|_{\omega,h}^{2}e^{-\psi}dV_{X,\omega}\leq
17\cdot\int_{Y}|f|_{\omega,h}^{2}dV_{Y,\omega}[\psi],$ (2.9)

where the pointwise norm is calculated with respect to $h^{L}$ and $\omega$.

Now, in the context of this article, the canonical line bundle and the
condition (2.8) do not appear naturally. Due to this, we will state an easy
consequence of Theorem 2.4, which will be better adapted for our needs.

We introduce first the notations for the following theorem. Let $X$ be a
projective manifold, $\omega$ is a Kähler metric on $X$ and $L$ is a fixed
ample line bundle on $X$. Let $Y$ be a closed submanifold of $X$. Let $h^{L}$
be a continuous psh metric on $L$.

######  Theorem 2.5.

There are $C>0$, $p_{0}\in\mathbb{N}$, such that for any $k\in\mathbb{N}$,
$k\geq p_{0}$, and any section $f\in H^{0}(Y,L|_{Y}^{k})$, there is a
holomorphic extension $\tilde{f}\in H^{0}(X,L^{k})$ of $f$, such that the
following $L^{2}$-bound is satisfied

$\|\tilde{f}\|_{L^{2}_{k}(X,h^{L},\omega)}\leq
C\|f\|_{L^{2}_{k}(Y,h^{L},\omega)}.$ (2.10)

###### Proof.

Since $L$ is assumed to be ample, by Kodaira theorem it carries a smooth
positive metric $h^{L}_{0}$. By Lemma 2.3 and positivity of $h^{L}_{0}$, we
see that there is $p_{0}\in\mathbb{N}$, such that the following
$(1,1)$-current is non-negative

$p_{0}\cdot
c_{1}(L,h^{L}_{0})-c_{1}(K_{X},h^{K_{X}}_{\omega})+\frac{\sqrt{-1}\alpha}{2\pi}\partial\overline{\partial}\delta_{Y},$
(2.11)

for any $\alpha\in[1,2]$, where $h^{K_{X}}_{\omega}$ is the metric on $K_{X}$
induced by $\omega$. Hence for any $k\in\mathbb{N}$, we may apply Theorem 2.4
for quasi-psh function $\delta_{Y}$ and line bundle $L^{k}\otimes
L^{p_{0}}\otimes K_{X}^{-1}$, endowed with the metric
$h_{1}\mathrel{\mathop{\ordinarycolon}}=(h^{L})^{k}\otimes(h^{L}_{0})^{p_{0}}\otimes(h^{K_{X}}_{\omega})^{-1}$.
In particular, from Theorem 2.4 and (2.6) we see that for any $f\in
H^{0}(Y,L|_{Y}^{k+p_{0}})$, there is an extension $\tilde{f}\in
H^{0}(X,L^{k+p_{0}})$ of $f$ to $X$, such that the following estimate is
satisfied

$\int_{X}\gamma(\delta_{Y})|\tilde{f}|_{1}^{2}e^{-\delta_{Y}}dV_{X,\omega}\leq
17\cdot\int_{Y}|f|_{1}^{2}dV_{Y,\omega|_{Y}},$ (2.12)

where the pointwise norm $|\cdot|_{1}$ above is taken with respect to $h_{1}$.
Remark now that since $h^{L}$ is bounded, there is $C>0$, such that

$C^{-1}h^{L}_{0}\leq h^{L}\leq Ch^{L}_{0}.$ (2.13)

Now, remark that by taking $r>0$ from Lemma 2.3 small, we may achieve that
$\delta_{Y}$ is non-positive. Then by an easy fact that for $x>0$, we have
$\frac{e^{x}}{1+4x^{2}}\geq\frac{1}{17},$ we deduce the following inequality

$\gamma(\delta_{Y})e^{-\delta_{Y}}\geq\frac{1}{17}.$ (2.14)

By combining (2.12), (2.13) and (2.14), we obtain (2.10). ∎

In particular, the previous result recovers the well-known fact that the
restriction morphism

${\rm{Res}}_{Y}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k})\to
H^{0}(Y,L|_{Y}^{k}),$ (2.15)

is surjective. The following result refines this statement on the metric level
in more regular setting.

######  Theorem 2.6 ([27, Theorem 1.1]).

Let $h^{L}$ be a smooth positive metric on $L$. There are $C>0$,
$p_{1}\in\mathbb{N}$, such that under (2.15), for any $k\geq p_{1}$, the
following relation between the $L^{2}$-norm on the total manifold $X$ and the
$L^{2}$-norm on the submanifold $Y$ is satisfied

$1-\frac{C}{k}\leq\frac{[{\rm{Hilb}}^{X}_{k}(h^{L})]}{{\rm{Hilb}}^{Y}_{k}(h^{L})}\cdot
k^{\frac{n-m}{2}}\leq 1+\frac{C}{k}.$ (2.16)

######  Remark 2.7.

See also [29, Theorem 1.3] for a refinement on the level of holomorphic jets.

###  2.2 Bernstein-Markov property and comparison of natural norms

The main goal of this section is to compare several natural norms on the
section ring associated to psh metrics. We will first recall the classical
construction due to Bedford-Taylor of Monge-Ampère operator on positive
$(1,1)$-currents with bounded potentials, thus extending the domain of
operators ${\rm{Ban}}^{1}$, ${\rm{Hilb}}$ to continuous psh functions. Then we
recall several results concerning the Bernstein-Markov property of the
Bedford-Taylor version of Monge-Ampère operator.

Assume that a holomorphic line bundle $F$ over a complex manifold $Y$ is
endowed with a bounded metric $h^{F}$. We define the $(1,1)$-current
$c_{1}(F,h^{F})$ as follows

$c_{1}(F,h^{F})=c_{1}(F,h^{F}_{0})+\frac{\partial\overline{\partial}}{2\pi\sqrt{-1}}\Big{[}\log\Big{(}\frac{h^{F}}{h^{F}_{0}}\Big{)}\Big{]},$
(2.17)

where $h^{F}_{0}$ is any smooth Hermitian product on $F$, and by $[g]$ we mean
the distribution associated to a locally integrable function $g$. Poincaré-
Lelong equation, cf. [22, Theorem V.13.9], shows that such a definition is
independent of the choice of $h^{F}_{0}$.

Now, adapting the construction of Bedford-Taylor [3], cf. [22, §III.3], in our
situation, we can define inductively the positive $(k,k)$-currents
$c_{1}(F,h^{F})^{k}$, $k\in\mathbb{N}$, for any bounded psh Hermitian metric
$h^{F}$ as follows

$c_{1}(F,h^{F})^{k+1}=c_{1}(F,h^{F}_{0})\wedge
c_{1}(F,h^{F})^{k}+\frac{\partial\overline{\partial}}{2\pi\sqrt{-1}}\Big{[}\log\Big{(}\frac{h^{F}}{h^{F}_{0}}\Big{)}\cdot
c_{1}(F,h^{F})^{k}\Big{]}.$ (2.18)

The term under the brackets in (2.18) is well-defined as a $(k,k)$-current
since the logarithm function in it is bounded by our assumption on $h^{F}$ and
$c_{1}(F,h^{F})^{k}$ is represented by a non-negative $(k,k)$-form with
measure coefficients. By (2.18), we see that the De Rham class of
$c_{1}(F,h^{F})^{k}$ coincides with $c_{1}(F)^{k}$, where the De Rham
cohomology is extended to distributions in the usual way.

Following the standard terminology, cf. [41], [51] and [5], we say that the
pair $(h^{L},\mu)$ satisfies the Bernstein-Markov property, if for any
$\epsilon>0$ there is $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$,
the following relation between the $L^{2}$ and $L^{\infty}$-norms holds

$\|\cdot\|_{L^{2}_{k}(X,h^{L},\mu)}\geq\exp(-\epsilon
k)\cdot\|\cdot\|_{L^{\infty}_{k}(X,h^{L})}.$ (2.19)

Following [51] and [5, Definition 1.9], we say that the pair $(h^{L},\mu)$ is
Bernstein-Markov with respect to psh weights if and only if for any
$\epsilon>0$, there is $C>0$, such that for any $p\geq 1$ and any upper semi-
continuous function
$\psi\mathrel{\mathop{\ordinarycolon}}X\to[-\infty,\infty[$, such that the
$(1,1)$-current
$\frac{\sqrt{-1}}{\pi}\partial\overline{\partial}\psi+c_{1}(L,h^{L})$ is
positive, we have

$\int e^{p\psi}d\mu\geq C\exp(-\epsilon p)\cdot\sup_{x\in X}e^{p\psi}(x).$
(2.20)

######  Lemma 2.8.

If a pair $(h^{L},\mu)$ is Bernstein-Markov with respect to psh weights, then
it has Bernstein-Markov property. Moreover, in the estimate (2.19) one can
replace the $L^{2}$-norm by the $L^{1}$-norm.

###### Proof.

As it was observed in [5, after Remark 1.10], to prove the first part of the
statement, it suffices to take $p=2k$ and $\psi=\frac{1}{k}\log|s|$ in (2.20),
where $s\in H^{0}(X,L^{k})$. The Poincaré-Lelong equation, cf. [22, (13.2)],
implies that the above $\psi$ verifies the positivity requirement stated
before (2.20) and inequality (2.20) reduces to (2.19). The proof of the second
part of the statement is analogous to the proof of the first one, one only
needs to put $p=k$ in (2.20) instead. ∎

######  Theorem 2.9 (Berman-Boucksom-Witt Nyström [5, Proposition 1.13 and
Theorem 1.14]).

For any continuous psh metric $h^{L}$ on $L$, the pair
$(h^{L},\frac{1}{n!}c_{1}(L,h^{L})^{n})$ satisfies Bernstein-Markov property
with respect to psh weights.

We will also need the following result, philosophically similar to Theorem
2.9, yet simpler.

######  Proposition 2.10.

The pair $(h^{L},\mu)$ satisfies Bernstein-Markov property with respect to psh
weights for any continuous psh metrics $h^{L}$ on $L$ and any measure $\mu$,
majorized from below by some strictly positive volume form on $X$.

###### Proof.

It follows directly from [5, Theorem 1.14] or from mean-value inequality. ∎

######  Corollary 2.11.

For any $h^{L}\in\mathcal{H}^{L}$, the following equivalences between the
graded norms on $R(X,L)$ hold

${\rm{Ban}}^{1}(h^{L})\sim{\rm{Hilb}}(h^{L})\sim{\rm{Ban}}^{\infty}(h^{L}).$
(2.21)

In particular, the maps (1.15) are all identical. Also, for any measure $\mu$,
majorized from below by some strictly positive volume form on $X$ and having
finite volume over $X$, we have

${\rm{Hilb}}(h^{L})\sim{\rm{Hilb}}(h^{L},\mu).$ (2.22)

###### Proof.

From the fact that the De Rham class of $c_{1}(L,h^{L})^{n}$ is fixed, we
readily deduce that

$\displaystyle{\rm{Ban}}^{1}(h^{L})\leq{\rm{Ban}}^{\infty}(h^{L})\cdot\frac{1}{n!}c_{1}(L)^{n},$
(2.23)
$\displaystyle{\rm{Hilb}}(h^{L})\leq{\rm{Ban}}^{\infty}(h^{L})\cdot\sqrt{\frac{1}{n!}c_{1}(L)^{n}},$
$\displaystyle{\rm{Hilb}}(h^{L},\mu)\leq{\rm{Ban}}^{\infty}(h^{L})\cdot\sqrt{{\rm{vol}}(\mu)}.$

This along with Theorem 2.9, Lemma 2.8 and Proposition 2.10 imply the result.
∎

As a sidenote, let us establish the following corollary. Compare it with
Theorem 2.6.

######  Corollary 2.12.

Let $h^{L}$ be a continuous psh metric on an ample line bundle $L$. Then under
(2.15), the following equivalence of norms holds

$[{\rm{Hilb}}^{X}(h^{L})]\sim{\rm{Hilb}}^{Y}(h^{L}).$ (2.24)

###### Proof.

From Corollary 2.11 and the obvious fact
$[{\rm{Ban}}^{\infty,X}(h^{L})]\geq{\rm{Ban}}^{\infty,Y}(h^{L})$, stipulating
that the $L^{\infty}$-norm of a section is always no smaller than the
$L^{\infty}$-norm of its restriction, we conclude that for any $\epsilon>0$
there is $p_{1}\in\mathbb{N}$, such that for $k\geq p_{1}$, we have

$[{\rm{Hilb}}^{X}(h^{L})]\geq\exp(-\epsilon k)\cdot{\rm{Hilb}}^{Y}(h^{L}).$
(2.25)

Now, following the terminology of Theorem 2.5, we can restate Theorem 2.5 in
the following way. There are $C>0$, $p_{1}\in\mathbb{N}$, such that for any
$k\geq p_{1}$, we have

${\rm{Hilb}}^{Y}(h^{L},\omega)\geq C[{\rm{Hilb}}^{X}(h^{L},\omega)].$ (2.26)

From Corollary 2.11, we see that it implies that for any $\epsilon>0$, there
is $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, the following
inequality is satisfied

${\rm{Hilb}}^{Y}(h^{L})\geq\exp(-\epsilon k)\cdot[{\rm{Hilb}}^{X}(h^{L})].$
(2.27)

But Corollary 2.12 means exactly that both inequalities (2.25) and (2.27) are
satisfied. ∎

######  Remark 2.13.

a) From Corollary 2.11, we see that the analogous result holds for $L^{1}$ and
$L^{\infty}$-norms.

b) For positive smooth metrics, the upper bound from (2.27) was independently
obtained by Randriam [46, Theorem 3.1.10] in a more general setting of reduced
submanifold (i.e. for holomorphic jets), see also Remark 2.7. The
$L^{\infty}$-version of this result, working more generally for certain non-
compact manifolds $X$ and non-smooth $Y$ but only for strictly positive
singular metrics $h^{L}$ was established by Randriam [47].

###  2.3 Fubini-Study operator and continuous psh metrics

In this section we recall the definition of the Fubini-Study operator. We will
then show that the Hilbert map can be used to approximate any continuous psh
metric.

First of all, recall that for $k$ sufficiently large so that $L^{k}$ is very
ample, Fubini-Study operator associates for any Hermitian norm
$N_{k}=\|\cdot\|_{k}$ on $H^{0}(X,L^{k})$, a positive metric $FS(N_{k})$ on
$L$, constructed in the following way. Consider the Kodaira embedding

${\rm{Kod}}_{k}\mathrel{\mathop{\ordinarycolon}}X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{*}),$
(2.28)

defined as $x\mapsto H^{0}(X,L^{k}\otimes\mathcal{J}_{x})$, where
$\mathcal{J}_{x}$ is the ideal sheaf of holomorphic functions vanishing at
$x$, and we implicitly used the identification between the hyperplanes in
$H^{0}(X,L^{k})$ and lines in $H^{0}(X,L^{k})^{*}$. The evaluation maps,
$ev_{x}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k})\to L^{k}_{x}$, $x\in
X$, provide the isomorphism

$L^{-k}\to{\rm{Kod}}_{k}^{*}\mathscr{O}(-1),$ (2.29)

where $\mathscr{O}(-1)$ is the tautological bundle over the projective space
$\mathbb{P}(H^{0}(X,L^{k})^{*})$. Let us now endow $H^{0}(X,L^{k})^{*}$ with
the dual norm $N_{k}^{*}$ and induce from it a metric $h^{FS}(N_{k})$ on the
tautological line bundle $\mathscr{O}(-1)$ over
$\mathbb{P}(H^{0}(X,L^{k})^{*})$. We define the metric $FS(N_{k})$ on $L^{k}$
through

$FS(N_{k})={\rm{Kod}}_{k}^{*}(h^{FS}(N_{k})^{*}).$ (2.30)

A statement below can be seen as an alternative definition.

