# GradeADreamer: Enhanced Text-to-3D Generation
Using Gaussian Splatting and Multi-View Diffusion

Trapoom Ukarapol
Tsinghua University
<EMAIL_ADDRESS>Kevin Pruvost
Tsinghua University
<EMAIL_ADDRESS>
###### Abstract

Text-to-3D generation has shown promising results, yet common challenges such
as the Multi-face Janus problem and extended generation time for high-quality
assets. In this paper, we address these issues by introducing a novel three-
stage training pipeline called GradeADreamer. This pipeline is capable of
producing high-quality assets with a total generation time of under 30 minutes
using only a single RTX 3090 GPU. Our proposed method employs a Multi-view
Diffusion Model, MVDream, to generate Gaussian Splats as a prior, followed by
refining geometry and texture using StableDiffusion. Experimental results
demonstrate that our approach significantly mitigates the Multi-face Janus
problem and achieves the highest average user preference ranking compared to
previous state-of-the-art methods. The project code is available at
https://github.com/trapoom555/GradeADreamer.

##  1 Introduction

3D Content Creation is pivotal in numerous fields, including virtual and
augmented reality, game design, head modeling, and beyond. One particularly
innovative area is text-to-3D generation, which enables the creation of
imaginative 3D models based on textual descriptions. This has emerged as a
rapidly growing research area in recent years. Considerable advancements were
achieved through deep learning techniques [36, 3, 8] in the beginning.

Currently, more novel methods [4, 21, 32, 35, 14, 17, 27, 24, 30, 40, 5]
utilize 3D representations, such as NeRF [26] or Gaussian Splatting [16], for
3D asset representation. These methods optimize the process by distilling a
pre-trained text-to-image generation model to guide 3D content generation.
Papers such as Dreamfusion [27] and Magic3D [21] employ StableDiffusion [29]
which use it as supervision for the optimization process of the 3D Model via
Score Distillation Sampling (SDS) [27].

Figure 1: High-quality assets generated by GradeADreamer

Apart from SDS [27], several other score distillation methods have been
proposed. For example, Variational Score Distillation (VSD) [35], first
introduced in ProlificDreamer [35], argues that SDS [27] leads to poor quality
because it requires a high classifier-free guidance (CFG) [12] weight. In
contrast, VSD [35] is compatible with the low CFG [12] used in text-to-image
generation models. Additionally, Interval Score Matching (ISM) [20] utilizes
Denoising Diffusion Implicit Models (DDIM) [33] inversion to obtain more
consistent pseudo-ground-truths during distillation. Score distillation
remains a debated and actively researched subfield. Our results demonstrate
that using a simple SDS [27] is sufficient to produce higher quality assets
than previously thought as shown in Figure 1.

A common problem in text-to-3D generation is the Multi-Face Janus Problem [6],
which involves view inconsistency in score-distilling text-to-3D generation,
leading to unrealistic and geometrically collapsed 3D objects as illustrated
in Figure LABEL:fig:multiface. A Multi-View Diffusion Model, such as MVDream
[32], addresses this problem by conditioning the generated image from the
text-to-image diffusion model with camera poses, thereby improving the model’s
viewpoint awareness. This method can significantly reduce the Multi-Face Janus
Problem [6]. However, despite these advancements, MVDream [32] often produces
poorer styles compared to StableDiffusion [29], due to the limitations of the
rendered dataset [32].

To achieve high-quality 3D asset generation with a low occurrence rate of the
Multi-Face Janus Problem [6] and low generation time, we propose
GradeADreamer, a solution with a pipeline described in three passes. To start,
we initialize random 3D Gaussian Splats [16] and optimize them with the Multi-
View Diffusion Model, MVDream [32], to establish a base for further
optimization. Next, we enter a Gaussian Splats Refinement phase, where we
refine the Gaussian Splats [16] using StableDiffusion [28] to achieve better
geometrical details. After this, we employ an efficient mesh extraction method
described in DreamGaussian [34]. Finally, we perform a Texture Optimization
pass, utilizing the Appearance Refinement technique outlined in Fantasia3D
[4].

All passes focus solely on using SDS [27] as the score distillation method to
optimize each stage of the 3D content generation pipeline. We leverage
Gaussian Splatting [16] and mesh as our 3D representations for geometry
optimization and texture optimization, respectively.

In summary, our contributions are as follows:

  1. 1.

We propose GradeADreamer, a high-quality text-to-3D generation model that
significantly reduces the occurrence of the Multi-face Janus Problem [6] and
achieves faster generation time.

  2. 2.

We introduce a three-stage optimization process that disentangles geometry and
texture optimization while utilizing the Multi-view diffusion model for prior
Gaussian Splats [16] generation.

  3. 3.

We evaluate GradeADreamer both qualitatively and quantitatively, demonstrating
that it outperforms previous state-of-the-art text-to-3D generation methods
[27, 21, 35] in several aspects, including user preference, Multi-face Janus
[6] occurrence rate, and generation time.

  4. 4.

We publicly open-source our text-to-3D generation code and integrate our
project in threestudio [11].

##  2 Related Works

###  2.1 3D Representations

In the current state of research for text-to-3D generation, there are multiple
3D Representations being used, the most popular ones are Neural Radiance
Fields (NeRF) [26] and Gaussian Splats [16]. They are particularly useful for
this field of research as they are both trainable and they also currently are
the most efficient representations to use for that purpose. NeRF trains a
neural network [26] whereas Gaussian Splatting optimizes 3D Gaussians [16].
The current trends exhibit a comparable prevalence of NeRF-based text-to-3D
studies [35, 39, 23, 18, 15] and those based on Gaussian Splatting [34, 38,
19, 25, 37].

The differences between NeRF [26], Gaussian Splats [16], and other more
traditional 3D representations lie in their optimization capabilities.
Techniques such as photogrammetry are not efficient for these tasks. NeRF [26]
and Gaussian Splats [16], on the other hand, are specifically designed to be
optimized in the tasks such as 3D rendering and reconstruction, offering more
advanced and efficient solutions for generating detailed 3D models from
textual descriptions.

The selection of NeRF [26] or Gaussian Splatting [16] for text-to-3D
generation depends largely on the specific requirements of the research task.
NeRF typically yields higher-quality results but requires more time to
converge compared to Gaussian Splatting methods [5, 34]. This trade-off
between quality and convergence time underscores the importance of choosing
the appropriate 3D generation technique based on the specific computational
resources and performance criteria of the research objectives.

###  2.2 Score Distillation Methods

Due to the limited availability of 3D data for training text-to-3D models,
Score Distillation Methods are widely used to effectively distill knowledge
from pre-trained text-to-image diffusion models, such as StableDiffusion [28].
This approach leverages the strengths of these well-trained text-to-image
models to achieve state-of-the-art performance in text-to-3D generation. By
transferring the expertise from 2D to 3D, Score Distillation Methods enhance
the capability to generate accurate and detailed 3D models from textual
descriptions, overcoming the data scarcity challenge in the 3D domain.

Score Distillation Sampling (SDS) [27] and Variational Score Distillation
(VSD) [35] have demonstrated significant promise in recent state-of-the-art
advancements in text-to-3D generation. While SDS and VSD share this
foundational approach, VSD sets itself apart from SDS by employing particle-
based variational inference [1, 7, 22]. This technique is perceived to offer
advantages by mitigating issues such as oversaturation and oversmoothing.
However, our findings show that a well-designed pipeline using SDS alone can
effectively address these challenges, demonstrating that SDS can achieve
comparable performance without the additional complexity introduced by VSD.

###  2.3 Mesh Extraction from Gaussian Splatting

Gaussian Splatting [16] represents 3D space using Gaussian Splats, which are
not directly compatible with the traditional 3D mesh formats commonly employed
in industry. Consequently, additional steps are required to convert these
splats into conventional mesh representations suitable for practical
applications.

For this task, studies such as SuGaR [10] initially utilize a traditional mesh
extraction method as outlined in the original Gaussian Splatting paper [16].
Subsequently, they refine this approach by employing a hybrid representation
of meshes and Gaussian Splats. This refinement enhances accuracy and
smoothness, enabling the achievement of state-of-the-art results.

Most of the research done on mesh extraction is based not just on Gaussian
Splats [16] as input but also on images [10, 2, 9, 31] from which the mesh
should be extracted from. A subtask that is not explored enough would be
efficient mesh extraction from Gaussian Splats alone.

##  3 Approach

Figure 3: Overview of GradeADreamer. The proposed method consists of three
optimization steps. The first step involves optimizing random Gaussian Splats
using MVDream to obtain a Gaussian Splats prior (see Section 3.2.1). In the
second step, this prior is refined using StableDiffusion (see Section 3.2.2).
Finally, the third step employs texture optimization on a mesh, guided by
StableDiffusion (see Section 3.2.3).

We introduce a novel text-to-3D training pipeline designed to maintain low
generation time while mitigating the Multi-face Janus Problem [6]. Our
pipeline is composed of three distinct stages:

  1. 1.

Gaussian Splats Prior Generation

  2. 2.

Gaussian Splats Refinement

  3. 3.

Texture Optimization

The training pipeline is executed sequentially. We follow the approach of
Fantasia3D [4] in disentangling geometry and texture optimization. The first
two stages focus on geometry optimization, while the last stage is responsible
for texture optimization. However, unlike Fantasia3D [4], which requires a
handcrafted mesh for initialization, our pipeline can produce 3D models
without any mesh prior. An overview of our proposed pipeline is shown in
Figure 3.

###  3.1 3D Representation Choices

For the first and second stages, we utilize Gaussian Splatting [16] as our 3D
representation. This choice is based on its significantly faster convergence
compared to Neural Radiance Fields (NeRF) [26] for 3D generation tasks, as
demonstrated by DreamGaussian [34]. To achieve high-quality textured meshes,
we extract meshes from the Gaussian Splats [16] and then proceed with texture
refinement during the later stages of our pipeline.

###  3.2 Training Stages

####  3.2.1 Gaussian Splats Prior Generation

Starting with random Gaussian Splats [16], we employ MVDream [32], a 2D multi-
view diffusion model, as our guidance model to optimize the random Gaussian
Splats [16] into a Gaussian Prior with coarse details of the result. This
approach effectively reduces the likelihood of encountering the Multi-face
Janus Problem [6].

####  3.2.2 Gaussian Splats Refinement

In this stage, we refine the Gaussian Splats [16] to enhance the geometry
using StableDiffusion 2.1 [29] as our guidance. This step is crucial because
MVDream [32] can only generate low-resolution images (256x256), whereas the
StableDiffusion 2.1 [29] can produce higher resolution outputs (512x512). We
also observe, similar to MVDream [32], that the StableDiffusion 2.1 [29]
yields better generation styles, likely due to the influence of the rendered
training dataset on MVDream [32] outputs.

Following the Gaussian Refinement stage, we extract the mesh using an
efficient mesh extraction algorithm as described by DreamGaussian [34].

####  3.2.3 Texture Optimization

In the final stage, we optimize the spatially varying Physically-Based
Rendering (PBR) material model on the mesh. Specifically, we focus on three
distinct components of the material model: the diffuse map
$k_{d}\in\mathbb{R}^{3}$, which represents the inherent color of the material
without any lighting effects; the roughness and metallic term
$k_{rm}\in\mathbb{R}^{2}$, which determines the micro-surface detail and
metalness of the material; and the normal variation term
$k_{n}\in\mathbb{R}^{3}$, which details surface texture by perturbing the
normal vectors of the surface. For this step, we follow the techniques
outlined in Fantasia3D [4] to achieve high-quality, detailed textures.

###  3.3 Score Distillation Choices

We examine the differences between Score Distillation Sampling (SDS) proposed
in Dreamfusion [27] and Variational Score Distillation (VSD) proposed in
ProlificDreamer [35]. Our observations are as follows:

  * •

Training Time : We observe that using Variational Score Distillation (VSD)
[35] extends the training time compared to using SDS [27].

  * •

Quality of Results : Our results demonstrate that using SDS [27] is sufficient
for generating high-quality 3D models in text-to-3D generation tasks.

Based on these findings, we employ SDS [27] as our objective for score
distillation in all three stages of our pipeline.

The gradient of the SDS loss, as described in [27], can be expressed by the
following equation:

$\nabla_{\theta}\mathcal{L}_{\text{SDS}}=\mathbb{E}_{\epsilon,t}\left[w(t)(\epsilon_{\phi}(x_{t}|y,t)-\epsilon)\frac{\partial
x}{\partial\theta}\right]$ (1)

where $\theta$ represents the parameters of a 3D representation, $x$ is an
image rendered from this 3D representation through a transformation function
$g$ expressed as $x=g(\theta)$. $x_{t}$is a residual noise image predicted by
a pretrained 2D diffusion model $\epsilon_{\phi}$ conditioned on a prompt
embedding $y$ and a timestep $t$. And $\epsilon$ is a noisy image drawn from a
normal distribution, $\epsilon\sim\mathcal{N}(0,I)$.

##  4 Experiments

In this section, we detail our experiment implementation and validate the
proposed approach’s effectiveness through both quantitative and qualitative
analyses by comparing GradeADreamer to previous state-of-the-art methods in
text-to-3D generation.

###  4.1 Implementation Details

####  4.1.1 Training Resources

Our entire pipeline can be run on a single RTX 3090 GPU, with a generation
time of only 30 minutes per 3D asset. In stage 1, it takes 12 GB of GPU RAM
and 4 minutes; stage 2 requires 15 GB and 6 minutes; and stage 3 uses 16 GB
and 16 minutes. The remaining 4 minutes are allocated for mesh extraction,
Gaussian Splats [16] export, and the final result validation step.

####  4.1.2 Hyperparameters

##### Initialization

We initialize 6000 points of Gaussian Splats [16]. We’ve found that using a
large number of initial Gaussian Splats [16] extends the optimization time and
may not converge in some cases.

##### Gaussian Splats Prior Generation

In the first stage, similar to Dreamtime [13], we employ a linear annealing
timestep from 0.98 to 0.02 for the entire 700 optimization steps. We’ve found
that applying the linear annealing timestep yields better results in geometry.
The densification and pruning interval is set to 55. Following DreamGaussian
[34], the gradient threshold for densification is set to 0.01, and we perform
an opacity reset at step 500. In this stage, we use a batch size of 1,
consisting of 4 image views as proposed in MVDream [32].

##### Gaussian Splats Refinement

In the second stage, there are 700 optimization steps. We utilize a linear
annealing timestep from 0.98 to 0.02 for the first 200 steps. For steps
200-300, we uniformly sample the timestep from the range 0.02 to 0.98,
$t\sim\mathcal{U}(0.02,0.98)$, and we anneal it to
$t\sim\mathcal{U}(0.02,0.50)$ for the last 400 steps to focus on distilling
elaborate details, following ProlificDreamer [35]. We set the densification
interval to 50 and the opacity reset interval to 300. The batch size is set to
2, as we observe that it converges faster than a batch size of 1 and yields a
reasonable geometric appearance.