######  Lemma 2.14.

For any $x\in X$, the metric $FS(N_{k})$ satisfies the identity

$\sum_{i=1}^{l}\big{|}s_{i}(x)\big{|}^{2}_{FS(N_{k})}=1,$ (2.31)

where $s_{1},\ldots,s_{l}$ is an orthonormal basis of
$(H^{0}(X,L^{k}),N_{k})$.

###### Proof.

An easy verification, cf. Ma-Marinescu [38, Theorem 5.1.3]. ∎

Explicit evaluation shows that $c_{1}(\mathscr{O}(-1),h^{FS}(N_{k}))$
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.

We will now discuss a result concerning quantization of continuous psh
metrics. For it, we claim no originality, but despite a huge amount of
literature, we weren’t able to find a precise reference for it. We present the
proof below for the convenience of the reader.

######  Theorem 2.15.

For any continuous psh metric $h^{L}$ on $L$, the sequence of metrics
$FS({\rm{Hilb}}_{k}(h^{L}))^{\frac{1}{k}}$ on $L$ converges uniformly, as
$k\to\infty$, to $h^{L}$.

######  Remark 2.16.

Theorem 2.15 generalizes the result of Tian [55], who proved the above
statement for smooth positive metric $h^{L}$ on $L$. Moreover, he proved in
this case that $FS({\rm{Hilb}}_{k}(h^{L}))^{\frac{1}{k}}$ converges to $h^{L}$
in $\mathscr{C}^{2}$-sense instead of $\mathscr{C}^{0}$. Later this was
refined by Zelditch [57], who proved that the convergence is
$\mathscr{C}^{\infty}$. For other related works, see Bouche [8], Catlin [11],
Lu [37], Wang [56] and Dai-Liu-Ma [15].

For the proof of Theorem 2.15 and later on, we will need the following
statement, which is a trivial consequence of Theorem 2.5.

######  Lemma 2.17.

Assume that the measure $\mu$ is bounded from above by a smooth volume form on
$X$. Then for any continuous psh metric $h^{L}$ on an ample line bundle $L$,
there are $C>0$, $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, $x\in
X$, there is $f\in H^{0}(X,L^{k})$, verifying

$|f(x)|\geq C\|f\|_{L^{2}_{k}(X,h^{L},\mu)},$ (2.32)

where the norm of $f(x)$ is taken with respect to $h^{L}$.

###### Proof of Theorem 2.15..

An easy verification, following from Lemma 2.14, shows that

$\frac{FS({\rm{Hilb}}_{k}(h^{L}))}{(h^{L})^{k}}=\rho_{k},$ (2.33)

where the function $\rho_{k}\mathrel{\mathop{\ordinarycolon}}X\to[0,+\infty[$
is defined as follows

$\rho_{k}(x)\mathrel{\mathop{\ordinarycolon}}=\sup\frac{|s(x)|}{\|s\|_{L^{2}_{k}(X,h^{L})}}.$
(2.34)

By Theorem 2.9 and Proposition 2.10, we see that for any $\epsilon>0$, there
is $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, we have

$\rho_{k}(x)\leq\exp(\epsilon k).$ (2.35)

Now, a combination of Theorem 2.9 and (2.23) implies that for any
$\epsilon>0$, there is $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$,
we have

$\|\cdot\|_{L^{2}_{k}(X,h^{L},\mu)}\geq\exp(-\epsilon
k)\cdot\|\cdot\|_{L^{2}_{k}(X,h^{L})}.$ (2.36)

From Lemma 2.17 and (2.36), we conclude that for any $\epsilon>0$, there is
$p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, we have

$\rho_{k}(x)\geq\exp(-\epsilon k).$ (2.37)

From (2.33), (2.35) and (2.37), we deduce Theorem 2.15. ∎

##  3 Synergy of metric and algebraic structures on section ring

The main goal of this section is to study the relation between metric and
algebraic structures of the section ring. In particular, we prove Theorem 1.1.

More precisely, this section is organized as follows. In Section 3.1, we make
a connection between the restriction map and the multiplication morphism and
prove Theorem 1.1 modulo some technical statements. Those statements will be
then established in Sections 3.2, 3.3 and 3.4.

###  3.1 Multiplication map as a restriction morphism, a proof of Theorem 1.1

The main goal of this section is to give a proof of Theorem 1.1 modulo some
technical results, which will be later established in this article. Our main
idea is to interpret the multiplication map of the section ring (1.2) as an
instance of the restriction morphism. This will allow us to use some of the
methods developed in [27], [28], [29] to prove Theorem 1.1.

More precisely, we conserve the notation from Section 1. Let us consider the
product manifold $X\times X$ and the diagonal submanifold in it, given by
$\\{(x,x)\mathrel{\mathop{\ordinarycolon}}x\in
X\\}=\mathrel{\mathop{\ordinarycolon}}\Delta\hookrightarrow X\times X$.
Clearly, we have a natural isomorphism

$\Delta\to X,\qquad(x,x)\mapsto x.$ (3.1)

Also, under the isomorphism (3.1), we have

$L^{k}\boxtimes L^{l}|_{\Delta}\to L^{k+l}.$ (3.2)

Künneth theorem allows us to interpret the tensor product of cohomologies as
the cohomology of the product manifold. In other words, it provides us with
the following isomorphism

$H^{0}(X\times X,L^{k}\boxtimes L^{l})\to H^{0}(X,L^{k})\otimes
H^{0}(X,L^{l}).$ (3.3)

A basic observation of this section is that those isomorphisms can be put into
the following commutative diagram

$\begin{CD}H^{0}(X\times X,L^{k}\boxtimes
L^{l})@>{{\rm{Res}}_{\Delta}}>{}>H^{0}(\Delta,L^{k}\boxtimes
L^{l}|_{\Delta})\\\ @V{}V{{}}V@V{}V{{}}V\\\ H^{0}(X,L^{k})\otimes
H^{0}(X,L^{l})@>{{\rm{Mult}}_{k,l}}>{}>H^{0}(X,L^{k+l}).\end{CD}$ (3.4)

A similar observation was already used by Demailly-Ein-Lazarsfeld [24, proof
of Theorem 2.3] in the proof of subadditivity theorem for multiplier ideals.

We will now explain how this observation allows us to establish Theorem 1.1.
Before that, let us use it to prove the following well-known result.

######  Proposition 3.1.

There is $p_{0}\in\mathbb{N}$, such that for any $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\in\mathbb{N}^{*}$, $k_{1}+\cdots+k_{r}=k$,
$k_{1},\cdots,k_{r}\geq p_{0}$, the map ${\rm{Mult}}_{k_{1},\cdots,k_{r}}$ is
surjetive.

###### Proof.

Clearly, it is enough to establish the result for $r=2$. We borrow the
notation from Section 2.1. We have a short exact sequence of sheaves

$0\to\mathscr{O}_{X\times X}(L^{k}\boxtimes
L^{l})\otimes\mathcal{J}_{\Delta}\to\mathscr{O}_{X\times X}(L^{k}\boxtimes
L^{l})\xrightarrow[]{{\rm{Res}}_{\Delta}}\mathscr{O}_{\Delta}((L^{k}\boxtimes
L^{l})|_{\Delta})\to 0.$ (3.5)

The associated long exact sequence in cohomology gives us

$\cdots\to H^{0}(X\times X,L^{k}\boxtimes
L^{l})\xrightarrow[]{{\rm{Res}}_{\Delta}}H^{0}(\Delta,(L^{k}\boxtimes
L^{l})|_{\Delta})\\\ \to H^{1}(X\times X,L^{k}\boxtimes
L^{l}\otimes\mathcal{J}_{\Delta})\to\cdots.$ (3.6)

Clearly, from (3.4), to prove the surjectivity of ${\rm{Mult}}_{k,l}$, it is
enough to establish the surjectivity of ${\rm{Res}}_{\Delta}$. From (3.6), we
see that for the surjectivity of ${\rm{Res}}_{\Delta}$ it is enough to
establish

$H^{1}(X\times X,L^{k}\boxtimes L^{l}\otimes\mathcal{J}_{\Delta})=\\{0\\},$
(3.7)

which follows directly from Nadel vanishing theorem [21, Theorem 5.11] and
Lemma 2.3. ∎

Next we will explain that (3.4) allows us to use our previous ideas from [27],
[28] and [29], studying the general restriction maps, to prove Theorem 1.1. We
decompose our proof into a series of more tractable statements. By the
isomorphism (3.3), we can endow the right-hand side of (3.3) with the natural
norms $\|\cdot\|_{L^{q}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}$,
$q=1,2,\infty$, constructed by the restriction of the corresponding
$L^{q}$-norms associated to the symplectic volume form of $\omega_{X\times X}$
to the vector space of holomorphic sections. Those norms will be of utmost
importance.

By using the interpretation (3.4) of the multiplication map as an instance of
restriction morphism and relying on some ideas from the proof of the
semiclassical trace theorem from [27, §4.3], we establish in Section 3.3 the
following result.

######  Theorem 3.2.

There are $C>0$, $p_{1}\in\mathbb{N}$, such that for any $k,l\geq p_{1}$,
$f\in H^{0}(X,L^{k})$, $g\in H^{0}(X,L^{l})$, we have

$\displaystyle\big{\|}f\cdot
g\big{\|}_{{\rm{Hilb}}_{k+l}(h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{\frac{n}{2}}\cdot\big{\|}f\big{\|}_{{\rm{Hilb}}_{k}(h^{L})}\cdot\big{\|}g\big{\|}_{{\rm{Hilb}}_{l}(h^{L})},$
(3.8) $\displaystyle\big{\|}f\cdot
g\big{\|}_{{\rm{Ban}}_{k+l}^{1}(h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\cdot
4^{n}\cdot\big{\|}f\big{\|}_{{\rm{Ban}}_{k}^{1}(h^{L})}\cdot\big{\|}g\big{\|}_{{\rm{Ban}}_{l}^{1}(h^{L})}.$

More precisely, we have the following estimates for operator norms

$\displaystyle\big{\|}{\rm{Mult}}_{k,l}\big{\|}_{L^{2}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})\to
L^{2}_{k+l}(X,h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{\frac{n}{2}},$ (3.9)
$\displaystyle\big{\|}{\rm{Mult}}_{k,l}\big{\|}_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})\to
L^{1}_{k+l}(X,h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\cdot 4^{n},$

where $\|\cdot\|_{L^{q}_{k,l}(X\times X,h^{L}\boxtimes h^{L})\to
L^{q}_{k+l}(X,h^{L})}$, $q=1,2,\infty$, is the operator norm of an operator
from $H^{0}(X,L^{k})\otimes H^{0}(X,L^{l})$ to $H^{0}(X,L^{k+l})$, calculated
when the domain is endowed with the norm $\|\cdot\|_{L^{q}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}$ and the image is endowed with the norm
$\|\cdot\|_{L^{q}_{k+l}(X,h^{L})}$.

######  Remark 3.3.

a) As the $L^{\infty}$-norm of a product of two sections is no bigger than the
product of the $L^{\infty}$-norms of the sections, we trivially have the
identity $\big{\|}{\rm{Mult}}_{k,l}\big{\|}_{L^{\infty}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})\to L^{\infty}_{k+l}(X,h^{L})}\leq 1$.

b) The constant $4^{n}$ in the second estimates from (3.8) and (3.9) is not
optimal. It is possible to improve it through our methods, but as this
improvement will not affect qualitatively the final result, we decided to omit
it for brevity.

c) In the proof we show that the first estimate from (3.9) is sharp.

By using again the interpretation of the multiplication map as a version of
the restriction morphism, cf. (3.4), and establishing a version of asymptotic
Ohsawa-Takegoshi theorem, following the methods from [27], [28], we prove in
Section 3.4 the following result.

######  Theorem 3.4.

There are $C>0$, $p_{0}\in\mathbb{N}$, such that for any $k,l\geq p_{0}$,
$f\in H^{0}(X,L^{k+l})$, there is $h\in H^{0}(X,L^{k})\otimes H^{0}(X,L^{l})$,
verifying ${\rm{Mult}}_{k,l}(h)=f$ and

$\displaystyle\big{\|}h\big{\|}_{L^{1}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n}\cdot 2^{n}\cdot\big{\|}f\big{\|}_{{\rm{Ban}}_{k+l}^{1}(h^{L})},$
(3.10) $\displaystyle\big{\|}h\big{\|}_{L^{2}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{\frac{n}{2}}\cdot\big{\|}f\big{\|}_{{\rm{Hilb}}_{k+l}(h^{L})},$
$\displaystyle\big{\|}h\big{\|}_{L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\big{\|}f\big{\|}_{{\rm{Ban}}_{k+l}^{\infty}(h^{L})}.$

######  Remark 3.5.

From Theorem 3.2, we see that the second and the third estimates in (3.10) are
optimal. Through some additional work, one can see that the first estimate is
optimal as well.

From Theorems 3.2 and 3.4, a natural question arises if it is possible to
interpret the norms $L^{q}_{k,l}(X\times X,h^{L}\boxtimes h^{L})$,
$k,l\in\mathbb{N}$, $q=1,2,\infty$, in terms of the corresponding norms
$L^{q}_{k}(X,h^{L})$, $k\in\mathbb{N}$. The following result, which we
establish in Section 3.2, shows that this is indeed the case.

######  Theorem 3.6.

The following estimate between the norms on $H^{0}(X,L^{k})\otimes
H^{0}(X,L^{l})$ holds

$\frac{1}{4^{n}}\cdot{\rm{Ban}}_{k}^{1}(h^{L})\otimes_{\pi}{\rm{Ban}}_{l}^{1}(h^{L})\leq\|\cdot\|_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes
h^{L})}\leq{\rm{Ban}}_{k}^{1}(h^{L})\otimes_{\pi}{\rm{Ban}}_{l}^{1}(h^{L}).$
(3.11)

Also, the following identity holds

$\|\cdot\|_{L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}={\rm{Ban}}_{k}^{\infty}(h^{L})\otimes_{\epsilon}{\rm{Ban}}_{l}^{\infty}(h^{L}).$
(3.12)

######  Remark 3.7.

Clearly, the following identity

$\langle f_{1}\boxtimes g_{1},f_{2}\boxtimes g_{2}\rangle_{L^{2}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}=\langle
f_{1},f_{2}\rangle_{L^{2}_{k}(X,h^{L})}\cdot\langle
g_{1},g_{2}\rangle_{L^{2}_{l}(X,h^{L})},$ (3.13)

shows that ${\rm{Hilb}}_{k}(h^{L})\otimes{\rm{Hilb}}_{l}(h^{L})$ coincides
with the norm $L^{2}_{k,l}(X\times X,h^{L}\boxtimes h^{L})$.