##### Gaussian Training

In the first two stages, we use a linear decaying positional learning rate for
Gaussian Splatting from 0.001 to $1.6\times 10^{-6}$ over 300 steps, and we
set the feature and opacity learning rates to 0.005 and 0.05, respectively.

##### Texture Optimization

For the last stage, we follow the hyperparameters provided by Fantasia3D [4],
optimizing for 2000 iterations with a batch size of 4 using an SDS [27] weight
strategy equal to 1. We employ SDS [27] as a score distillation objective for
all stages with a classifier-free guidance (CFG [12]) value of 100.

###  4.2 Baselines

We utilize three previous state-of-the-art text-to-3D generation models as our
baselines: DreamFusion [27], Magic3D [21], and ProlificDreamer [35]. Our
results will be compared to these baselines both qualitatively and
quantitatively in the following sections.

###  4.3 Qualitative Evaluations

Figure 4: Qualitative results

We qualitatively assess the performance of our proposed method, GradeADreamer,
against the established baselines. As illustrated in Figure 4, our results
significantly improve the geometric features of the generated assets,
addressing the geometry flaws caused by the Multi-face Janus Problem [6] that
other baselines [27, 21, 35] encounter. Additionally, our method produces
high-frequency, high-quality textures, whereas DreamFusion [27] and Magic3D
[21] struggle with issues of over-saturation and over-smoothing. For more
qualitative results, please refer to the Appendix B.

###  4.4 Quantitative Evaluations

####  4.4.1 User Study

We conducted a user study to evaluate our text-to-3D model against other
baselines [27, 21, 35]. The study involved 54 participants who were asked to
rank 3D models generated from 15 different prompts. These results were non-
cherry-picked, meaning all data was selected without selectively showcasing
only the best outcomes. This approach ensures a fair comparison and avoids the
bias and misleading outcomes that come from cherry-picking. Participants
ranked the results from best to worst for each prompt. As shown in Figure 5,
Our model decisively outperformed the others, achieving a significantly higher
win rate for the top rank compared to the baselines [27, 21, 35].
Additionally, the average rankings presented in Table 1 demonstrate that our
model gets the highest average ranking among the four selected methods. This
study shows that the 3D models created by our pipeline were consistently
preferred by users.

Figure 5: Rank distribution comparison from user study Name | Average Ranking ($\downarrow$)
---|---
DreamFusion | 3.29
Magic3D | 3.23
ProlificDreamer | 2.02
GradeADreamer (ours) | 1.49
Table 1: Average ranking from the user study

####  4.4.2 3D-FID Score

We also performed an additional evaluation method utilizing a variant of the
Fréchet Inception Distance (FID) score. The FID score is a widely used metric
for comparing two datasets, specifically quantifying how well generated
content captures the statistical properties of the original content. This
metric is predominantly used in the context of 2D image generation, where it
requires the generated images and a set of ground truth images or approximate
target results for comparison.

In our study, GradeADreamer operates on 3D generated content, necessitating
certain modifications to the standard FID calculation. To address this, we
employed a 3D-FID score. The process for calculating the 3D-FID score involves
generating a reference image using StableDiffusion [28], followed by
calculating a 2D-FID score for a series of multiple views of the 3D generated
model. Specifically, we used 10 different views by rotating the model along
the yaw axis. The final 3D-FID score is obtained by averaging the scores from
each of these views.

In our comparison table of 3D-FID scores (see Table 2), we computed the
average score from 15 randomly generated content samples. Each Text-to-3D
model was tasked with generating the same content, and the 3D-FID score was
calculated for each model, followed by averaging the scores from the 15
samples. This rigorous evaluation demonstrated the robustness of
GradeADreamer, as evidenced by its 3D-FID score.

Name | 3D-FID ($\downarrow$)
---|---
DreamFusion | 68.41
Magic3D | 75.96
ProlificDreamer | 58.75
GradeADreamer (ours) | 47.69
Table 2: 3D-FID Score

####  4.4.3 Multi-face Janus Problem

We measured the occurrence in percentage of the Multi-face Janus Problem [6]
over 15 prompts using our model, compared to other baselines [27, 21, 35]. As
shown in Table 3, by using MVDream [32] as a guidance model for the first
stage, we achieved a significantly lower occurrence rate of the Multi-face
Janus Problem [6], approximately 6 to 7 times lower than the baselines [27,
21, 35].

Name | % Multi-face ($\downarrow$)
---|---
DreamFusion | 35.71
Magic3D | 35.71
ProlificDreamer | 40.00
GradeADreamer (ours) | 6.67
Table 3: Percentage of Multi-face Janus problem [6] occurrence

####  4.4.4 Runtime

We evaluated the generation time per prompt for the selected four methods to
assess the efficiency of our proposed method. The results are summarized in
Table 4.

Our model achieves a significant reduction in generation time compared to
Magic3D [21] and ProlificDreamer [35], making it highly suitable for
applications requiring quick asset generation. The equivalence in speed to
DreamFusion [27], one of the fastest models, highlights the effectiveness of
our pipeline without compromising on the quality of the generated assets, as
proven in Table 1 and Table 2. This efficiency is crucial for the applications
and scenarios where rapid prototyping and iteration are necessary.

Name | Generation Time ($\downarrow$)
---|---
DreamFusion | 30 mins
Magic3D | 1 hour
ProlificDreamer | 10 hours
GradeADreamer (ours) | 30 mins
Table 4: Generation time per prompt

##  5 Conclusion

In this paper, we present GradeADreamer, a novel text-to-3D generation
pipeline designed to produce high-quality 3D assets while addressing the
Multi-face Janus Problem [6] and maintaining low generation time. Our pipeline
comprises three stages: Gaussian Splats Prior Generation, Gaussian Splats
Refinement, and Texture Optimization. The first two stages leverage Gaussian
Splatting as a 3D representation to optimize geometry, while the third stage
focuses on texture optimization using Mesh representation to ensure high-
quality textures.

To mitigate the Multi-face Janus Problem [6], we employ a Multi-view Diffusion
model, MVDream [32], in the initial stage to generate priors, and subsequently
use StableDiffusion [29] for enhanced generation quality. Our experimental
results demonstrate that GradeADreamer achieves the highest average ranking in
user studies and the lowest generation time, around 30 minutes per asset,
compared to all baseline models. Additionally, the occurrence rate of the
Multi-face Janus Problem [6] significantly decreased.

##### Limitations

Despite these advancements, GradeADreamer has some limitations. For example,
the performance is constrained by the StableDiffusion [29] Text Encoder, and
there are still instances of the Multi-face problem since we still use
StableDiffusion [29] as guidance for the last two stages. Furthermore, there
is potential for improvement in the mesh extraction method from Gaussian
Splats to obtain higher quality mesh.

In summary, GradeADreamer represents a significant step forward in the field
of text-to-3D generation, combining efficiency with high-quality output and
effectively addressing key challenges in the domain. Future work will focus on
overcoming the current limitations and further enhancing the performance and
quality of the generated assets.

##  6 Contribution

##### Trapoom Ukarapol

Propose the main idea of GradeADreamer, implementing the pipeline, generate
results, qualitative evaluation, Multi-face Janus Evaluation, user study,
presentation and paper work.

##### Kevin Pruvost

Mesh extraction experiments, geometry optimization experiments, generate
results, 3D-FID evaluation, user study, presentation and paper work.

##  7 Acknowledgement

We would like to express our sincere gratitude to the Deep Learning 2024
Spring course at Tsinghua University for providing us with a robust
foundational knowledge of deep learning and offering us the opportunity to
undertake a project aligned with our interests. We are particularly grateful
to Professor Xiaolin Hu and Professor Jun Zhu, the course lecturers, for their
invaluable guidance. Additionally, we extend our appreciation to Professor
Hualiu Ping and Professor Liu Yongjin for supplying the necessary
computational resources. Finally, we would like to thank the 54 participants
in our user study for their valuable contributions.

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##  Appendix A Evaluation Prompts

We selected 15 commonly used prompts for text-to-3D generation from several
research papers to ensure fair evaluations. The prompts are listed in Table 5

# | Prompt
---|---
1 | A DSLR photo of a panda
2 | A high quality photo of a furry corgi
3 |  A blue jay standing on a large basket of rainbow macarons
4 | A ripe strawberry
5 | A DSLR photo of an ice cream sundae
6 | A DSLR photo of car made out of sushi
7 | A pineapple
8 | A DSLR photo of an astronaut is riding a horse
9 | A model of a house in Tudor style
10 | A highly detailed sand castle
11 | A plate piled high with chocolate chip cookies
12 |  A zoomed out DSLR photo of a baby bunny sitting on top of a stack of pancakes
13 | A photo of a horse walking
14 |  A DSLR photo of a plate of fried chicken and waffles with maple syrup on them
15 | A bulldog wearing a black pirate hat
Table 5: Prompts used for Evaluations

##  Appendix B Additional Qualitative Results

We provide more text-to-3D qualitative comparisons of our proposed method
against the other baselines in Figure 6, 7, and 8.

Figure 6: More qualitative results Figure 7: More qualitative results Figure
8: More qualitative results


# Infinite-dimensional flats in the space of positive metrics on an ample line
bundle

Rémi Reboulet  and  David Witt Nyström

###### Abstract.

We show that any continuous positive metric on an ample line bundle $L$ lies
at the apex of many infinite-dimensional Mabuchi-flat cones. More precisely,
given any bounded graded filtration $\mathcal{F}$ of the section ring of $L$,
the set of bounded decreasing convex functions on the support of the
Duistermaat–Heckman measure of $\mathcal{F}$ embeds $L^{p}$-isometrically into
the space of bounded positive metrics on $L$ with respect to Darvas’ $d_{p}$
distance for all $p\in[1,\infty)$, and in particular with respect to the
Mabuchi metric ($p=2$).

###### Contents

  1. 1 Quantisation of positive metrics.
  2. 2 Filtrations and Duistermaat–Heckman measures.
  3. 3 Embedding the flat cone.

## Introduction.

We consider the space of bounded positive metrics on an ample line bundle $L$
over a projective manifold $X$. Mabuchi [Mab87] defined a distance between two
smooth positive metrics $\phi_{0}$, $\phi_{1}$ on $L$, with

$d_{2}(\phi_{0},\phi_{1})^{2}:=\inf_{\\{\gamma_{t}\\}}\int_{0}^{1}\int_{X}|\dot{\gamma}_{t}|^{2}\mathrm{MA}(\gamma_{t})\,dt,$

where the infimum is taken over smooth paths $\\{\gamma_{t}\\}$ joining
$\phi_{0}$ and $\phi_{1}$. This distance then extends by continuity to bounded
positive metrics.

By work of Chen [C+00] and Darvas [Dar15], this distance in fact makes it into
a geodesic metric space, which is by itself remarkable: indeed, the Mabuchi
distance is highly nonlinear, and geodesics are given as solutions to a
degenerate complex Monge–Ampère equation [Sem92].

Endowed with the Mabuchi distance, the set $L^{\infty}\cap\mathrm{PSH}(X,L)$
of positive bounded metrics on $L$ furthermore has negative curvature, in the
sense of Busemann, i.e. the distance function between two geodesics is convex
[CC21]. In this paper, we will be interested in finding flat subspaces of
$L^{\infty}\cap\mathrm{PSH}(X,L)$, i.e. isometric embeddings of (subsets of) a
real vector space into it. Geodesics are of course examples of one-dimensional
flats, but in this paper, we show that this space in fact admits infinite-
dimensional flats.

To construct such subspaces, we will use the method of quantisation. Recall
that the space of continuous positive metrics is closely related to the spaces
of Hermitian norms on $H^{0}(X,kL)$. Indeed, fixing a smooth volume form $dV$
on $X$, a continuous positive metric $\phi$ induces a Hermitian norm
$\left\|\cdot\right\|_{k\phi}$ on the space of sections $H^{0}(X,kL)$ by

$\left\|s\right\|_{k\phi}^{2}:=\int_{X}|s|^{2}_{k\phi}\,dV.$

Conversely, a Hermitian norm induces a metric via the Fubini–Study map:
letting $(s_{k,i})_{i}$ be an orthonormal basis for
$\left\|\cdot\right\|_{k\phi}$, one defines the smooth positive metric

$\phi_{k}:=k^{-1}\log\sum_{i}|s_{k,i}|^{2}.$

It is well-known from results on quantisation [Bou90, Cat99, Tia90, Zel98]
that, at the limit, these metrics recover $\phi$:

$\lim_{k\to\infty}\phi_{k}=\phi.$

Chen–Sun [CS12] indeed show that the space of positive metrics is a
Gromov–Hausdorff limit of these spaces of Hermitian norms.

Such spaces of norms admit many flat subspaces: fixing a basis of
$H^{0}(X,kL)$, the subclass of Hermitian norms admitting it as an orthogonal
basis is in fact isometric to $\mathbb{R}^{\dim H^{0}(X,kL)}$. Those subspaces
are called apartments and fit more generally into the structure of an
Euclidean building. It has been conjectured for a long time, see Donaldson
[Don99, Section 6(iii)] and Codogni [Cod19, Section 7], that these structures
should reflect to an asymptotic building structure on the space of Kähler
metrics. Our result is in this direction.

We now consider a bounded graded filtration $\mathcal{F}$ of the section ring
of $L$ (in the sense of [BC11]), which means that on each graded piece
$H^{0}(X,kL)$, we have an induced decreasing, left-continuous filtration
$\mathcal{F}^{\lambda}H^{0}(X,kL)$ which satisfies a submultiplicativity
property, in that

$\mathcal{F}^{\lambda}H^{0}(X,kL)\cdot\mathcal{F}^{\gamma}H^{0}(X,mL)\subseteq\mathcal{F}^{\lambda+\gamma}H^{0}(X,(k+m)L).$

To such a filtration are associated its jumping numbers $\lambda_{i,k}$, which
are the (ordered) values of $\lambda$ at which the filtration changes
dimension. Such filtrations were shown by the second author [Nys12] to arise
from test configurations of $(X,L)$, i.e. $\mathbb{C}^{*}$-equivariant one-
parameter degenerations of $(X,L)$, although they are strictly more general
than test configurations [Szé15].