###### Proof of Theorem 1.1.

Once we unravel the definition of the quotient norm, we conclude directly from
Theorems 3.2 and 3.4 that there are $C>0$, $p_{1}\in\mathbb{N}$, such that for
any $k,l\geq p_{1}$, we have

$\displaystyle\frac{1}{4^{n}}\cdot\Big{(}1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\leq$
$\displaystyle\frac{[\|\cdot\|_{L^{1}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}]}{{\rm{Ban}}_{k+l}^{1}(h^{L})}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\cdot\frac{1}{2^{n}}$ (3.14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)},$
$\displaystyle 1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\leq$
$\displaystyle\frac{[\|\cdot\|_{L^{2}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}]}{{\rm{Hilb}}_{k+l}(h^{L})}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{\frac{n}{2}}\leq 1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)},$
$\displaystyle 1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\leq$
$\displaystyle\frac{[\|\cdot\|_{L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}]}{{\rm{Ban}}_{k+l}^{\infty}(h^{L})}\leq
1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}.$

The result now follows from Theorem 3.6 and (3.14). ∎

###  3.2 Norms on the product manifold, injective and projective tensor norms

The main goal of this section is to study the functional-analytic properties
of the normed spaces $(H^{0}(X,L^{k}),{\rm{Ban}}_{k}^{1}(h^{L}))$,
$(H^{0}(X,L^{k}),{\rm{Ban}}_{k}^{\infty}(h^{L}))$. From this analysis, we
establish Theorem 3.6.

Our main idea of the proof of Theorem 3.6 is to first use the general theory
of injective tensor products to settle down the corresponding part of Theorem
3.6. Then we will obtain the version for the projective tensor products from
the already established version for the injective tensor products and a
careful study of duality on the normed spaces
$(H^{0}(X,L^{k}),{\rm{Ban}}_{k}^{1}(h^{L}))$,
$(H^{0}(X,L^{k}),{\rm{Ban}}_{k}^{\infty}(h^{L}))$. More precisely, the main
result of this section goes as follows.

######  Theorem 3.8.

There are constants $c>0$, $p_{0}\in\mathbb{N}$, such that for any $k\geq
p_{0}$, the following estimate between the norms on $H^{0}(X,L^{k})$ holds

$\frac{1}{2^{n}}\cdot\Big{(}1-\frac{C}{k}\Big{)}\cdot{\rm{Ban}}_{k}^{1}(h^{L})\leq({\rm{Ban}}_{k}^{\infty}(h^{L}))^{*}\leq{\rm{Ban}}_{k}^{1}(h^{L}),$
(3.15)

where we implicitly identified $H^{0}(X,L^{k})$ and $H^{0}(X,L^{k})^{*}$ using
the $L^{2}$-scalar product. Similarly, for the norms over $H^{0}(X\times
X,L^{k}\boxtimes L^{l})$, under a similar identification, we have the
following comparison

$\frac{1}{4^{n}}\cdot\Big{(}1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot\|\cdot\|_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}\leq\|\cdot\|_{L^{\infty}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}^{*}\leq\|\cdot\|_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}.$ (3.16)

######  Remark 3.9.

We remark that on the left-hand side of (3.15), the constant $\frac{1}{2^{n}}$
coincides with the analogous constant obtained in the study of dual of
$L^{p}$-Bargmann space by Gryc-Kemp [31, Theorem 1.2], cf. also Janson-Peetre-
Rochberg [33] and Zhu [58] for related results. It seems, however, that the
constant is not optimal, see [32, Theorem 1.5 and discussion before].

Before establishing Theorem 3.8, let us see how it implies Theorem 3.6.

###### Proof of Theorem 3.6..

Let us first establish the result for the ${\rm{Ban}}^{\infty}$ norm. Remark
that we have the following isometric embedding

$(H^{0}(X,L^{k}),{\rm{Ban}}_{k}^{\infty}(h^{l}))\hookrightarrow(\mathscr{C}^{0}(X,L^{k}),\|\cdot\|_{L^{\infty}_{k}(X,h^{L})}).$
(3.17)

The statement (3.12) now follows from Lemmas B.5 and B.9.

Let us establish the upper bound of (3.11). For any $f\in H^{0}(X,L^{k})$,
$g\in H^{0}(X,L^{l})$, we have

$\|f\otimes g\|_{L^{1}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}=\|f\|_{L^{1}_{k}(X,h^{L})}\cdot\|g\|_{L^{1}_{l}(X,h^{L})}.$ (3.18)

By the triangle inequality and the definition of the projective tensor norm,
the above inequality implies the upper bound of (3.11).

Now, by taking the dual of (3.12) and using Lemma B.3, we obtain

$\|\cdot\|_{L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}^{*}={\rm{Ban}}_{k}^{\infty}(h^{L})^{*}\otimes_{\pi}{\rm{Ban}}_{l}^{\infty}(h^{L})^{*}.$
(3.19)

The lower bound of (3.11) now follows directly from Theorem 3.8, monotonicity
of the projective tensor norm and (3.19). ∎

To establish Theorem 3.8, we need to recall some basic properties of the
Bergman projector $B_{k}^{X}$, which is the orthogonal projection from
$L^{2}_{k}(X,h^{L})$ to $H^{0}(X,L^{k})$. Recall that the Bergman kernel is
defined as the Schwartz kernel, $B_{k}^{X}(x_{1},x_{2})\in
L^{k}_{x_{1}}\otimes(L^{k}_{x_{2}})^{*}$, $x_{1},x_{2}\in X$, of the Bergman
projection $B_{k}^{X}$, evaluated with respect to the Riemannian volume form
$dv_{X}$, i.e.

$(B_{k}^{X}s)(x_{1})=\int_{X}B_{k}^{X}(x_{1},x_{2})\cdot
s(x_{2})dv_{X}(x_{2}),\qquad s\in L^{2}_{k}(X,h^{L}).$ (3.20)

The first result we will need concerns the study of the Bergman kernel for
distant parameters. It says that the Bergman kernel has an exponential decay.

######  Theorem 3.10 (Ma-Marinescu [39, Theorem 1]).

There are $c,C>0$, $p_{1}\in\mathbb{N}^{*}$ such that for any $k\geq p_{1}$,
$x_{1},x_{2}\in X$, we have

$\Big{|}B_{k}^{X}(x_{1},x_{2})\Big{|}\leq
Ck^{n}\cdot\exp\big{(}-c\sqrt{k}\cdot{\rm{dist}}(x_{1},x_{2})\big{)}.$ (3.21)

The next result we will need is about the near-diagonal behavior of the
Bergman kernel. To state it, we will need to fix some further notation.

We will first introduce a special coordinate system in a neighborhood of a
given point. Let us fix $x_{0}\in X$ and an orthonormal frame
$(e_{1},\ldots,e_{2n})$ of $(T_{x_{0}}X,g^{TX}_{x_{0}})$. We identify
$Z\in{}^{2n}$ with an element from $T_{x_{0}}X$ using this frame and define
the exponential coordinates
$\phi_{x_{0}}^{X}\mathrel{\mathop{\ordinarycolon}}{}^{2n}\to X$, $x_{0}\in X$,
as follows

$\phi_{x_{0}}^{X}(Z)\mathrel{\mathop{\ordinarycolon}}=\exp^{X}_{x_{0}}(Z).$
(3.22)

Let us now construct an orthonormal trivialization of $(L,h^{L})$ in a
neighborhood of $x_{0}$. We fix an orthonormal vector $l_{0}\in L_{x_{0}}$,
and define the local orthonormal frame $\tilde{l}^{\prime}_{0}$ of $L$ around
$x_{0}$ by the parallel transport of $l_{0}$ with respect to the Chern
connection $\nabla^{L}$, done along the curve $\phi_{x_{0}}^{X}(tZ)$,
$t\in[0,1]$, $Z\in T_{x_{0}}X$, $|Z|<r_{X}$, where $r_{X}$ is the infimum of
the injectivity radius of $X$. Now, for $Z,Z^{\prime}\in{}^{2n}$,
$|Z|,|Z^{\prime}|\leq\epsilon$, we will denote by
$B_{k}^{X}\big{(}\phi_{x_{0}}(Z),\phi_{x_{0}}(Z^{\prime})\big{)}\in\real$ the
Bergman kernel, evaluated at $(\phi_{x_{0}}(Z),\phi_{x_{0}}(Z^{\prime}))$, and
viewed as a real number using the frame $\tilde{l}^{\prime}_{0}$.

Define the function $\mathscr{P}_{n}$, cf. [38, (4.1.84)], as follows

$\mathscr{P}_{n}(Z,Z^{\prime})=\exp\Big{(}-\frac{\pi}{2}\sum_{i=1}^{n}\big{(}|z_{i}|^{2}+|z^{\prime}_{i}|^{2}-2z_{i}\overline{z}^{\prime}_{i}\big{)}\Big{)},\quad\text{for
}Z,Z^{\prime}\in\mathbb{C}^{n}.$ (3.23)

From [38, §4.1.6], $\mathscr{P}_{n}$ is the Schwartz kernel of the orthogonal
projection from the space of $L^{2}$-integrable functions with respect to the
Gaussian measure onto the Bargmann space.

Next result shows that the behavior of the Bergman kernel on any manifold $X$
for close parameters is governed by the function $\mathscr{P}_{n}$.

######  Theorem 3.11.

There are $\epsilon,c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any
$x_{0}\in X$, $k\geq p_{1}$, $Z,Z^{\prime}\in{}^{2n}$,
$|Z|,|Z^{\prime}|\leq\epsilon$, the following estimate holds

$\bigg{|}\frac{1}{k^{n}}B_{k}^{X}\big{(}\phi_{x_{0}}^{X}(Z),\phi_{x_{0}}^{X}(Z^{\prime})\big{)}-\mathscr{P}_{n}(\sqrt{k}Z,\sqrt{k}Z^{\prime})\bigg{|}\\\
\leq
Ck^{-1}\Big{(}1+\sqrt{k}|Z|+\sqrt{k}|Z^{\prime}|\Big{)}^{3n+2}\exp\Big{(}-c\sqrt{k}|Z-Z^{\prime}|\Big{)}.$
(3.24)

###### Proof.

The result follows from the first two terms of the asymptotic expansion of
Dai-Liu-Ma [15] and a result of Ma-Marinescu [38, Remark 4.1.26], which states
that the second term of the asymptotic expansion vanishes. The $\kappa$
function, see [38, (4.1.28)] for a definition, doesn’t appear in our expansion
since our expansion only uses the two first terms of the expansion of Dai-Liu-
Ma, and according to [38, (1.2.19)], the appearance of the $\kappa$-term
doesn’t affect them. ∎

We are now ready to draw the first consequence of those statements.

######  Proposition 3.12.

For $k\in\mathbb{N}$ big enough, the operator $B_{k}^{X}$ extends using the
Bergman kernel as a bounded operator to the space of $L^{p}$-integrable
sections of $L^{k}$, for any $p\in[1,+\infty]$. Moreover, there is $C>0$, such
that for any $k\in\mathbb{N}^{*}$, $p\in[1,+\infty]$, the following bound
holds

$\big{\|}B_{k}^{X}\big{\|}_{L^{p}_{k}(X,h^{L})\to L^{p}_{k}(X,h^{L})}\leq
2^{n}\Big{(}1+\frac{C}{k}\Big{)},$ (3.25)

where $\|\cdot\|_{L^{p}_{k}(X,h^{L})\to L^{p}_{k}(X,h^{L})}$ is the operator
norm of an operator from the space of $L^{p}$-integrable sections of $L^{k}$
to itself.

######  Remark 3.13.

Similar results were obtained for Bargmann space, see [33]. In fact, according
to [31, Theorem 1.1], the constant $2^{n}$ in this case is sharp for $p=1$.

###### Proof.

From Theorems 3.10, 3.11 and (3.48), we conclude that there is $C>0$, such
that for any $x_{0}\in X$, for $k\geq p_{1}$, we have

$\displaystyle\int_{X}\big{|}B_{k}^{X}(x_{0},x)\big{|}dv_{X}(x)\leq
2^{n}\Big{(}1+\frac{C}{k}\Big{)},$ (3.26)
$\displaystyle\int_{X}\big{|}B_{k}^{X}(x,x_{0})\big{|}dv_{X}(x)\leq
2^{n}\Big{(}1+\frac{C}{k}\Big{)}.$

Directly from (3.26) and Young’s inequality for integral operators, cf. [52,
Theorem 0.3.1] applied for $q=p$, $r=1$ in the notations of [52], we deduce
(3.25). ∎

###### Proof of Theorem 3.8..

Instead of (3.15), let us establish the following inequalities

$\frac{1}{2^{n}}{\rm{Ban}}_{k}^{\infty}(h^{L})\leq({\rm{Ban}}_{k}^{1}(h^{L}))^{*}\leq{\rm{Ban}}_{l}^{\infty}(h^{L}).$
(3.27)

By taking duals, and using the fact that any norm on a finitely dimensional
vector space is equal to its double dual, we see that (3.27) is equivalent to
(3.15).

First of all, by the definition of the dual norm, for any $f\in
H^{0}(X,L^{k})$, we have

$\|f\|_{L^{1}_{k}(X,h^{L})}^{*}=\sup_{g\in H^{0}(X,L^{k})}\frac{|\langle
f,g\rangle_{L^{2}_{k}(X,h^{L})}|}{\|g\|_{L^{1}_{k}(X,h^{L})}}.$ (3.28)

From this, the fact that $H^{0}(X,L^{k})$ is a subspace of $L^{1}(X,L^{k})$
and the well-known fact that the dual of
$(L^{1}(X,L^{k}),\|\cdot\|_{L^{1}_{k}(X,h^{L})})$ coincides with
$(L^{\infty}(X,L^{k}),\|\cdot\|_{L^{\infty}_{k}(X,h^{L})})$, we deduce the
right estimate from (3.27).

Now, for a fixed $\alpha>0$ and $f\in H^{0}(X,L^{k})$, consider an element
$g_{\alpha}\mathrel{\mathop{\ordinarycolon}}=B_{k}^{X}(|f|^{\alpha}\cdot f)\in
H^{0}(X,L^{k})$. Then according to Proposition 3.12, we have

$\|g_{\alpha}\|_{L^{1}_{k}(X,h^{L})}\leq
2^{n}\Big{(}1+\frac{C}{k}\Big{)}\cdot\int_{X}|f|^{1+\alpha}(x)dv_{X}(x).$
(3.29)

Remark also that we have

$\langle f,g_{\alpha}\rangle_{L^{2}_{k}(X,h^{L})}=\langle f,|f|^{\alpha}\cdot
f\rangle_{L^{2}_{k}(X,h^{L})}=\int_{X}|f|^{2+\alpha}(x)dv_{X}(x).$ (3.30)

In particular, from (3.30), we see that $g_{\alpha}\neq 0$. Recall, finally,
that for any continuous non-zero function
$h\mathrel{\mathop{\ordinarycolon}}X\to\real$, we have

$\lim_{\alpha\to+\infty}\frac{\int_{X}|h|^{2+\alpha}(x)dv_{X}(x)}{\int_{X}|h|^{1+\alpha}(x)dv_{X}(x)}=\sup_{x\in
X}|h|(x).$ (3.31)

The left estimate from (3.27) now follows from (3.29), (3.30) and (3.31) by
putting $g\mathrel{\mathop{\ordinarycolon}}=g_{\alpha}$ in (3.28) and letting
$\alpha\to\infty$.

The second estimate from Theorem 3.8 is obtained in an analogous way. ∎

###  3.3 Bounds on products of holomorphic sections, a proof of Theorem 3.2

The main goal of this section is to give a bound on the norm of a product of
two holomorphic sections in terms of their norms. In particular, we prove
Theorem 3.2. The main caveat of the proof is an introduction of a sequence of
operators, called multiplicative defect. They would allow us to extend our
methods from [27], [28], [29] in the current setting.

Before all, let us prove that Theorem 3.2 holds for $L^{1}$-norm. For this we
will rely solely on the results on the study of the asymptotic behavior of the
Bergman kernel from Section 3.2.