Furthermore, to a filtration $\mathcal{F}$ one can associate its
Duistermaat–Heckman measure $\sigma(\mathcal{F})$ [BC11], which is given as
the weak limit of the measures

$\sigma_{k}(\mathcal{F}):=k^{-n}\sum_{i}\delta_{\lambda_{i,k}/k}.$

Let us now pick a function $\mathfrak{f}$ on the support of the
Duistermaat–Heckman measure that is bounded, convex and decreasing. Note that
one can always choose an orthogonal basis $(s_{k,i})$ for
$\left\|\cdot\right\|_{k\phi}$ which is adapted to $\mathcal{F}$, in that
$\mathcal{F}^{\lambda_{i}}H^{0}(X,kL)$ is spanned by $\\{s_{k,j}:j\geq i\\}$.
We then define a modified Fubini–Study metric:

$\phi_{k}^{\mathfrak{f}}:=k^{-1}\log\sum_{i}|s_{k,i}|^{2}e^{k\cdot\mathfrak{f}(\lambda_{k,i}/k)}.$

Our main result is then the following:

Main Theorem (3.5). Let $\phi$ be a continuous positive metric on $L$, and let
$\mathcal{F}$ be a bounded graded filtration on $L$. Then, we have that:

  1. (1)

for any $\mathfrak{f}$ bounded, convex and decreasing on the support of
$\sigma(\mathcal{F})$, the limit

$\phi^{\mathfrak{f}}:=\mathrm{usc}\,\lim_{k\to\infty}\phi_{k}^{\mathfrak{f}}$

exists, and defines a bounded positive metric on $L$;

  2. (2)

given two such functions $\mathfrak{f}$ and $\mathfrak{g}$, we have that

$d_{2}(\phi^{\mathfrak{f}},\phi^{\mathfrak{g}})=\|f-g\|_{L^{2}(\sigma(\mathcal{F}))}.$

Remark 1. The second statement implies that the map
$\mathfrak{f}\mapsto\phi^{\mathfrak{f}}$ is an isometric embedding of the cone
of bounded, convex and decreasing functions on the support of
$\sigma(\mathcal{F})$ equipped with the $L^{2}_{\sigma(\mathcal{F})}$-metric
into the cone of bounded positive metrics on $L$ equipped with the Mabuchi
metric. Therefore, for any bounded graded filtration $\mathcal{F}$, there
exists an associated infinite-dimensional flat cone with apex $\phi$.

Remark 2. Phong–Sturm [PS07] (in the case of test configurations) and
Ross–Witt Nyström [RN14] (in general) constructed a geodesic ray, i.e. a one-
dimensional flat cone, associated to the data of a bounded graded filtration
on the section ring of $L$. One recovers those results by taking the family of
functions $\mathfrak{f}_{t}:x\mapsto-t\cdot x$, so that
$\\{\phi^{\mathfrak{f}_{t}}\\}$ coincides with that geodesic ray. In fact, our
construction can be seen as first modifying the filtration $\mathcal{F}$ by
the data of the function $\mathfrak{f}$, in a way that is flat in a certain
sense of the metric geometry of filtrations, and then taking $t=1$ in this
geodesic ray construction.

Remark 3. Later on, Darvas [Dar15] introduced Finsler structures $d_{p}$,
$p\in[1,\infty)$, on the space of positive metrics, recovering Mabuchi’s
metric for $p=2$, and this circle of ideas has blossomed into a field of its
own [Dar19]. Our result also covers the case $p\in[1,\infty)$, in that

$d_{p}(\phi^{\mathfrak{f}},\phi^{\mathfrak{g}})=\|f-g\|_{L^{p}(\sigma(\mathcal{F}))}.$

Ideas for the proof. Let us fix a continuous positive metric $\phi$ on $L$ and
a bounded graded filtration $\mathcal{F}$ of the section ring of $L$. While
the language of filtrations is likely more familiar to the reader, we prefer
to consider instead bounded graded non-Archimedean norms
$\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$, which are in one-to-one
correspondence with filtrations: indeed, on each piece $H^{0}(X,kL)$ of the
section ring, a filtration $\mathcal{F}^{\lambda}_{k}$ induces a non-
Archimedean vector space norm by setting

$\|v\|_{k}^{\mathrm{NA}}=\sup\\{\lambda\in\mathbb{R},\,v\in\mathcal{F}^{\lambda}_{k}H^{0}(X,kL)\\};$

and this construction can be reversed. Due to the fact that we also work with
norms in quantisation, it will make proofs and parallels more clear.

We then consider the class of bounded, decreasing convex functions
$\mathfrak{f}$ on the support of the Duistermaat–Heckman measure of
$\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ (i.e. that of its associated
filtration). This class is naturally infinite-dimensional. The convexity
property is an analogue of submultiplicativity, and the decreasing property is
important to preserve the order of the jumping numbers of the filtration,
which become values of the logarithm of the non-Archimedean norm at an
orthogonal basis. Those two properties then allow us to construct another
bounded graded non-Archimedean norm on the section ring of $L$. The crucial
aspect of this article is that the construction associating a new norm to such
a function $\mathfrak{f}$ can in fact be made in a flat manner, in a certain
sense of metric geometry of graded norms. The flatness on the non-Archimedean
side, which can be seen as the boundary at infinity of the space of positive
metrics on $L$, in turn reflects on the space of positive metrics itself, via
Phong–Sturm’s construction. We are greatly helped by recent results of Finski
[Fin22, Fin23] that give algebraic conditions allowing us to understand the
metric geometry of positive metrics via the metric geometry of norms on the
section ring of $L$, which appear e.g. in quantisation of geodesic rays.

Structure of the paper. The first section gathers results from the literature
concerning spaces of norms and quantisation of positive metrics. It is
essentially expository.

The second section recalls notions on non-Archimedean norms and
Duistermaat–Heckman measures. We explain how to modify a filtration
(equivalently a non-Archimedean norm) using the data of a convex decreasing
function on the support of the Duistermaat–Heckman measure, then prove our
construction yields a new bounded graded norm (Theorem 2.1).

The third section is devoted to the proof of our main theorem. We first study
how to modify complex norms, and how this operation preserves
submultiplicativity. Finally, we prove the main result (Theorem 3.5). In the
last subsection, we discuss the results and list some open questions, in
particular related to Okounkov bodies.

Acknowledgements. The authors warmly thank Siarhei Finski for many
enlightening discussions and comments on a draft version of this article. He
in particular suggested that our results should work in a more general setting
than the one we initially considered, described in the last section on open
questions.

##  1\. Quantisation of positive metrics.

Let $X$ be a compact projective complex manifold, endowed with an ample line
bundle $L$. The principle of quantisation allows one to construct positive
Hermitian metrics on $L$ coming from sequences of norms on the spaces of
sections $H^{0}(X,kL)$. The ideal type of sequence of norms is a
submultiplicative sequence of Hermitian norms, as submultiplicative sequences
have good asymptotic behaviour, while Hermitian norms are easier to analyse,
notably because they always admit orthogonal bases.

Unfortunately, in Kähler geometry, most sequences used in quantisation fail to
be both submultiplicative and Hermitian. An important class of examples is
given by taking $L^{2}$ or $L^{\infty}$ norms associated to continuous metrics
on the line bundle, as we will explain below. The problem of relating
submultiplicativity and Hermitian-ness is at the heart of various problems,
including the construction of Mabuchi geodesics [PS07, Ber09, Fin22]. Indeed,
while one can obtain partial submultiplicativity for sequences of Hermitian
norms (coming from the Bernstein–Markov property [BB10, Nys14]), this is not
in general sufficient to yield convergence of the quantisation process, and is
harder to control under perturbations. Work of Berman–Berndtsson [BB10] and
recently Finski [Fin22] provides a way to bridge the gap between those two
notions.

###  1.1. The space of norms on a complex vector space.

We begin by recollecting facts about spaces of norms. Throughout this section,
$V$ will be a finite-dimensional vector space of dimension $N$ over
$\mathbb{C}$. We will denote by $\mathcal{N}(V)$ the space of norms on $V$,
and by $\mathcal{N}^{H}(V)\subset\mathcal{N}(V)$ the space of Hermitian norms
on $V$.

Those spaces can be endowed with quasi-metric structures. To that end, we
consider two norms $\left\|\cdot\right\|$,
$\left\|\cdot\right\|^{\prime}\in\mathcal{N}(V)$, and define their successive
minima:

(1.1)
$\lambda_{i}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}):=\sup_{W\in\mathrm{Grass}_{i}(V)}\inf_{w\in
W^{\times}}\log\frac{\left\|w\right\|^{\prime}}{\left\|w\right\|},$

$\mathrm{Grass}_{i}(V)$ denoting the Grassmannian of $i$-dimensional subspaces
of $V$. If both norms admit a common orthogonal basis $\underline{v}=(v_{i})$,
which is always the case if for example both norms are Hermitian, then up to
reordering one has

(1.2)
$\lambda_{i}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})=\log\frac{\left\|v_{i}\right\|^{\prime}}{\left\|v_{i}\right\|},$

corresponding to the logarithms of the eigenvalues of the transition matrix
between the two norms. For $p\in[1,\infty)$, one then defines

$d_{p}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})^{p}:=N^{-1}\sum_{i}|\lambda_{i}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})|^{p},$

and

$d_{\infty}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}):=\max_{i}|\lambda_{i}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})|.$

It is then clear that, for $p<q\in[1,\infty]$,

(1.3) $d_{1}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})\leq
d_{p}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})\leq
d_{q}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})\leq
d_{\infty}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}).$

The following fact is well-known for the Hermitian case (e.g. [BE21, Theorem
3.1]), and is explained in [Fin23, Eq (3.9)] in the non-Hermitian case:

######  Proposition 1.1.

For $p\in[1,\infty]$, $d_{p}$ defines a metric on $\mathcal{N}^{H}(V)$. It
satisfies a quasi-triangle inequality on $\mathcal{N}(V)$ in general: for
$\left\|\cdot\right\|$, $\left\|\cdot\right\|^{\prime}$,
$\left\|\cdot\right\|^{\prime\prime}$,

$d_{p}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})\leq
d_{p}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime\prime})+d_{p}(\left\|\cdot\right\|^{\prime\prime},\left\|\cdot\right\|^{\prime})+\log
N.$

The spaces $(\mathcal{N}^{H}(V),d_{p})$ are geodesic for $p\in[1,\infty]$ and
uniquely geodesic for $p\in(1,\infty)$. The unique geodesic is given by
picking a common orthogonal basis $\underline{v}$ for two norms
$\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}$, and defining for
$t\in[0,1]$:

$\left\|\sum
c_{i}v_{i}\right\|_{t}:=\sum|c_{i}|\left\|v_{i}\right\|^{1-t}\left\|v_{i}\right\|^{\prime}{}^{t}.$

We also define the relative volume of two norms by

$\mathrm{vol}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}):=N^{-1}\sum_{i}\lambda_{i}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}).$

This is a quantity that measures the discrepancy between the volumes of the
unit balls of two given norms. It is closely related to the $d_{1}$ distance:
indeed, if $\left\|v\right\|\leq\left\|\cdot\right\|^{\prime}[v]$ for all
$v\in V$, then

$d_{1}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})=\mathrm{vol}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}).$

One can show that this is a particular case of a more general formula using
the max operator in the Hermitian case. Given two Hermitian norms diagonalised
in a common basis ${\underline{v}}$, define
$\vee(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})$ as the unique norm
diagonalised in the basis ${\underline{v}}$ and such that

$\vee(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})(v_{i})=\max(\left\|v_{i}\right\|,\left\|v_{i}\right\|^{\prime}).$

A simple computation then shows that

(1.4)
$d_{1}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})=d_{1}(\left\|\cdot\right\|,\vee(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime}))+d_{1}(\left\|\cdot\right\|^{\prime},\vee(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})).$

###  1.2. Spaces of sequences of norms.

Let, as in the beginning of this section, $X$ be a compact projective complex
manifold endowed with an ample line bundle $L$. For clarity, we will denote
$V_{k}:=H^{0}(X,kL)$ and $N_{k}=\dim H^{0}(X,kL)=h^{0}(X,kL)$. We now take our
interest to sequences
$\left\|\cdot\right\|_{\bullet}:k\mapsto\left\|\cdot\right\|_{k}$, where
$\left\|\cdot\right\|_{k}$ is a norm on $V_{k}$.

Let us pick two such sequences $\left\|\cdot\right\|_{\bullet}$ and
$\left\|\cdot\right\|^{\prime}_{\bullet}$. Let us define, for
$p\in[1,\infty]$,

$d_{p}(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet}):=\limsup_{k}k^{-1}d_{p}(\left\|\cdot\right\|_{k},\left\|\cdot\right\|^{\prime}_{k}).$

We will say that $\left\|\cdot\right\|_{\bullet}$ is $p$-bounded relative to
$\left\|\cdot\right\|^{\prime}_{\bullet}$ if the limit above is finite. We
note that this only defines a pseudo-distance on the set of equivalence
classes of relatively $p$-bounded sequences of norms; in general, one defines
an equivalence relation whereby two such sequences are $p$-equivalent if and
only if their $d_{p}$ distance is zero.

We also define the relative volume of
$\left\|\cdot\right\|_{\bullet}$,$\left\|\cdot\right\|^{\prime}_{\bullet}$ as

$\mathrm{vol}(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet}):=\limsup_{k}k^{-1}\mathrm{vol}(\left\|\cdot\right\|_{k},\left\|\cdot\right\|^{\prime}_{k}).$

A quick computation shows that $\mathrm{vol}$ is $1$-Lipschitz with respect to
$d_{1}$, so that the relative volume is well-defined between two relatively
$1$-bounded sequences of norms. Likewise, defining the max operator for
sequences of Hermitian norms by

$\vee(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|_{\bullet}^{\prime}):k\mapsto\vee(\left\|\cdot\right\|_{k},\left\|\cdot\right\|^{\prime}_{k})$

shows that the asymptotic version of (1.4) also holds:

(1.5)
$d_{1}(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet})=d_{1}(\left\|\cdot\right\|_{\bullet},\vee(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet}))+d_{1}(\left\|\cdot\right\|^{\prime}_{\bullet},\vee(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet})).$

###  1.3. Quantisation.

Given a continuous positive metric $\phi$ on $L$, one can associate to it two
different norms on $V_{k}$: a Hermitian norm $\left\|\cdot\right\|_{k\phi}$,
given by

$\|s\|_{k\phi}^{2}=\int_{X}|s|e^{-k\phi}\,dV$

for a smooth volume form $dV$, and a non-necessarily Hermitian sup-norm

$\|s\|_{k\phi,\infty}:=\sup_{X}|s|e^{-k\phi}.$

To any norm $\left\|\cdot\right\|\in\mathcal{N}(V_{k})$, one can associate its
Fubini–Study metric, a bounded positive metric on $L$, defined at
$\ell\in(kL)_{x}$, $x\in X$, by

$|\ell|_{\mathrm{FS}(\left\|\cdot\right\|)}:=\inf_{s\in
V_{k},\,s(x)=\ell}\|s\|.$