###### Proof of Theorem 3.2 for the $L^{1}$-norm.

Let us first recall that the Bergman projector $B_{k,l}^{X\times X}$ is
defined as the orthogonal projection from $L^{2}(X\times X,L^{k}\boxtimes
L^{l})$ to $H^{0}(X\times X,L^{k}\boxtimes L^{l})$. Clearly, under the
identification

$L^{2}(X\times X,L^{k}\boxtimes L^{l})\to L^{2}(X,L^{k})\otimes
L^{2}(X,L^{l}),$ (3.32)

where $\otimes$ means here the tensor product of Hilbert spaces, and the
identification (3.3), we can write

$B_{k,l}^{X\times X}=B_{k}^{X}\otimes B_{l}^{X}.$ (3.33)

We extend ${\rm{Mult}}_{k,l}$ to $L^{2}(X,L^{k})\otimes L^{2}(X,L^{l})$ by
precomposing it with the Bergman projection. Then, under the isomorphisms from
(3.4) and (3.32), we have the following identity

${\rm{Mult}}_{k,l}={\rm{Res}}_{\Delta}\circ B_{k,l}^{X\times X}.$ (3.34)

Similarly to (3.26), from (3.34), we conclude that there are $C>0$,
$p_{1}\in\mathbb{N}$, such that for any $x_{0},x_{1}\in X$, for $k,l\geq
p_{1}$, we have

$\displaystyle\int_{X}\Big{|}{\rm{Mult}}_{k,l}\big{(}x,(x_{0},x_{1})\big{)}\Big{|}dv_{X}(x)\leq
2^{n}\Big{(}\frac{k\cdot
l}{\max\\{k,l\\}}\Big{)}^{n}\cdot\Big{(}1+\frac{C}{k}+\frac{C}{l}\Big{)},$
(3.35) $\displaystyle\int_{X\times
X}\Big{|}{\rm{Mult}}_{k,l}\big{(}x_{0},(x,x^{\prime})\big{)}\Big{|}dv_{X}(x)dv_{X}(x^{\prime})\leq
4^{n}\Big{(}1+\frac{C}{k}+\frac{C}{l}\Big{)}.$

Directly from (3.35) and Young’s inequality for integral operators, cf. [52,
Theorem 0.3.1] applied for $q=p=1$, $r=1$ in the notations of [52], we deduce
the second inequality from (3.9).

Let us now show that the first part of the statement for $L^{1}$-norm follows
from the second one. Remark that for any $f\in H^{0}(X,L^{k})$, $g\in
H^{0}(X,L^{l})$, we have

$\big{\|}f\otimes g\big{\|}_{L^{1}_{k,l}(X\times X,h^{L}\boxtimes
h^{L})}=\big{\|}f\big{\|}_{L^{1}_{k}(X,h^{L})}\cdot\big{\|}g\big{\|}_{L^{1}_{l}(X,h^{L})}.$
(3.36)

By the definition of the operator norm and (3.36), we deduce that

$\big{\|}f\cdot
g\big{\|}_{L^{1}_{k+l}(X,h^{L})}\leq\|{\rm{Mult}}_{k,l}\|_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})\to L^{1}_{k+l}(X,h^{L})}\cdot\big{\|}f\otimes
g\big{\|}_{L^{1}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}.$ (3.37)

We conclude by (3.9), (3.36) and (3.37). ∎

Now, to prove Theorem 3.2 for the $L^{2}$-norm, more work has to be done. This
is due to the fact that the bound provided by the Young’s inequality will no
longer coincide asymptotically with the optimal bound. To remedy this problem,
using the surjectivity statement from Proposition 3.1, we define the optimal
decomposition map

${\rm{Dec}}_{k,l}\mathrel{\mathop{\ordinarycolon}}H^{0}(X,L^{k+l})\to
H^{0}(X,L^{k})\otimes H^{0}(X,L^{l}),$ (3.38)

as follows. We let ${\rm{Dec}}_{k,l}(f)=h$ if $h$ satisfies
${\rm{Mult}}_{k,l}(h)=f$, and $h$ has the minimal $L^{2}$-norm among all $h\in
H^{0}(X,L^{k})\otimes H^{0}(X,L^{l})$, verifying such an identity (the
uniqueness of such $h$ follows from the non-degeneracy of the
$L^{2}$-Hermitian product). Clearly, from the commutative square (3.4), we see
that the optimal decomposition map is just a version of the optimal extension
map, studied in [27], associated for the pair $(X\times X,\Delta)$. The
difference between our situation here from [27] is that there we consider only
the tensor powers and not $\boxtimes$ products. This being said, most of our
techniques from [29] transfer in the current setting almost directly.

We will now use the methods of [29] to study ${\rm{Dec}}_{k,l}$. Our first
goal is to prove the existence of a sequence of operators, relating the
restriction maps with optimal decomposition maps.

######  Theorem 3.14.

There is $p_{1}\in\mathbb{N}^{*}$, such that for any $k,l\geq p_{1}$, there is
a unique operator $A_{k,l}\in\text{End}{(H^{0}(X,L^{k+l}))}$, verifying under
the isomorphisms (3.1) and (3.2) the following identity

$({\rm{Res}}_{\Delta}\circ B_{k,l}^{X\times X})^{*}={\rm{Dec}}_{k,l}\circ
A_{k,l}.$ (3.39)

######  Remark 3.15.

The sequence of operators $A_{k,l}$, will be later called multiplicative
defect. An analogous sequence of operators for more general immersions (and
tensor powers of the line bundles instead of $\boxtimes$-products) was
introduced in [28, Theorem 4.3] in their relation with the problem of
transitivity of optimal holomorphic extensions. It was later studied in [28,
Theorem 4.1] in a more general situation of holomorphic jets along
submanifold, and used as a tool to extend our previous work [27] on the
semiclassical extension theorem for holomorphic jets instead of sections.

###### Proof.

It follows from the equality of kernels and images of
$({\rm{Res}}_{\Delta}\circ B_{k,l}^{X\times X})^{*}$ and ${\rm{Dec}}_{k,l}$,
which relies on Proposition 3.1. See [28, Theorem 4.3] or [29, Theorem 4.1]
for details. ∎

Now, we will show that, $A_{k,l}$ coincides asymptotically with the identity
operator up to a multiplication by an explicit constant. At least for $k=l$,
this is quite an expected result.

In fact, from the commutative diagram (3.4) and (3.39), the operator $A_{k,l}$
relates the extension and the adjoint of the restriction operators. But from
the calculations in [28, (3.10)] or [29, (3.15)], we know that those operators
coincide (in a non-asymptotic sense) for the model Bargmann space. Moreover,
by the results of [27, Theorem 1.6] and [29, Lemma 5.9], we know that the
extension and restriction operators for general manifolds coincide
asymptotically with those model operators after an appropriate rescaling. So
over general manifolds the extension and the adjoint of the restriction
operators should asymptotically coincide as well.

The following result shows that despite the fact that we do not posses a
general version of extension theorem, which would cover not only the tensor
powers, but also line bundles of the form $L^{k}\boxtimes L^{l}$, the property
of $A_{k,l}$, described above, persists.

######  Theorem 3.16.

There are $p_{1}\in\mathbb{N}^{*}$, $C>0$, such that for any $k,l\geq p_{1}$,
we have

$\Big{\|}A_{k,l}-\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\cdot{\rm{Id}}_{H^{0}(X,L^{k+l})}\Big{\|}\leq
C\cdot\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n},$ (3.40)

where the operator norm of the operators on $H^{0}(X,L^{k+l})$ is taken with
respect to ${\rm{Hilb}}_{k+l}(h^{L})$.

In the proof of Theorem 3.16, the following result will be of utmost
importance. To state it, we extend the domain of $A_{k,l}$ to
$L^{2}_{k+l}(X,h^{L})$ by precomposing it with the Bergman projection. Then
$A_{k,l}$ is a smoothing operator and hence has a smooth Schwartz kernel,
$A_{k,l}(x_{1},x_{2})$, $x_{1},x_{2}\in X$, with respect to $dv_{X}$. Next
lemma provides a comprehensive study of this Schwartz kernel.

######  Lemma 3.17.

There are $c,C>0$, $p_{1}\in\mathbb{N}^{*}$ such that for any $k,l\geq p_{1}$,
$x_{1},x_{2}\in X$, we have

$\Big{|}A_{k,l}(x_{1},x_{2})\Big{|}\leq C(k\cdot
l)^{n}\cdot\exp\big{(}-c(\sqrt{k}+\sqrt{l})\cdot{\rm{dist}}(x_{1},x_{2})\big{)}.$
(3.41)

Moreover, there are $\epsilon,c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for
any $x_{0}\in X$, $k,l\geq p_{1}$, $Z,Z^{\prime}\in{}^{2n}$,
$|Z|,|Z^{\prime}|\leq\epsilon$, we have

$\bigg{|}A_{k,l}\big{(}\phi_{x_{0}}^{X}(Z),\phi_{x_{0}}^{X}(Z^{\prime})\big{)}-\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}B_{k+l}^{X}\big{(}\phi_{x_{0}}^{X}(Z),\phi_{x_{0}}^{X}(Z^{\prime})\big{)}\bigg{|}\\\
\leq C(k\cdot
l)^{n}\cdot\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}1+\sqrt{k+l}|Z|+\sqrt{k+l}|Z^{\prime}|\Big{)}^{6n+4}\cdot\\\
\cdot\exp\Big{(}-c(\sqrt{k}+\sqrt{l})\cdot|Z-Z^{\prime}|\Big{)}.$ (3.42)

###### Proof.

First of all, from (3.39), we see that the following identity holds

$A_{k,l}={\rm{Res}}_{\Delta}\circ({\rm{Res}}_{\Delta}\circ B_{k,l}^{X\times
X})^{*}.$ (3.43)

Remark also the trivial fact that the embedding $\Delta\hookrightarrow X\times
X$ is totally geodesic once we endow $X\times X$ with the product metric. From
this, we deduce that

$d_{X\times X}(z,z^{\prime})=d_{\Delta}(z,z^{\prime}),\qquad\text{for
}z,z^{\prime}\in\Delta,$ (3.44)

where the metric on $\Delta$ is taken with respect to the Kähler form
$\omega_{1,1}$. Also, for any $x_{0}\in X$, $Z\in{}^{2n}$, $|Z|<r_{X}$, we
have

$\phi_{x_{0}\times x_{0}}^{X\times
X}(Z,Z)=(\phi_{x_{0}}^{X}(Z),\phi_{x_{0}}^{X}(Z)).$ (3.45)

From Theorem 3.10, (3.33), (3.43) and (3.44), we deduce (3.41).

Let us now prove (3.42). From Theorem 3.11, (3.33), (3.43) and (3.45), we see
that there are $\epsilon,c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any
$x_{0}\in X$, $k,l\geq p_{1}$, $Z,Z^{\prime}\in{}^{2n}$,
$|Z|,|Z^{\prime}|\leq\epsilon$, the following estimate holds

$\bigg{|}A_{k,l}\big{(}\phi_{x_{0}}^{X}(Z),\phi_{x_{0}}^{X}(Z^{\prime})\big{)}-(k\cdot
l)^{n}\cdot\mathscr{P}_{n}(\sqrt{k}Z,\sqrt{k}Z^{\prime})\cdot\mathscr{P}_{n}(\sqrt{l}Z,\sqrt{l}Z^{\prime})\bigg{|}\\\
\leq C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot(k\cdot
l)^{n}\cdot\Big{(}1+\sqrt{k+l}|Z|+\sqrt{k+l}|Z^{\prime}|\Big{)}^{6n+4}\cdot\\\
\cdot\exp\Big{(}-c(\sqrt{k}+\sqrt{l})\cdot|Z-Z^{\prime}|\Big{)}.$ (3.46)

We now remark that from (3.23), we have

$\mathscr{P}_{n}(\sqrt{k}Z,\sqrt{k}Z^{\prime})\cdot\mathscr{P}_{n}(\sqrt{l}Z,\sqrt{l}Z^{\prime})=\mathscr{P}_{n}(\sqrt{k+l}Z,\sqrt{k+l}Z^{\prime}).$
(3.47)

From Theorem 3.11, (3.46) and (3.47), we conclude that (3.42) holds. ∎

###### Proof of Theorem 3.16.

Remark first, cf. [27, Proposition 3.2], that there is $C>0$, such that for
any $x_{0}\in X$, we have the following bound

$\int_{X}\exp\big{(}-c(\sqrt{k}+\sqrt{l})\cdot{\rm{dist}}(x_{0},x)\big{)}dv_{X}(x)<\frac{C}{(k+l)^{n}}.$
(3.48)

From Theorem 3.10, (3.42) and (3.48), we conclude that there are $C>0$,
$p_{1}\in\mathbb{N}^{*}$, such that for any $x_{0}\in X$, for $k,l\geq p_{1}$,
we have

$\displaystyle\int_{X}\Big{|}\Big{(}A_{k,l}-\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}B_{k+l}^{X}\Big{)}(x_{0},x)\Big{|}dv_{X}(x)\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n},$ (3.49)
$\displaystyle\int_{X}\Big{|}\Big{(}A_{k,l}-\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}B_{k+l}^{X}\Big{)}(x,x_{0})\Big{|}dv_{X}(x)\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}.$

Directly from (3.49) and Young’s inequality for integral operators, cf. [52,
Theorem 0.3.1], applied for $p,q=2$, $r=1$ in the notations of [52], we deduce
the result. ∎

We are now finally ready to finish the proof of the main result of this
section.

###### Proof of Theorem 3.2 for the $L^{2}$-norm.

In this proof all the operator norms are taken with respect to the
$L^{2}$-norm. First of all, similarly to the proof of Theorem 3.2 for the
$L^{1}$-norm, it suffices to establish the second part of the statement. We
will concentrate on this from now on. Directly from Theorem 3.16, we see that
there are $p_{1}\in\mathbb{N}^{*}$, $C>0$, such that for any $k,l\geq p_{1}$,
we have

$\Big{|}\big{\|}A_{k,l}\big{\|}-\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}\Big{|}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k\cdot
l}{k+l}\Big{)}^{n}.$ (3.50)

From Theorem 3.14, under the isomorphisms from (3.4), we have the following
identity

${\rm{Mult}}_{k,l}\circ({\rm{Mult}}_{k,l})^{*}=A_{k,l}.$ (3.51)

Clearly, we have
$\|{\rm{Mult}}_{k,l}\circ({\rm{Mult}}_{k,l})^{*}\|=\|{\rm{Mult}}_{k,l}\|^{2}$.
The first estimate from (3.9) now follows from this observation, (3.50) and
(3.51). ∎

###  3.4 Asymptotics of optimal decomposition map, a proof of Theorem 3.4

The main goal of this section is to study more precisely the asymptotics of
optimal decomposition map, introduced in (3.38). Through this study we will
establish Theorem 3.4.

We will begin by showing that for the $L^{2}$-norm, Theorem 3.4 already
follows from the analysis on the multiplicative defect from Section 3.3.

###### Proof of Theorem 3.4 for the $L^{2}$-norm.

Directly from Theorem 3.16, we see that there is $p_{1}\in\mathbb{N}^{*}$,
such that $A_{k,l}$ is invertible on $H^{0}(X,L^{k+l})$ for any $k,l\geq
p_{1}$. We, moreover, deduce that there are $p_{1}\in\mathbb{N}^{*}$, $C>0$,
such that for any $k,l\geq p_{1}$, the following inequality holds

$\Big{|}\big{\|}A_{k,l}^{-1}\big{\|}-\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n}\Big{|}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n},$ (3.52)

where the operator norm here (as well as below in this proof) is taken with
respect to the $L^{2}$-norm. From Theorem 3.14, under the isomorphisms from
(3.4), we have the following identity

$({\rm{Dec}}_{k,l})^{*}\circ{\rm{Dec}}_{k,l}=\big{(}(A_{k,l})^{*}\big{)}^{-1}.$
(3.53)

Clearly, we have
$\|({\rm{Dec}}_{k,l})^{*}\circ{\rm{Dec}}_{k,l}\|=\|{\rm{Dec}}_{k,l}\|^{2}$.
Theorem 3.4 for the $L^{2}$-norm now follows from this observation, (3.52) and
(3.53). ∎

To establish Theorem 3.4 for $L^{1}$ and $L^{\infty}$-norms, we have to do
more refined analysis. This is due to the fact that $L^{1}$ and
$L^{\infty}$-norms are not Hermitian ones, and so the above argument using the
dual mappings wouldn’t work.