If $\left\|\cdot\right\|$ is furthermore Hermitian, diagonalised in a basis
${\underline{v}}_{k}=(v_{1,k},\dots,v_{N_{k},k})$ one recovers [Fin22, Lemma
2.1] the usual definition

$\mathrm{FS}(\left\|\cdot\right\|)=\log\sum_{i}|v_{i,k}|^{2}e^{-2\left\|v_{i,k}\right\|}.$

Classical results on quantisation [Bou90, Cat99, Tia90, Zel98] show that

$\lim_{k}\mathrm{FS}(\left\|\cdot\right\|_{k\phi})=\lim_{k}\mathrm{FS}(\left\|\cdot\right\|_{k\phi,\infty})=\phi.$

The space of bounded positive metrics on $L$ can also be endowed with
distances closely related to the $d_{p}$ distances on spaces of norms [Dar15]:
given $\phi,\psi$ that are positive with respect to a Kähler form $\omega$
representing $c_{1}(L)$, one defines

$d_{p}(\phi,\psi):=\inf_{t\mapsto\gamma_{t}}\int_{0}^{1}\left(\int_{X}|\dot{\gamma}_{t}|^{p}\,\mathrm{MA}(\gamma_{t})\right)^{\frac{1}{p}}dt,$

where the infimum is taken over smooth curves joining the two metrics. The
case $p=2$ corresponds to the distance introduced by Mabuchi [Mab87]. The
general case was treated by Darvas [Dar15]. As it turns out, by results of
Berndtsson [Ber09], Darvas–Lu–Rubinstein [DLR20], and Finski [Fin22, Theorem
1.3, Corollary 3.6], this quantisation procedure can also be metrised:

######  Theorem 1.2.

Let $\phi,\psi$ be bounded positive metrics on $L$. Then,

$d_{p}(\phi,\psi)=d_{p}(\left\|\cdot\right\|_{\bullet\phi},\left\|\cdot\right\|_{\bullet\psi})=d_{p}(\left\|\cdot\right\|_{\bullet\phi,\infty},\left\|\cdot\right\|_{\bullet\psi,\infty})$

for $p\in[1,\infty)$, and also for $p=\infty$ if both metrics are continuous
(or more generally bounded and regularisable from below, [Fin22, Proposition
2.15]).

As explained at the start of the section, one easily sees that the sup-norms
associated to a continuous positive metric yield a submultiplicative sequence
of norms on the algebra of sections of $L$. This means that, given $s_{m}\in
V_{m}$, $s_{n}\in V_{n}$, one has that

$\left\|s_{m}\cdot
s_{n}\right\|_{(m+n)\phi,\infty}\leq\left\|s_{m}\right\|_{m\phi,\infty}\cdot\left\|s_{n}\right\|_{n\phi,\infty}.$

More generally, picking a submultiplicative sequence of norms
$\left\|\cdot\right\|_{\bullet}$ always ensures that the Fubini–Study operator

$\mathrm{FS}(\left\|\cdot\right\|_{\bullet}):=\lim_{k}{}^{*}\mathrm{FS}(\left\|\cdot\right\|_{k})$

converges, by Fekete’s lemma; furthermore, Fekete’s lemma ensures that we are
taking the usc regularisation of the supremum of a family of positive metrics,
which itself will be positive. To ensure that this operator takes finite
values, one must add another condition:

######  Definition 1.3.

Let $\left\|\cdot\right\|_{\bullet}$ be a submultiplicative sequence of norms
on the algebra of sections of $L$. We say that
$\left\|\cdot\right\|_{\bullet}$ is a bounded graded norm on $R(X,L)$ if there
exists a bounded positive metric on $L$ such that

$d_{\infty}(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|_{\bullet\phi,\infty})<\infty.$

In particular, as a consequence of the previous theorem and the classification
results for sequences of norms from [Fin22], one has

######  Theorem 1.4 ([Fin22, Corollary 3.6]).

If $\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet}$
are two bounded graded norms on $L$, then for all $p\in[1,\infty)$,

$d_{p}(\mathrm{FS}(\left\|\cdot\right\|_{\bullet}),\mathrm{FS}(\left\|\cdot\right\|^{\prime}_{\bullet}))=d_{p}(\left\|\cdot\right\|_{\bullet},\left\|\cdot\right\|^{\prime}_{\bullet}).$

Importantly, this allows us to work directly with general sequences of norms,
which do not a priori come from bounded metrics on $L$. The case $p=\infty$ is
more subtle, but will not be necessary for the purpose of this article.

###  1.4. More general sequences of norms.

We explain here how the aforementioned results on quantisation generalise to
more abstract norms than those that a priori come from positive metrics.

######  Definition 1.5.

Let $\left\|\cdot\right\|_{\bullet}$ be a bounded graded norm. We say that a
non-necessarily submultiplicative sequence of norms
$\left\|\cdot\right\|_{\bullet}^{\prime}$ is Bernstein–Markov with respect to
$\left\|\cdot\right\|_{\bullet}$ if, for all $\varepsilon>0$, there exists
$C_{\varepsilon}>1$ such that for all $k$,

$C_{\varepsilon}^{-1}e^{-\varepsilon
k}\left\|\cdot\right\|_{k}\leq\left\|\cdot\right\|^{\prime}_{k}\leq
C_{\varepsilon}e^{\varepsilon k}\left\|\cdot\right\|_{k},$

or equivalently

$d_{\infty}(\left\|\cdot\right\|_{k},\left\|\cdot\right\|^{\prime}_{k})\leq
k\varepsilon+\log C_{\varepsilon}.$

######  Example 1.6 ([BB10, Lemma 3.2]).

Given a continuous positive metric $\phi$ on $L$, the sequence of
$L^{2}$-norms associated to $\phi$ (with respect to a smooth volume form) is
Bernstein–Markov with respect to the sequence of sup-norms associated to
$\phi$.

######  Proposition 1.7.

Let $\left\|\cdot\right\|_{\bullet}$ be a bounded graded norm, and
$\left\|\cdot\right\|^{\prime}_{\bullet}$ be a sequence of norms that is
Bernstein–Markov with respect to $\left\|\cdot\right\|_{\bullet}$. Then,
$\mathrm{FS}(\left\|\cdot\right\|_{\bullet})$ exists as a positive bounded
metric, and

$\mathrm{FS}(\left\|\cdot\right\|_{\bullet})=\mathrm{FS}(\left\|\cdot\right\|^{\prime}_{\bullet}).$

###### Proof.

Because $\left\|\cdot\right\|_{\bullet}$ is submultiplicative, the limit
$\lim_{k}k^{-1}\mathrm{FS}(\left\|\cdot\right\|_{k})$ exists by Fekete’s
lemma, and equals $\sup_{k}k^{-1}\mathrm{FS}(\left\|\cdot\right\|_{k})$.
Boundedness is easily seen to ensure that this supremum is finite, hence
$\mathrm{FS}(\left\|\cdot\right\|_{\bullet})$ is the regularised supremum of a
bounded above and below family of positive metrics, hence is itself a bounded
positive metric. The equality then follows from the fact that
$\left\|\cdot\right\|\mapsto k^{-1}\mathrm{FS}(\left\|\cdot\right\|)$ is
obviously $1$-Lipschitz with respect to $d_{\infty}$ on
$\mathcal{N}^{H}(H^{0}(X,kL))$ and the supremum norm on
$L^{\infty}\cap\mathrm{PSH}(X,L)$, while the Bernstein–Markov property implies
that
$k^{-1}d_{\infty}(\left\|\cdot\right\|_{k},\left\|\cdot\right\|_{k}^{\prime})\to
0$. ∎