To overcome this, we develop analysis similar in spirit to [29, proof of
Theorem 1], yet technically a bit more involved. In particular, we will give
an asymptotic formula for the optimal decomposition map (and not only for its
norm or the norm of the inverse as it was done in proof of Theorems 3.2 and
3.4 for the $L^{2}$-norm). To describe this asymptotics, we introduce some
further notation to study the geometry of the diagonal embedding.

For $\alpha,\beta\geq 0$, not vanishing at the same time, let us consider the
orthogonal complement $N_{\alpha,\beta}$ of $T\Delta$ in $T(X\times X)$ with
respect to the non-negative $(1,1)$-form

$\omega_{\alpha,\beta}\mathrel{\mathop{\ordinarycolon}}=\alpha\pi_{1}^{*}\omega+\beta\pi_{2}^{*}\omega$
(3.54)

on $X\times X$, where $\pi_{1},\pi_{2}$ are the projections from $X\times X$
to the first and the second factor respectively. Clearly, $N_{\alpha,\beta}$
is well-defined even if either $\alpha$ or $\beta$ is equal to $0$, as
$\omega_{\alpha,\beta}$ is always positive-definite along $T\Delta$ as long as
$\alpha,\beta$ do not vanish at the same time. Moreover, $N_{\alpha,\beta}$
has constant dimension along $\Delta$ and hence forms a (smooth) vector bundle
over $\Delta$.

For $y=(x_{0},x_{0})\in\Delta$, $Z_{N}\in N_{\alpha,\beta}$, let $\real\ni
t\mapsto\exp_{y}^{X\times X}(tZ_{N})\in X$ be the geodesic in $X\times X$ in
direction $Z_{N}$. The geodesic is taken with respect to the Kähler form
$\omega_{1,1}$. In other words, for $Z_{N}=(Z_{1},Z_{2})$, we have
$\exp_{y}^{X\times
X}(Z_{N})=(\exp_{x_{0}}^{X}(Z_{1}),\exp_{x_{0}}^{X}(Z_{2}))$.

Clearly, for a certain $r_{\perp}>0$, the above exponential map induces a
diffeomorphism of $r_{\perp}$-neighborhood of the zero section in
$N_{\alpha,\beta}$ (with respect to the norm $\|\cdot\|$ induced by
$\omega_{1,1}$) with a tubular neighborhood $U_{\alpha,\beta}$ of $\Delta$ in
$X\times X$. Such $r_{\perp}$ can be chosen independently of $\alpha,\beta\geq
0$ since $N_{\alpha,\beta}$ can be seen as a piecewise smooth vector bundle
over $\Delta\times\Delta_{1}$, where
$\Delta_{1}\mathrel{\mathop{\ordinarycolon}}=\\{(\gamma,\delta)\in{}^{2}\mathrel{\mathop{\ordinarycolon}}|\gamma|+|\delta|=1\\}$
and the projection over $\Delta_{1}$ is induced by
$(\alpha,\beta)\mapsto\frac{1}{|\alpha|+|\beta|}(\alpha,\beta)$.

We denote by
$\pi_{\alpha,\beta}\mathrel{\mathop{\ordinarycolon}}U_{\alpha,\beta}\to\Delta$
the natural projection $(y,Z_{N})\mapsto y$. Over $U_{\alpha,\beta}$, we
identify $\pi_{i}^{*}L$, $i=1,2$, to
$\pi_{\alpha,\beta}^{*}(\pi_{i}^{*}L|_{\Delta})$ by the parallel transport
with respect to the respective Chern connections along the geodesic $[0,1]\ni
t\mapsto(y,tZ_{N})\in X$, $|Z_{N}|<r_{\perp}$, previously described. This
gives us a local trivialization in normal directions of the vector bundle
$L^{k}\boxtimes L^{l}$ in the neighborhood of $\Delta$.

Using the above identifications, we define the operator
${\rm{Dec}}_{k,l}^{0}\mathrel{\mathop{\ordinarycolon}}L^{2}(X,L^{k+l})\to
L^{2}(X\times X,L^{k}\boxtimes L^{l})$ as follows. For $g\in
L^{2}(X,L^{k+l})$, we let $({\rm{Dec}}_{k,l}^{0}g)(x)=0$, $x\notin U_{k,l}$,
and in $U_{k,l}$, we put

$({\rm{Dec}}_{k,l}^{0}g)(y,Z_{N})=(B_{k+l}^{X}g)(y)\exp\Big{(}-\frac{\pi}{2}|Z_{N}|_{k,l}^{2}\Big{)}\rho\Big{(}\frac{|Z_{N}|}{r_{\perp}}\Big{)},$
(3.55)

where $|\cdot|_{\alpha,\beta}$ is the norm induced on $N_{\alpha,\beta}$ by
the Kähler form $\omega_{\alpha,\beta}$ and
$\rho\mathrel{\mathop{\ordinarycolon}}{}_{+}\to[0,1]$ satisfies

$\rho(x)=\begin{cases}1,\quad\text{for $x<\frac{1}{4}$},\\\ 0,\quad\text{for
$x>\frac{1}{2}$}.\end{cases}$ (3.56)

Now, for $g\in H^{0}(X,L^{k+l})$, the section ${\rm{Dec}}_{k,l}^{0}g$
satisfies $({\rm{Dec}}_{k,l}^{0}g)|_{\Delta}=g$, but ${\rm{Dec}}_{k,l}^{0}g$
is not holomorphic over $X\times X$ in general. The main result of this
sections, nevertheless, says that this operator approximates well the optimal
decomposition map.

######  Theorem 3.18.

There are $C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any $k,l\geq p_{1}$,
we have

$\displaystyle\Big{\|}{\rm{Dec}}_{k,l}-{\rm{Dec}}_{k,l}^{0}\Big{\|}_{L^{1}_{k+l}(X,h^{L})\to
L^{1}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n},$ (3.57)
$\displaystyle\Big{\|}{\rm{Dec}}_{k,l}-{\rm{Dec}}_{k,l}^{0}\Big{\|}_{L^{\infty}_{k+l}(X,h^{L})\to
L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}.$

######  Remark 3.19.

For the $L^{2}$-norm, an analogous result can be proved by similar methods. It
would give us an alternative proof of Theorem 3.4 for the $L^{2}$-norm.

Before establishing Theorem 3.18, let us see how useful it is in proving
Theorem 3.4.

###### Proof of Theorem 3.4 for $L^{1}$ and $L^{\infty}$-norms.

Directly from (3.55), the fact that $({\rm{Dec}}_{k,l}^{0}g)|_{\Delta}=g$ and
$\exp(-x)\leq 1$ for $x\geq 0$, we see that for any $g\in H^{0}(X,L^{k+l})$,
we have

$\big{\|}{\rm{Dec}}_{k,l}^{0}g\big{\|}_{L^{\infty}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}=\big{\|}g\big{\|}_{L^{\infty}_{k+l}(X,h^{L})}.$
(3.58)

Then from the second bound of (3.57) and (3.58), we establish that Theorem 3.4
holds for the $L^{\infty}$-norm, and one can moreover take
$h\mathrel{\mathop{\ordinarycolon}}={\rm{Dec}}_{k,l}f$ in the notations of
Theorem 3.4.

Let us now treat the $L^{1}$-norm. Remark first that the volume form
$dv_{k,l}$ on $X\times X$ associated to $\omega_{k,l}$ relates to $dv_{1,1}$
as follows

$dv_{k,l}=(k\cdot l)^{n}\cdot dv_{1,1}.$ (3.59)

Remark also that the restriction of $\omega_{k,l}$ to $\Delta$ coincides with
$(k+l)\cdot\omega$ under the isomorphism (3.1). Finally, we see that for the
Lebesgue measure $d_{k,l}Z_{N}$ on $N_{k,l,y}$, calculated with respect to the
norm on $N_{k,l}$ induced by $\omega_{k,l}$, we have

$\int_{Z_{N}\in
N_{k,l,y}}\exp\Big{(}-\frac{\pi}{2}|Z_{N}|_{k,l}^{2}\Big{)}d_{k,l}Z_{N}=2^{n}.$
(3.60)

Directly from those remarks, we see that there are $p_{1}\in\mathbb{N}$,
$C>0$, such that for any $k,l\geq p_{1}$, $g\in H^{0}(X,L^{k+l})$, we have

$\Big{(}1-C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot
2^{n}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n}\cdot\big{\|}g\big{\|}_{L^{1}_{k+l}(X,h^{L})}\leq\big{\|}{\rm{Dec}}_{k,l}^{0}g\big{\|}_{L^{1}_{k,l}(X\times
X,h^{L}\boxtimes h^{L})}\\\
\leq\Big{(}1+C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\Big{)}\cdot
2^{n}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n}\cdot\big{\|}g\big{\|}_{L^{1}_{k+l}(X,h^{L})}.$ (3.61)

We conclude by the first inequality from Theorem 3.18 and (3.61) that Theorem
3.4 holds for the $L^{1}$-norm, and one can moreover take
$h\mathrel{\mathop{\ordinarycolon}}={\rm{Dec}}_{k,l}f$ in the notations of
Theorem 3.4. ∎

To establish Theorem 3.18, we proceed in several steps. First, we show that
the Schwartz kernels of both operators, ${\rm{Dec}}_{k,l}$ and
${\rm{Dec}}_{k,l}^{0}$, have exponential decay. Second, we study the near-
diagonal behavior of the Schwartz kernels of those operators and infer that
they have the same principal asymptotic terms. This would imply Theorem 3.18
by some standard techniques.

More precisely, in the first step we prove the following exponential decay
estimates.

######  Lemma 3.20.

There are $c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any $k,l\geq
p_{1}$, $x_{0},x_{1},x_{2}\in X$, the following estimate holds

$\Big{|}R_{k,l}\big{(}(x_{1},x_{2}),x_{0}\big{)}\Big{|}\leq
C(k+l)^{n}\cdot\exp\Big{(}-c\cdot\big{(}\sqrt{k}{\rm{dist}}(x_{1},x_{0})+\sqrt{l}{\rm{dist}}(x_{2},x_{0})\big{)}\Big{)},$
(3.62)

where $R_{k,l}$ is ether ${\rm{Dec}}_{k,l}$ or ${\rm{Dec}}_{k,l}^{0}$.

###### Proof.

Directly from Theorem 3.16, we see that there is $p_{1}\in\mathbb{N}^{*}$,
such that $A_{k,l}$ is invertible on $H^{0}(X,L^{k+l})$ for any $k,l\geq
p_{1}$. From this and Theorem 3.14, we conclude that

${\rm{Dec}}_{k,l}=({\rm{Res}}_{\Delta}\circ B_{k,l}^{X\times X})^{*}\circ
A_{k,l}^{-1}.$ (3.63)

From Theorem 3.10, Lemma 3.17 and the usual laws of composition and inversion
of operators with exponential decay, cf. [27, Lemma 3.1], [28, Lemma 4.5], we
deduce (3.62) for $R_{k,l}={\rm{Dec}}_{k,l}$.

Directly from Theorem 3.10 and (3.55), we see that

$\Big{|}{\rm{Dec}}_{k,l}^{0}\big{(}(x_{1},x_{2}),x_{0}\big{)}\Big{|}\\\ \leq
C(k+l)^{n}\cdot\exp\Big{(}-c\cdot{\rm{dist}}_{k,l}\big{(}(x_{1},x_{2}),x_{0}\big{)}\Big{)},$
(3.64)

where ${\rm{dist}}_{k,l}((x_{1},x_{2}),x_{0})$ is the distance between points
$(x_{1},x_{2})$ and $(x_{0},x_{0})$ in $X\times X$ with respect to the Kähler
form $\omega_{k,l}$. An easy verification shows that

$2{\rm{dist}}_{k,l}((x_{1},x_{2}),x_{0})\geq\sqrt{k}{\rm{dist}}(x_{1},x_{0})+\sqrt{l}{\rm{dist}}(x_{2},x_{0}).$
(3.65)

This with (3.64) clearly finish the proof. ∎

Theorem 3.20 implies that to understand fully the asymptotics of the Schwartz
kernel of the optimal decomposition map, it suffices to do so in a
neighborhood of a fixed point $(x_{0},x_{0})\in\Delta\subset X\times X$. Our
next result calculates the asymptotics of this Schwartz kernel.

To state this result precisely, we need to introduce a version of Fermi
coordinate system in $X\times X$ around $\Delta$. We fix
$k,l\in\mathbb{N}^{*}$ and $y_{0}\in X$. Let
$P^{N}_{k,l}\mathrel{\mathop{\ordinarycolon}}T(X\times X)|_{\Delta}\to
N_{k,l}$, be the orthogonal projection. For $Z=(Z_{H},Z_{N})$, $Z_{H}\in
T_{y_{0}}X$, $Z_{N}\in N_{k,l,y_{0}}$, $|Z_{H}|\leq r_{X}$, $|Z_{N}|\leq
r_{\perp}$, we define a coordinate system
$\psi_{y_{0}}^{k,l}\mathrel{\mathop{\ordinarycolon}}B_{0}^{T_{y_{0}}X}(r_{X})\times
B_{0}^{N_{k,l,y_{0}}}(r_{\perp})\to X\times X$ by

$\psi_{y_{0}}^{k,l}(Z_{H},Z_{N})\mathrel{\mathop{\ordinarycolon}}=\exp_{\exp_{y_{0}}^{X}(Z_{H})}^{X\times
X}(Z_{N}(Z_{H})),$ (3.66)

where $Z_{N}(Z_{H})$ is the parallel transport of $Z_{N}\in N_{k,l,y_{0}}$
along the path $(\exp_{y_{0}}^{X}(tZ_{H}),\exp_{y_{0}}^{X}(tZ_{H}))$,
$t=[0,1]$, with respect to the connection
$\nabla^{N}_{k,l}\mathrel{\mathop{\ordinarycolon}}=P^{N}_{k,l}\nabla^{T(X\times
X)}P^{N}_{k,l}$ on $N_{k,l}$, and the exponential map on $X\times X$ is taken
with respect to the Kähler form $\omega_{X\times X}$. We will implicitly fix
an orthonormal basis in $T_{y_{0}}X$ and trivialize $N_{k,l,y_{0}}$ using it.
In this way, we may view $\psi_{y_{0}}^{k,l}$ as a map defined on an open
neighborhood of $0$ in ${}^{n}\times{}^{n}$.

Let us now fix an element $f_{0}\in L_{y_{0}}$ of unit norm. We define the
local orthonormal frame $\tilde{f}$ of $L$ around $y_{0}$ by the parallel
transport of $f$ with respect to $\nabla^{L}$, done first along
$\psi^{k,l}_{y_{0}}(tZ_{H},0)$, $t\in[0,1]$, and then along
$\psi^{k,l}_{y_{0}}(Z_{H},tZ_{N})$, $t\in[0,1]$, $Z_{H}\in T_{y_{0}}X$,
$Z_{N}\in N_{k,l,y_{0}}$, $|Z_{H}|<r_{X}$, $|Z_{N}|<r_{\perp}$.

Recall that the exponential coordinates $\phi^{X}$ and the frame
$\tilde{f}^{\prime}$ of $L$ in a neighborhood of $y_{0}\in X$ were defined in
(3.22). For $g\in\mathscr{C}^{\infty}(X\times X,L^{k}\boxtimes L^{l})$, by an
abuse of notation, we write
$g(\phi_{y_{0}}^{X}(Z),\phi_{y_{0}}^{X}(Z^{\prime}))\in L_{y_{0}}^{k+l}$,
$Z,Z^{\prime}\in Z_{H}\in T_{y_{0}}X$, $|Z|,|Z^{\prime}|\leq R$, for
coordinates of $g$ in the frame $\tilde{f}^{\prime}$, identified with an
element in $L_{y_{0}}^{k+l}$ using the frame $f$. Similarly, we denote by
$g(\psi_{y_{0}}^{k,l}(Z_{H},Z_{N}))\in L_{y_{0}}^{k+l}$ the coordinates in the
frame $\tilde{f}$, identified with an element from $L_{y_{0}}^{k+l}$.

Our first result gives a near-diagonal expansion of the optimal decomposition
map.

######  Lemma 3.21.

There are $\epsilon,c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any
$y_{0}\in X$, $k,l\geq p_{1}$, $Z_{0},Z_{1},Z_{2}\in T_{y_{0}}X$,
$|Z_{0}|,|Z_{1}|,|Z_{2}|\leq\epsilon$, we have

$\bigg{|}\frac{1}{(k+l)^{n}}{\rm{Dec}}_{k,l}\big{(}(\phi_{y_{0}}^{X}(Z_{1}),\phi_{y_{0}}^{X}(Z_{2})),\phi_{y_{0}}^{X}(Z_{0})\big{)}-\mathscr{P}_{n}\big{(}\sqrt{k}Z_{1},\sqrt{k}Z_{0}\big{)}\cdot\mathscr{P}_{n}\big{(}\sqrt{l}Z_{2},\sqrt{l}Z_{0}\big{)}\bigg{|}\\\
\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}1+\sqrt{k}(|Z_{1}|+|Z_{0}|)+\sqrt{l}(|Z_{2}|+|Z_{0}|)\Big{)}^{12n+8}\cdot\\\
\cdot\exp\Big{(}-c\big{(}\sqrt{k}|Z_{1}-Z_{0}|+\sqrt{l}|Z_{2}-Z_{0}|\big{)}\Big{)},$
(3.67)

where $\mathscr{P}_{n}$ was defined in (3.23).

###### Proof.

It follows directly from Theorems 3.10, 3.11, Lemma 3.17, (3.63) and the usual
composition rules for the operators with Taylor-type expansions and
exponential decay of the Schwartz kernel, cf. [27, Lemma 3.5]. ∎

We will now state the analogous result for the trivial decomposition map.

######  Lemma 3.22.

There are $\epsilon,c,C>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any
$y_{0}\in X$, $k,l\geq p_{1}$, $Z_{H},Z_{0}\in T_{y_{0}}X$, $Z_{N}\in
N_{k,l,y_{0}}$, $|Z_{H}|,|Z_{N}|,|Z_{0}|\leq\epsilon$, we have

$\bigg{|}\frac{1}{(k+l)^{n}}{\rm{Dec}}_{k,l}^{0}\big{(}\psi_{y_{0}}^{k,l}(Z_{H},Z_{N}),\phi_{y_{0}}^{X}(Z_{0})\big{)}-\mathscr{P}_{n}\big{(}\sqrt{k+l}Z_{H},\sqrt{k+l}Z_{0}\big{)}\cdot\\\
\cdot\exp\Big{(}-\frac{\pi}{2}|Z_{N}|^{2}_{k,l}\Big{)}\bigg{|}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}1+\sqrt{k+l}(|Z_{H}|+|Z_{0}|)\Big{)}^{3n+2}\cdot\\\
\cdot\exp\Big{(}-c\cdot\sqrt{k+l}\cdot|Z_{H}-Z_{0}|-c\cdot\frac{kl}{k+l}\cdot|Z_{N}|^{2}\Big{)}.$
(3.68)

###### Proof.

The proof follows directly from Theorem 3.11 and (3.55). ∎

We, finally need the following general result for the estimation of operator’s
norms. Let assume that a sequence of operators
$R_{k,l}\mathrel{\mathop{\ordinarycolon}}L^{2}(X,L^{k+l})\to L^{2}(X\times
X,L^{k}\boxtimes L^{l})$, $k,l\in\mathbb{N}$, is such that its Schwartz kernel
satisfies both the estimate (3.62) and the estimate below. There are
$\epsilon,c,C,Q>0$, $p_{1}\in\mathbb{N}^{*}$, such that for any $y_{0}\in X$,
$k,l\geq p_{1}$, $Z_{0},Z_{1},Z_{2}\in T_{x_{0}}X$,
$|Z_{0}|,|Z_{1}|,|Z_{2}|\leq\epsilon$, we have

$\bigg{|}\frac{1}{(k+l)^{n}}R_{k,l}\big{(}(\phi_{y_{0}}^{X}(Z_{1}),\phi_{y_{0}}^{X}(Z_{2})),\phi_{y_{0}}^{X}(Z_{0})\big{)}\bigg{|}\\\
\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}1+\sqrt{k}(|Z_{1}|+|Z_{0}|)+\sqrt{l}(|Z_{2}|+|Z_{0}|)\Big{)}^{Q}\cdot\\\
\cdot\exp\Big{(}-c\big{(}\sqrt{k}|Z_{1}-Z_{0}|+\sqrt{l}|Z_{2}-Z_{0}|\big{)}\Big{)}.$
(3.69)

######  Lemma 3.23.

There is $p_{1}\in\mathbb{N}$, such that for $k,l\geq p_{1}$, the operator
$R_{k,l}$ extends using Schwartz kernel as an operator
$R_{k,l}\mathrel{\mathop{\ordinarycolon}}L^{q}(X,L^{k+l})\to L^{q}(X\times
X,L^{k}\boxtimes L^{l})$ for $q=1,\infty$, and we have

$\displaystyle\big{\|}R_{k,l}\big{\|}_{L^{1}_{k+l}(X,h^{L})\to
L^{1}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}\cdot\Big{(}\frac{k+l}{k\cdot
l}\Big{)}^{n},$ (3.70)
$\displaystyle\big{\|}R_{k,l}\big{\|}_{L^{\infty}_{k+l}(X,h^{L})\to
L^{\infty}_{k,l}(X\times X,h^{L}\boxtimes h^{L})}\leq
C\Big{(}\frac{1}{k}+\frac{1}{l}\Big{)}.$

###### Proof.

First of all, a calculation of Gaussian integrals shows that there is $C>0$,
such that for any $k,l\in\mathbb{N}^{*}$, $f\in L^{1}(\mathbb{C}^{n})$, we
have

$\int_{Z_{2}\in\mathbb{C}^{n}}\int_{Z_{1}\in\mathbb{C}^{n}}\int_{Z_{0}\in\mathbb{C}^{n}}\exp\Big{(}-c\big{(}\sqrt{k}|Z_{1}-Z_{0}|+\sqrt{l}|Z_{2}-Z_{0}|\big{)}\Big{)}\cdot|f(Z_{0})|dZ_{0}dZ_{1}dZ_{2}\\\
\leq\frac{C}{(k\cdot l)^{n}}\|f\|_{L^{1}}.$ (3.71)

Now, remark that the triangle inequality implies

$\sqrt{k}|Z_{1}-Z_{0}|+\sqrt{l}|Z_{2}-Z_{0}|\geq\big{(}\sqrt{k}+\sqrt{l}\big{)}\cdot\Big{|}\frac{\sqrt{k}Z_{1}+\sqrt{l}Z_{2}}{\sqrt{k}+\sqrt{l}}-Z_{0}\Big{|}.$
(3.72)

This with a calculation of Gaussian integrals shows that there is $C>0$, such
that for any $k,l\in\mathbb{N}^{*}$, $g\in L^{\infty}(\mathbb{C}^{n})$, we
have

$\sup_{Z_{2}\in\mathbb{C}^{n}}\sup_{Z_{1}\in\mathbb{C}^{n}}\int_{Z_{0}\in\mathbb{C}^{n}}\exp\Big{(}-c\big{(}\sqrt{k}|Z_{1}-Z_{0}|+\sqrt{l}|Z_{2}-Z_{0}|\big{)}\Big{)}\cdot|g(Z_{0})|dZ_{0}\\\
\leq\frac{C}{(k+l)^{n}}\cdot\|g\|_{L^{\infty}}.$ (3.73)

Exponential decay assumption on the kernel of $R_{k,l}$ implies that only
local contributions of the Schwartz kernel of $R_{k,l}$ matter in the study of
the $L^{1}$ and $L^{\infty}$-operator norms of $R_{k,l}f$ for $f\in
H^{0}(X,L^{k+l})$. According to (3.70), estimates (3.71) and (3.73), done in
local coordinates, show that $R_{k,l}$ extends to an operator
$R_{k,l}\mathrel{\mathop{\ordinarycolon}}L^{q}(X,L^{k+l})\to L^{q}(X\times
X,L^{k}\boxtimes L^{l})$ for $q=1,\infty$, and the announced bounds on the
resulting norms hold. ∎

###### Proof of Theorem 3.18.

First of all, let us denote by
$R_{k,l}\mathrel{\mathop{\ordinarycolon}}L^{2}(X,L^{k+l})\to L^{2}(X\times
X,L^{k}\boxtimes L^{l})$ the operator, given by the difference

$R_{k,l}\mathrel{\mathop{\ordinarycolon}}={\rm{Dec}}_{k,l}-{\rm{Dec}}_{k,l}^{0}.$
(3.74)

Then Lemma 3.20 implies that the Schwartz kernel of $R_{k,l}$ has exponential
decay as in (3.62).

Now, for a fixed $x_{0}\in X$ and arbitrary $Z_{1},Z_{2}\in T_{x_{0}}X$, we
decompose the vector $(Z_{1},Z_{2})\in T_{x_{0}\times x_{0}}(X\times X)$ into
tangential and normal part

$(Z_{1},Z_{2})=Z_{H}+Z_{N},$ (3.75)

where $Z_{H}=(Z_{H}^{0},Z_{H}^{0})\in T\Delta$ and $Z_{N}\in N_{k,l}$ for a
given $k,l\in\mathbb{N}^{*}$. Clearly, we then have

$Z_{H}^{0}=\frac{kZ_{1}+lZ_{2}}{k+l},\qquad
Z_{N}=\Big{(}\frac{l(Z_{1}-Z_{2})}{k+l},\frac{k(Z_{2}-Z_{1})}{k+l}\Big{)}.$
(3.76)

An easy calculation, using the formula for $\mathscr{P}_{n}$ and the above
formulas for $Z_{H}^{0},Z_{N}$ shows that

$\mathscr{P}_{n}\big{(}\sqrt{k}Z_{1},\sqrt{k}Z_{0}\big{)}\cdot\mathscr{P}_{n}\big{(}\sqrt{l}Z_{2},\sqrt{l}Z_{0}\big{)}\\\
=\mathscr{P}_{n}\big{(}\sqrt{k+l}Z_{H}^{0},\sqrt{k+l}Z_{0}\big{)}\cdot\exp\Big{(}-\frac{\pi}{2}|Z_{N}|^{2}_{k,l}\Big{)}.$
(3.77)

Also, up to the second order, the following identity holds

$\psi_{y_{0}}^{k,l}(Z_{H}^{0},Z_{N})=(\phi_{y_{0}}^{X}(Z_{1}),\phi_{y_{0}}^{X}(Z_{2})).$
(3.78)

Hence, by (3.77) and Lemmas 3.20, 3.21, 3.22, we see that all the assumptions
of Lemma 3.23 are satisfied for the operator $R_{k,l}$. An application of
Lemma 3.23, hence, finishes the proof. ∎

##  4 The set of multiplicatively generated norms

The main goal of this section is to study asymptotically the set of
multiplicatively generated norms on the section ring. In particular, we
establish Theorems 1.5, 1.6, 1.7 and 1.8. The main tool in this study is the
Fubini-Study operator from Section 2.3.

The section is organized as follows. In Section 4.1, we establish that
plurisubharmonic metrics give rise to multiplicatively generated norms. More
precisely, we show Theorem 1.5.

In Section 4.2, we study the metric properties of the maps ${\rm{Ban}}^{1}$,
${\rm{Hilb}}$, ${\rm{Ban}}^{\infty}$. More precisely, we prove Theorem 1.7. We
will also explain how this result falls into a more general picture of
approximations of $\mathcal{H}^{L}$ by the space of norms on $H^{0}(X,L^{k})$,
as $k\to\infty$. This section is essentially independent from the rest of the
text, except for Section 4.6.

In Section 4.3, we study the convergence properties of Fubini-Study operator
for multiplicatively generated norms and establish the following result.

######  Theorem 4.1.

For any graded multiplicatively generated norm
$N\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}N_{k}$ on $R(X,L)$, the
sequence of metrics $FS(N_{k})^{\frac{1}{k}}$ converges uniformly, as
$k\to\infty$, to a (continuous psh) metric on $L$, which we denote by
$FS(N)\in\mathcal{H}^{L}$.

Theorem 4.1 motivates the following natural question: does the convergence of
the Fubini-Study operator of a graded norm of the section ring determines the
equivalence class of the norm? The answer is negative, see Section 4.4 for an
explicit example.

Nevertheless, the following result from Section 4.5 shows that the answer
becomes positive if we restrict our attention to a special class of graded
norms.

######  Theorem 4.2.

Assume that for a graded norm
$N\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}N_{k}$ on $R(X,L)$, the
sequence of metrics $FS(N_{k})^{\frac{1}{k}}$ converges uniformly, as
$k\to\infty$, to a (continuous psh) metric $FS(N)$ on $L$, and there is
$p_{0}\in\mathbb{N}$ and a function
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{N}_{\geq p_{0}}\to\real$, verifying
$f(k)=o(k)$, as $k\to\infty$, such that for any $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, under the map (1.2),
we have the following bound

$N_{k}\leq\big{[}N_{k_{1}}\otimes\cdots\otimes
N_{k_{r}}\big{]}\cdot\exp\Big{(}f(k_{1})+\cdots+f(k_{r})+f(k)\Big{)}.$ (4.1)

Then the following equivalence of graded norms on $R(X,L)$ holds
$N\sim{\rm{Hilb}}(FS(N))$.

Clearly, Theorems 4.1 and 4.2 imply Theorem 1.6. We then show in Section 4.6,
that from Theorem 4.2 and some elementary facts from the theory of
interpolation of Hilbert spaces, one can establish Theorem 1.8.

###  4.1 Multiplicatively generated norms from plurisubharmonic metrics

The main goal of this section is to establish that plurisubharmonic metrics
give rise to multiplicatively generated norms through the Hilbert map. In
other words, we prove Theorem 1.5.

The proof of this result is decomposed into two parts: establishing the lower
bound and the upper bound. For the upper bound, the following simple
observation is crucial.

######  Lemma 4.3.

For any $h^{L}\in\mathcal{H}^{L}$, $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, under the map (1.2),
the following inequality holds

${\rm{Ban}}_{k}^{\infty}(h^{L})\leq\Big{[}{\rm{Ban}}_{k_{1}}^{\infty}(h^{L})\otimes_{\epsilon}\cdots\otimes_{\epsilon}{\rm{Ban}}_{k_{r}}^{\infty}(h^{L})\Big{]}.$
(4.2)

###### Proof.

From Theorem 3.6, we see that under the Künneth isomorphism

$H^{0}(X\times\cdots\times X,L^{k_{1}}\boxtimes\cdots\boxtimes L^{k_{r}})\to
H^{0}(X,L^{k_{1}})\otimes\cdots\otimes H^{0}(X,L^{k_{r}}),$ (4.3)

the norm
${\rm{Ban}}_{k_{1}}^{\infty}(h^{L})\otimes_{\epsilon}\cdots\otimes_{\epsilon}{\rm{Ban}}_{k_{r}}^{\infty}(h^{L})$
gets pulled-back to the $L^{\infty}$-norm on $H^{0}(X\times\cdots\times
X,L^{k_{1}}\boxtimes\cdots\boxtimes L^{k_{r}})$. The result now follows
directly from this, the fact that multiplication operator can be seen as the
the restriction operator, cf. (3.4), and the trivial fact that the
$L^{\infty}$-norm of the restriction of a given section is always no bigger
than the $L^{\infty}$-norm of the section. ∎

For the lower bound, we will need the following slightly more complicated
result.

######  Lemma 4.4.

Assume that the measure $\mu$ on $X$ can be majorized from below and above by
some strictly positive volume forms on $X$. Then there are $C>0$,
$p_{0}\in\mathbb{N}$, such that for any $h^{L}\in\mathcal{H}^{L}$,
$r\in\mathbb{N}^{*}$, $k;k_{1},\ldots,k_{r}\geq p_{0}$,
$k_{1}+\cdots+k_{r}=k$, under the map (1.2), the following inequality holds

$\Big{[}{\rm{Hilb}}_{k_{1}}(h^{L},\mu)\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L},\mu)\Big{]}\leq
C^{r}\cdot{\rm{Hilb}}_{k}(h^{L},\mu).$ (4.4)

###### Proof.

Clearly, by Corollary B.7, it is enough to establish this result for $r=2$, as
for other $r$ it would follow by induction. Once the definition of the
quotient norm is unraveled, it becomes clear that it suffices to prove that
for any $f\in H^{0}(X,L^{k_{1}+k_{2}})$, there is a tensor $h\in
H^{0}(X,L^{k_{1}})\otimes H^{0}(X,L^{k_{2}})$, verifying
${\rm{Mult}}_{k_{1},k_{2}}(h)=f$, such that

$\|h\|_{L^{2}_{k_{1}}(X,h^{L})\otimes L^{2}_{k_{2}}(X,h^{L})}\leq
C\|f\|_{L^{2}_{k_{1}+k_{2}}(X,h^{L})}.$ (4.5)

This, however, follows directly from the interpretation of the multiplication
map of the section ring as an instance of the restriction map, see (3.4), and
a statement similar to Theorem 2.5, where instead of $L^{k}$, we consider
$L^{k_{1}}\boxtimes L^{k_{2}}$. The proof of the latter statement is done
analogously to the proof of Theorem 2.5, and is left to the interested reader.
∎

###### Proof of Theorem 1.5..

From Corollary 2.11, Lemma 4.3, (2.23) and the fact that the injective tensor
norm is majorized from above by the Hermitian tensor norm, cf. Lemma B.2, we
conclude that there is a function
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{N}\to\real$, $f(k)=o(k)$, such that
in the notations of Lemma 4.4, we have

${\rm{Hilb}}_{k}(h^{L})\leq\Big{[}{\rm{Hilb}}_{k_{1}}(h^{L})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L})\Big{]}\cdot\exp\Big{(}f(k_{1})+\cdots+f(k_{r})\Big{)}.$
(4.6)

Similarly, from Lemma 4.4 and the second part of Corollary 2.11, we conclude
that there is a function
$g\mathrel{\mathop{\ordinarycolon}}\mathbb{N}\to\real$, $g(k)=o(k)$, such that
in the notations of Lemma 4.4 we have

${\rm{Hilb}}_{k}(h^{L})\geq\Big{[}{\rm{Hilb}}_{k_{1}}(h^{L})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L})\Big{]}\cdot\exp\Big{(}g(k_{1})+\cdots+g(k_{r})+g(k)\Big{)}.$
(4.7)

We deduce Theorem 1.5 from (4.6) and (4.7). ∎

###  4.2 Metric properties of the Hilbert map and Mabuchi supremum geometry

The main goal of this section is to study the asymptotic metric properties of
the map ${\rm{Hilb}}$. More precisely, we compare the norms
${\rm{Hilb}}_{k}(h^{L}_{0}),{\rm{Hilb}}_{k}(h^{L}_{1})$ for
$h^{L}_{0},h^{L}_{1}\in\mathcal{H}^{L}$, as $k\to\infty$.

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

By Corollary 2.11, we can study the distance between the images of the map
${\rm{Ban}}^{\infty}$ instead of ${\rm{Hilb}}^{\infty}$. Trivially, for any
metrics $h^{L}_{0}$, $h^{L}_{1}$ on $L$, we obtain that

$d_{\infty}({\rm{Ban}}^{\infty}(h^{L}_{0}),{\rm{Ban}}^{\infty}(h^{L}_{1}))\leq
d_{\infty}(h^{L}_{0},h^{L}_{1}).$ (4.8)

Let us now establish the opposite bound. Our idea is to construct, using the
Ohsawa-Takegoshi extension theorem, a holomorphic section of $L^{k}$, with the
mass concentrated in a neighborhood of a fixed point. More precisely, by
Theorem 2.9, Proposition 2.10 and Lemmas 2.8, 2.17, we conclude that for any
$\epsilon>0$, there is $p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$,
$x\in X$, there is $f\in H^{0}(X,L^{k})$, verifying the following inequality

$|f(x)|_{1}\geq\exp(-\epsilon k)\|f\|_{L^{\infty}_{k}(X,h^{L}_{1})},$ (4.9)

where $|\cdot|_{i}$, $i=1,2$, are the pointwise norms induced by $h^{L}_{i}$.
Remark, however, that

$\|f\|_{L^{\infty}_{k}(X,h^{L}_{0})}\geq|f(x)|_{0},$ (4.10)

and we clearly have

$|f(x)|_{0}=\Big{(}\frac{h^{L}_{0}}{h^{L}_{1}}\Big{)}^{\frac{k}{2}}(x)\cdot|f(x)|_{1}.$
(4.11)

From (4.9), (4.10) and (4.11), we conclude that for any $\epsilon>0$, there is
$p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, $x\in X$, there is
$f\in H^{0}(X,L^{k})$, such that we have

$\|f\|_{L^{\infty}_{k}(X,h^{L}_{0})}\geq\exp(-\epsilon
k)\cdot\Big{(}\frac{h^{L}_{0}}{h^{L}_{1}}\Big{)}^{\frac{k}{2}}(x)\cdot\|f\|_{L^{\infty}_{k}(X,h^{L}_{1})}.$
(4.12)

Moreover, one can interchange the roles of $h^{L}_{0},h^{L}_{1}$ in (4.12). In
particular, we obtain that

$d_{\infty}({\rm{Ban}}^{\infty}(h^{L}_{0}),{\rm{Ban}}^{\infty}(h^{L}_{1}))\geq
d_{\infty}(h^{L}_{0},h^{L}_{1}).$ (4.13)

From (4.8) and (4.13), we deduce Theorem 1.7. ∎

Theorem 1.7 goes in line with the general philosophy that the geometry of
$\mathcal{H}^{L}$ can be approximated by the geometry of the space of norms on
$H^{0}(X,L^{k})$, as $k\to\infty$, see Donaldson [25]. To explain this
precisely, we recall the basics of the geometry on the space of Kähler
potentials. To make our presentation compatible with the notations introduced
previously, we will only work with Kähler potentials in the integral class
$c_{1}(L)$, but everything we say below can be trivially extended to the case
of a general (non-integral) Kähler class.

First of all, remark that the tangent space $T\mathcal{H}^{L}$ of continuous
psh metrics $\mathcal{H}^{L}$ on $L$ can be seen as a subbundle of the trivial
vector bundle

$T\mathcal{H}^{L}\subset\mathcal{H}^{L}\times\mathscr{C}^{0}(X),$ (4.14)

as follows. Let $h_{t}^{L}$, $t\in[0,1]$, be a $\mathcal{C}^{1}$-path in
$\mathcal{H}^{L}$. Then
$\dot{h}_{t}^{L}\mathrel{\mathop{\ordinarycolon}}=(h_{t}^{L})^{-1}\frac{d}{dt}h_{t}^{L}\in\mathscr{C}^{0}(X)$
gives the needed tangent vector in the identification.

For $p\in[1,+\infty]$, let us define the $L^{p}$-norm $\|\cdot\|_{p}$ for
$u\in T\mathcal{H}^{L}$ as follows

$\|u\|_{p}=\Big{(}\frac{1}{\int c_{1}(L)^{n}}\int_{X}|v|^{p}(x)\cdot
c_{1}(L,h^{L})^{n}\Big{)}^{\frac{1}{p}},$ (4.15)

where $u=(h^{L},v)$ in the identification (4.14) and $c_{1}(L,h^{L})^{n}$ is
the measure constructed from the procedure of Bedford-Taylor, see (2.17).

In particular, in the most relevant case for this article, $p=+\infty$, we
have

$\|u\|_{\infty}={\rm{ess\,sup}}|v|,$ (4.16)

where the essential supremum over $X$ is taken with respect to the measure
$c_{1}(L,h^{L})^{n}$.

For $p=2$, this (Hermitian) norm in the context of Kähler potentials was
defined by Mabuchi. For other $p$, it was studied by Darvas in [16].

We define the $L^{\infty}$-length ${\rm{len}}(l)$, of a piece-wise
$\mathscr{C}^{1}$-curve
$l(t)\mathrel{\mathop{\ordinarycolon}}=h^{L}_{t}\in\mathcal{H}^{L}$,
$t\in[0,1]$, as

${\rm{len}}(l)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{2}\int_{0}^{1}\big{\|}\dot{h}_{t}^{L}\big{\|}_{\infty}dt.$
(4.17)

The normalizing factor $\frac{1}{2}$, usually omitted in this setting, is here
to make things compatible with our previous normalizations, adapted to the
study of norms and not metrics. We define the supremum
($L^{\infty}$)-pseudodistance ${\rm{dist}}(h^{L}_{0},h^{L}_{1})$ between
$h^{L}_{0},h^{L}_{1}\in\mathcal{H}^{L}$ as follows

${\rm{dist}}(h^{L}_{0},h^{L}_{1})=\inf{\rm{len}}(l),$ (4.18)

where the infimum is taken among all possible piece-wise
$\mathscr{C}^{1}$-paths $l$ from $h^{L}_{0}$ to $h^{L}_{1}$.

The fact that ${\rm{dist}}$ is actually a distance (i.e. separates distinct
points) is a nontrivial result in this infinite dimensional setting. Since
$L^{\infty}$-distance introduced above obviously dominates $L^{p}$-distances
for any $p\in[1,\infty[$, for smooth endpoints, the fact that ${\rm{dist}}$ is
actually a distance follows from the work of Chen [13], who proved that the
$L^{2}$-pseudodistance is actually a distance. For general endpoints it
follows from Darvas [16], who refined this statement for $p=1$.

Our interest in this metric comes from the following result.

######  Theorem 4.5.

The distance ${\rm{dist}}$ from (4.18), and the $d_{\infty}$ from (1.18),
coincide.

######  Remark 4.6.

If we combine Theorems 1.7 and 4.5, we see that the limit of the Goldman-
Iwahori distances on the space of norms on $H^{0}(X,L^{k})$ coincides with the
$L^{\infty}$-Mabuchi distance on $\mathcal{H}^{L}$. It complements the results
of Chen-Sun [14], who proved a similar result relating the distance induced by
the Hilbert-Shmidt norm on the space of norms on $H^{0}(X,L^{k})$ and the
Mabuchi distance on $\mathcal{H}^{L}$, Berndtsson [6] who generalized the last
statement for the $L^{p}$-distances with smooth positive exteremities, $1\leq
p<+\infty$, and Darvas-Lu-Rubinstein [18], who generalized the statement of
Berndtsson for paths with non-smooth extremities.

The proof of this result is decomposed into a number of statements, which
essentially follow from the works of Berndtsson, Darvas and Lempert. To
explain it in details, we need to recall the notion of weak geodesics. Let
$h^{L}_{0},h^{L}_{1}$ be two smooth positive metrics on $L$. A geodesic
equation (with respect to $L^{2}$-metric) for a smooth curve $h^{L}_{t}$
connecting $h^{L}_{0}$ and $h^{L}_{1}$ was found by Mabuchi [40]. Later,
Semmes [50] and Donaldson [25] independently discovered that this equation can
be understood as a complex Monge-Ampère equation.

More precisely, they proved that if $h^{L}_{t}$, $t\in[0,1]$, is a geodesic
segment in the sense of Mabuchi, then for a Hermitian metric $\tilde{h}^{L}$
on $L$, over $X\times S_{1,2}$,
$S_{1,2}\mathrel{\mathop{\ordinarycolon}}=\\{s\in\mathbb{C}\mathrel{\mathop{\ordinarycolon}}1\leq\operatorname{Re}s\leq
2\\}$, defined as $\tilde{h}^{L}|_{X\times
u}\mathrel{\mathop{\ordinarycolon}}=h^{L}_{\operatorname{Re}u-1}$, $u\in
S_{1,2}$, the following identity over $X\times S_{1,2}$ holds

$c_{1}(L,\tilde{h}^{L})^{n+1}=0.$ (4.19)

Analogously to the smooth setting, a segment $h^{L}_{t}\in\mathcal{H}^{L}$,
$t\in[0,1]$, is called weak geodesic if the complexification of it,
$\tilde{h}^{L}$, constructed as above, satisfies (4.19) in the Bedford-Taylor
sense.

The existence and uniqueness of weak geodesics $h^{L}_{t}\in\mathcal{H}^{L}$,
$t\in[0,1]$, with smooth positive extremities $h^{L}_{0}$, $h^{L}_{1}$, was
established by Chen [13]. Chen, moreover, proved that in the above notations,
$c_{1}(L,\tilde{h}^{L})$ is a bounded $(1,1)$-form. By standard regularity
results, it implies that the path $h^{L}_{t}$ viewed as a metric over
$X\times[0,1]$ is $\mathscr{C}^{1,\alpha}$, for any $0\leq\alpha<1$. From the
general theory of Dirichlet problem for complex Monge-Ampère equations of
Bedford-Taylor [2, Theorem D], weak geodesics exist more generally for
$h^{L}_{0},h^{L}_{1}\in\mathcal{H}^{L}$. Those geodesics are moreover unique
by the “minimum principle”, see [2, Theorem A]. The following result explains
their relevance for us.

######  Lemma 4.7.

Let $h^{L}_{t}\in\mathcal{H}^{L}$, $t\in[0,1]$, be a weak geodesic. Then
${\rm{dist}}(h^{L}_{0},h^{L}_{1})=\frac{1}{2}\|\dot{h}_{0}^{L}\|_{\infty}$.

###### Proof.

It is a direct consequence of Lempert [35, Theorem 8.1]. In fact, in the
notations of Lempert, the Lagrangian $\|\cdot\|_{\infty}$ is convex and
invariant under strict rearrangements, see [35, Definition 2.1] for necessary
definitions. Hence, [35, Theorem 8.1] applies. ∎

The next result we need relates the right-hand side from Lemma 4.7 with weak
geodesics.

######  Lemma 4.8.

Let $h^{L}_{t}\in\mathcal{H}^{L}$, $t\in[0,1]$, be a weak geodesic with smooth
positive extremities $h^{L}_{0}$, $h^{L}_{1}$. Then the essential supremums of
the velocities of $h^{L}_{t}$ are constant in $t\in[0,1]$, and, despite the
fact that $h^{L}_{t}$ are not necessarily strictly positive, they coincide
with the supremums. More precisely, for any $t_{0},t_{1}\in[0,1]$, we have

$\|\dot{h}_{t_{0}}^{L}\|_{\infty}=\sup|\dot{h}_{t_{0}}^{L}|=\sup|\dot{h}_{t_{1}}^{L}|=\|\dot{h}_{t_{1}}^{L}\|_{\infty}=2\cdot
d_{\infty}(h^{L}_{0},h^{L}_{1}).$ (4.20)

###### Proof.

First of all, by an estimate of Darvas [17, Theorem 3.4], for any $t\in[0,1]$,
we have

$2\cdot d_{\infty}(h^{L}_{0},h^{L}_{1})\geq\sup|\dot{h}_{t}^{L}|.$ (4.21)

Now, since the path $h^{L}_{t}$ is $\mathscr{C}^{1}$, for any $x\in X$, we
have

$\log\frac{h^{L}_{1}(x)}{h^{L}_{0}(x)}=\int_{0}^{1}\dot{h}_{t}^{L}(x).$ (4.22)

In particular, we obtain that

$\Big{|}\log\frac{h^{L}_{1}(x)}{h^{L}_{0}(x)}\Big{|}\leq\sup_{t\in[0,1]}|\dot{h}_{t}^{L}(x)|.$
(4.23)

By combining (4.21) with (4.23) and using the fact that $\sup_{x\in
X}|\dot{h}_{t}^{L}(x)|$ is continuous for $t\in[0,1]$, we obtain that for any
$t\in[0,1]$, we have

$2\cdot d_{\infty}(h^{L}_{0},h^{L}_{1})=\sup_{x\in X}|\dot{h}_{t}^{L}(x)|.$
(4.24)

On the other hand, since $h^{L}_{0}$ is smooth and positive,
$c_{1}(L,h^{L}_{0})^{n}$ is a smooth (strictly) positive volume form. Hence
the essential supremum of $|\dot{h}_{0}^{L}|$ coincides with the supremum. In
other words, we have the following identity

$\sup_{x\in X}|\dot{h}_{0}^{L}(x)|=\|\dot{h}_{0}^{L}\|_{\infty}.$ (4.25)

Recall, however, that by a result of Berndtsson [6, Proposition 2.2], for any
function $f\in\mathscr{C}^{0}(\real)$, and any $t_{1},t_{2}\in[0,1]$, the
following conservation law holds

$\int_{X}f(\dot{h}_{t_{1}}^{L})c_{1}(L,h_{t_{1}}^{L})^{n}=\int_{X}f(\dot{h}_{t_{2}}^{L})c_{1}(L,h_{t_{2}}^{L})^{n}.$
(4.26)

In particular, by choosing a sequence of functions $f$ appropriately, we
obtain that

$\|\dot{h}_{t_{1}}^{L}\|_{\infty}=\|\dot{h}_{t_{2}}^{L}\|_{\infty}.$ (4.27)

A combination of (4.24), (4.25) and (4.27) finishes the proof. ∎

###### Proof of Theorem 4.5.

A combination of Lemmas 4.7 and 4.8 gives the result for smooth positive
extremities. Remark, however, that any $h^{L}\in\mathcal{H}^{L}$ can be
approximated uniformly by smooth positive metrics $h^{L}_{k}$,
$k\in\mathbb{N}$. To see this, consider the approximation
$h^{L}_{k}\mathrel{\mathop{\ordinarycolon}}=FS({\rm{Hilb}}_{k}(h^{L}))$, and
use Theorem 2.15. The result now follows by the fact that Theorem 4.5 holds
for $h^{L}_{k}$, by approximation and triangle inequality. ∎

###  4.3 Fubini-Study operator on graded multiplicatively generated norms

The main goal of this section is to study the convergence of the Fubini-Study
operator associated to graded pieces of multiplicatively generated norms, i.e.
to prove Theorem 4.1.

The proof of this result decomposes into a series of small statements. Our
first lemma, which is a generalization of Theorem 1.1, was the starting point
for the formulation of Definition 1.3.

######  Lemma 4.9.

For any smooth positive metric $h^{L}$ on $L$, there are $p_{0}\in\mathbb{N}$,
$C>0$, such that for any $r\in\mathbb{N}^{*}$, $k;k_{1},\ldots,k_{r}\geq
p_{0}$, $k_{1}+\cdots+k_{r}=k$, under the map (1.2), we have

$\prod_{i=1}^{r}\Big{(}1-\frac{C}{k_{i}}\Big{)}\leq\frac{[{\rm{Hilb}}_{k_{1}}(h^{L})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L})]}{{\rm{Hilb}}_{k}(h^{L})}\cdot\Big{(}\frac{k_{1}\cdots
k_{r}}{k}\Big{)}^{\frac{n}{2}}\leq\prod_{i=1}^{r}\Big{(}1+\frac{C}{k_{i}}\Big{)}.$
(4.28)

######  Remark 4.10.

In particular, the graded norms ${\rm{Hilb}}(h^{L})$ are multiplicatively
generated for smooth positive $h^{L}$, and we can take
$f(k)\mathrel{\mathop{\ordinarycolon}}=f^{\prime}(k)=\frac{n}{2}\log(k)+2+\log\dim
H^{0}(X,L^{k})$ in Definition 1.3.

###### Proof.

It follows from Theorem 1.1 and Corollary B.7 by induction. ∎

In the next lemma we study how the Fubini-Study maps associated for different
parameters $k$ relate to each other.

######  Lemma 4.11.

Assume that $p_{0}\in\mathbb{N}$ is such that $L^{p_{0}}$ is very ample. Then
for any Hermitian graded norm $N=\sum_{k=1}^{\infty}N_{k}$ on $R(X,L)$ and for
any $r\in\mathbb{N}^{*}$, $k;k_{1},\ldots,k_{r}\geq p_{0}$,
$k_{1}+\cdots+k_{r}=k$, the following identity is satisfied

${\rm{FS}}(N_{k_{1}})\cdots{\rm{FS}}(N_{k_{r}})={\rm{FS}}([N_{k_{1}}\otimes\cdots\otimes
N_{k_{r}}]).$ (4.29)

###### Proof.

For simplicity, let us establish the result for $N=2$, as the general case is
an easy adaptation of this particular one. To simplify the notation, let us
fix two sufficiently big numbers $k,l\in\mathbb{N}$. Recall that the Segre
embedding

${\rm{Seg}}_{k,l}\mathrel{\mathop{\ordinarycolon}}\mathbb{P}(H^{0}(X,L^{k})^{*})\times\mathbb{P}(H^{0}(X,L^{l})^{*})\to\mathbb{P}(H^{0}(X,L^{k})^{*}\otimes
H^{0}(X,L^{l})^{*}),$ (4.30)

is defined as follows $[x]\times[y]\mapsto[x\otimes y]$. Under (4.30), the
tautological line bundle over $\mathbb{P}(H^{0}(X,L^{k})^{*}\otimes
H^{0}(X,L^{l})^{*})$ gets pulled back to the $\boxtimes$-product of the
tautological line bundles on $\mathbb{P}(H^{0}(X,L^{k})^{*})$ and
$\mathbb{P}(H^{0}(X,L^{l})^{*})$. By Lemma B.2, this map is an isometry once
the tautological line bundle over $\mathbb{P}(H^{0}(X,L^{k})^{*}\otimes
H^{0}(X,L^{l})^{*})$ is endowed with the metric induced by the Fubini-Study
metric associated to $N_{k}^{*}\otimes N_{l}^{*}$ and the tautological line
bundles on $\mathbb{P}(H^{0}(X,L^{k})^{*})$ and
$\mathbb{P}(H^{0}(X,L^{l})^{*})$ are induced by the Fubini-Study metrics
associated to $N_{k}^{*}$ and $N_{l}^{*}$ respectively.

An easy verification shows that the following diagram is commutative

$\begin{CD}X@>{{\rm{Kod}}_{k}\times{\rm{Kod}}_{l}}>{}>\mathbb{P}(H^{0}(X,L^{k})^{*})\times\mathbb{P}(H^{0}(X,L^{l})^{*})\\\
@V{}V{{{\rm{Kod}}_{k+l}}}V@V{}V{{{\rm{Seg}}_{k,l}}}V\\\
\mathbb{P}(H^{0}(X,L^{k+l})^{*})@>{({\rm{Mult}}_{k,l})^{*}}>{}>\mathbb{P}(H^{0}(X,L^{k})^{*}\otimes
H^{0}(X,L^{l})^{*}).\end{CD}$ (4.31)

Moreover, the map $({\rm{Mult}}_{k,l})^{*}$ induces the isomorphism of the
corresponding tautological line bundles. This isomorphism becomes an isometry
if the tautological line bundle over $\mathbb{P}(H^{0}(X,L^{k+l})^{*})$ is
induced by the Fubini-Study metric associated to the metric on
$H^{0}(X,L^{k+l})^{*}$ obtained from $N_{k}^{*}\otimes N_{l}^{*}$ and the
inclusion $({\rm{Mult}}_{k,l})^{*}$, and the tautological line bundle over
$\mathbb{P}(H^{0}(X,L^{k})^{*}\otimes H^{0}(X,L^{l})^{*})$ is induced by the
Fubini-Study metric associated to $N_{k}^{*}\otimes N_{l}^{*}$. Remark,
however, that by Lemmas B.1 and B.3, the dual of the above norm on
$H^{0}(X,L^{k+l})^{*}$ coincides with $[N_{k}\otimes N_{l}]$. This now
finishes the proof by the commutativity of square (4.31) and the above
isometries. ∎

Now, to finally establish Theorem 4.1, we need the following simple lemma.

######  Lemma 4.12.

Assume that $k\in\mathbb{N}^{*}$ is such that $L^{k}$ is very ample and the
norms $N_{k}$, $N^{\prime}_{k}$ over $H^{0}(X,L^{k})$ satisfy the inequality
$N_{k}\leq C^{k}\cdot N^{\prime}_{k}$ for a given constant $C>0$. Then we have

${\rm{FS}}(N_{k})\leq C^{k}\cdot{\rm{FS}}(N^{\prime}_{k}).$ (4.32)

If the graded norms $N=\sum_{k=1}^{\infty}N_{k}$,
$N^{\prime}=\sum_{k=1}^{\infty}N^{\prime}_{k}$ on $R(X,L)$ are moreover
equivalent, then, as $k\to\infty$, the following convergence holds uniformly

$\Big{(}\frac{FS(N_{k})}{FS(N^{\prime}_{k})}\Big{)}^{\frac{1}{k}}\to 1.$
(4.33)

###### Proof.

It follows trivially from the definition of the Fubini-Study operator from
(2.30). ∎

###### Proof of Theorem 4.1.

Let us denote by $f$ the function from Definition 1.3 associated to the norm
$N$. From Lemmas 4.11 and 4.12, we conclude that there is
$p_{0}\in\mathbb{N}$, such that for any $r\in\mathbb{N}^{*}$,
$k;k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, we have

${\rm{FS}}(N_{k_{1}})\cdots{\rm{FS}}(N_{k_{r}})\leq{\rm{FS}}(N_{k})\cdot\exp\Big{(}f(k_{1})+\cdots+f(k_{r})+f(k)\Big{)}.$
(4.34)

Now, let $h^{L}$ be an arbitrary smooth positive Hermitian metric on $L$.
According to Lemmas 4.9 and 4.12, for the function $f^{\prime}$ as in Remark
4.10, we see that there is $p_{1}\geq p_{0}$, such that for any
$r\in\mathbb{N}^{*}$, $k;k_{1},\ldots,k_{r}\geq p_{1}$,
$k_{1}+\cdots+k_{r}=k$, we have

${\rm{FS}}({\rm{Hilb}}_{k_{1}}(h^{L}))\cdots{\rm{FS}}({\rm{Hilb}}_{k_{r}}(h^{L}))\\\
\geq{\rm{FS}}({\rm{Hilb}}_{k}(h^{L}))\cdot\exp\Big{(}f^{\prime}(k_{1})+\cdots+f^{\prime}(k_{r})+f^{\prime}(k)\Big{)}.$
(4.35)

From (4.34) and (4.35), we conclude that the sequence of functions
$g_{k}\mathrel{\mathop{\ordinarycolon}}X\to]-\infty,+\infty[$,
$k\in\mathbb{N}$, defined as follows

$g_{k}\mathrel{\mathop{\ordinarycolon}}=\log\Big{(}\frac{{\rm{FS}}({\rm{Hilb}}_{k}(h^{L}))}{{\rm{FS}}(N_{k})}\Big{)},$
(4.36)

is uniformly almost sub-additive in the sense of Section A.2. By symmetry, the
sequence $g_{k}$ is also uniformly almost super-additive. By a version of
Fekete’s subadditive lemma and Dini’s lemma, see Lemma A.7, we conclude that
there is a continuous function
$g\mathrel{\mathop{\ordinarycolon}}X\to]-\infty,+\infty[$, such that
$\frac{g_{k}}{k}$ converges to $g$ uniformly. By this and Tian’s theorem, cf.
Remark 2.16, we conclude from (4.36) that, as $k\to\infty$, the following
uniform convergence of metrics on $L$ holds

${\rm{FS}}(N_{k})^{\frac{1}{k}}\to e^{-g}\cdot h^{L}.$ (4.37)

Since ${\rm{FS}}(N_{k})$ are psh, we conclude that $e^{-g}\cdot h^{L}$ is psh
as well. ∎

Finally, motivated by the methods of this section, let us establish that
graded multiplicatively generated norms decart at most exponentially one from
another.

######  Lemma 4.13.

Let $N\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}N_{k}$,
$N^{\prime}\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}N^{\prime}_{k}$
be two graded multiplicatively generated norms on $R(X,L)$. Then there is
$C>1$, such that for any $k\in\mathbb{N}$, we have

$C^{-k}\cdot N^{\prime}_{k}\leq N_{k}\leq C^{k}\cdot N^{\prime}_{k}.$ (4.38)

######  Remark 4.14.

In particular, we have $d_{\infty}(N,N^{\prime})<+\infty$.

###### Proof.

Assume for simplicity that we can take $p_{0}=1$ in Definition 1.3. Then we
see that there is a function
$f\mathrel{\mathop{\ordinarycolon}}\mathbb{N}\to\real$, verifying $f(k)=o(k)$,
as $k\to\infty$, and such that

$N_{k}\leq N_{1}\otimes\cdots\otimes N_{1}\cdot\exp(f(n)+nf(1)).$ (4.39)

Similarly, for $N^{\prime}$ we can write the opposite inequality

$N^{\prime}_{1}\otimes\cdots\otimes N^{\prime}_{1}\leq
N^{\prime}_{k}\cdot\exp(f^{\prime}(n)+nf^{\prime}(1)),$ (4.40)

where the function
$f^{\prime}\mathrel{\mathop{\ordinarycolon}}\mathbb{N}\to\real$ verifies
$f^{\prime}(k)=o(k)$, as $k\to\infty$. In particular, by the assumption on the
growth of $f$, and the fact that all norms on $H^{0}(X,L)$ are equivalent, we
see that there is a constant $C>0$, such that for any $k\in\mathbb{N}$, the
left side of the inequality (4.38) holds. By symmetry, we establish also the
inequality from the right side. ∎