##  2\. Filtrations and Duistermaat–Heckman measures.

###  2.1. Bounded graded non-Archimedean norms.

If $V$ is as before a complex vector space of dimension $N$, one can also
consider the space of non-Archimedean norms on $V$. Namely, the trivial
absolute value $|\cdot|_{0}$ on $\mathbb{C}$, defined by $|z|_{0}=1$ if and
only if $z\neq 0$, is non-Archimedean: trivially, one sees that

$|z+z^{\prime}|\leq\max(|z|,|z^{\prime}|).$

A non-Archimedean norm on $V$ is a function
$\left\|\cdot\right\|^{\mathrm{NA}}:V\to\mathbb{R}$ satisfying the usual norm
axioms, but with respect to $|\cdot|_{0}$, and satisfying the ultrametric
triangle inequality:

$\left\|v+w\right\|^{\mathrm{NA}}\leq\max(\left\|v\right\|^{\mathrm{NA}},\left\|w\right\|^{\mathrm{NA}}).$

An exercise in linear algebra shows that such norms can all be constructed by
fixing a basis $\underline{v}=(v_{i})$ of $V$, real numbers
$\underline{a}=(a_{i})$, and setting

$\left\|\sum c_{i}v_{i}\right\|^{\mathrm{NA}}:=\max_{i,c_{i}\neq
0}e^{-a_{i}}.$

A basis $\underline{v}$ in which $\left\|\cdot\right\|^{\mathrm{NA}}$ can be
expressed as above is called an orthogonal basis, in the non-Archimedean
sense. Conversely, any non-Archimedean norm on $V$ admits an orthogonal basis
[BE21, Example 1.11].

There is also a trivial norm $\left\|\cdot\right\|_{0}$ on $V$, defined by
$\|v\|_{0}=1$ if and only if $v\neq 0$. It has the property that any basis of
$V$ is an orthogonal basis.

We denote the space of such norms by $\mathcal{N}^{\mathrm{NA}}(V)$. We note
that this space can also be metrised in a similar way as in the complex case,
but this will not be required for the purpose of this article. In fact, in a
strong sense, $\mathcal{N}^{\mathrm{NA}}(V)$ with such metric structures can
be understood as the boundary at infinity of the negatively curved space
$(\mathcal{N}^{H}(V),d_{p})$.

Returning now to the geometric setting, where $V_{k}=H^{0}(X,kL)$ for an ample
line bundle on $X$, and $\dim V_{k}=N_{k}$, one can also define an appropriate
notion of a sequence of norms on the section ring of $L$.

We define a bounded graded non-Archimedean norm on $L$ to be a sequence of
non-Archimedean norms $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$, where
$\left\|\cdot\right\|_{k}^{\mathrm{NA}}$ is a non-Archimedean norm on $V_{k}$,
satisfying the following two properties:

  1. (1)

submultiplicativity: for all $s_{m}\in H^{0}(X,mL)$ and $s_{n}\in
H^{0}(X,nL)$,

$\left\|s_{m}\cdot
s_{n}\right\|^{NA}_{m+n,\mathfrak{f}}\leq\left\|s_{m}\right\|^{\mathrm{NA}}_{m,\mathfrak{f}}\left\|s_{n}\right\|^{\mathrm{NA}}_{n,\mathfrak{f}};$

  2. (2)

boundedness: there exists $\delta>0$, such that for all $k$,

$e^{-\delta
k}\left\|\cdot\right\|_{0,k}\leq\left\|\cdot\right\|^{\mathrm{NA}}_{k}\leq
e^{\delta k}\left\|\cdot\right\|_{0,k}.$

Here, $\left\|\cdot\right\|_{0,k}$ is the trivial norm on $V_{k}$.

Bounded graded norms are in bijection with bounded graded filtrations of the
section ring, in the sense of [Nys12, BC11, BJ21], as in the introduction. In
particular, test configurations [BJ21] and divisorial valuations [BJ23] induce
examples of bounded graded norms. The setting of filtrations is more likely to
be familiar to the reader, but our results will be more easily stated in terms
of non-Archimedean norms, justifying our choice of notation.

###  2.2. The Duistermaat–Heckman measure and convex functions.

Let $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ be a bounded graded non-
Archimedean norm. To each $k$, one can define a measure
$\sigma_{k}(\left\|\cdot\right\|_{k}^{\mathrm{NA}})$ on the real numbers as
follows. Let $(s_{i,k})_{i}$ be an orthogonal basis for
$\left\|\cdot\right\|_{k}^{\mathrm{NA}}$. We then set

$\sigma_{k}(\left\|\cdot\right\|_{k}^{\mathrm{NA}}):=k^{-n}_{*}\sum_{i=1}^{N_{k}}\delta_{-k^{-1}\log\left(\left\|s_{i,k}\right\|^{\mathrm{NA}}\right)}.$

By [BC11, CM15], the measures
$\sigma_{k}(\left\|\cdot\right\|_{k}^{\mathrm{NA}})$ converge weakly to a
measure $\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}})$ supported on a
closed interval of $\mathbb{R}$, independently of the choice of orthogonal
basis for each $\left\|\cdot\right\|_{k}^{\mathrm{NA}}$. We define the
Duistermaat–Heckman measure of $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$
to be this weak limit.

Our main result in this section is the observation that, given a bounded
graded non-Archimedean norm and a bounded decreasing convex function
$\mathfrak{f}$ on the support of the Duistermaat–Heckman measure, one can
construct a different bounded graded non-Archimedean norm. This construction
will be essential in the proof of our main theorem.

The construction. Fix $\left\|\cdot\right\|^{\mathrm{NA}}_{\bullet}$, and
$(s_{i,k})_{i}$ a sequence of orthogonal bases as above. Let $\mathfrak{f}$ be
a bounded convex, decreasing function on the support of
$\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}})$.

Define now, for each $k$, a non-Archimedean norm
$\left\|\cdot\right\|^{\mathrm{NA}}_{k,\mathfrak{f}}$ as the unique norm
satisfying the following two properties:

  1. (1)

$\left\|\cdot\right\|^{\mathrm{NA}}_{k,\mathfrak{f}}$ admits $(s_{i,k})_{i}$
as an orthogonal basis;

  2. (2)

for each $i=1,\dots,N_{k}$, one has

$\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k,\mathfrak{f}}=\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k}e^{k\cdot
f\left(\frac{-\log\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k}}{k}\right)}.$

######  Theorem 2.1.

The sequence $\left\|\cdot\right\|_{\bullet,\mathfrak{f}}^{\mathrm{NA}}$ is a
bounded graded non-Archimedean norm.

###### Proof.

It is easy to see that, if $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ is
bounded, then its modification by $\mathfrak{f}$ is also bounded, with
additional constant $\delta=\sup\mathfrak{f}$. We thus only have to show
submultiplicativity: for $s_{m}\in H^{0}(X,mL)$ and $s_{n}\in H^{0}(X,nL)$, we
want to prove that

$\left\|s_{m}\cdot
s_{n}\right\|_{m+n,\mathfrak{f}}^{\mathrm{NA}}\leq\left\|s_{m}\right\|_{m,\mathfrak{f}}^{\mathrm{NA}}\left\|s_{n}\right\|_{n,\mathfrak{f}}^{\mathrm{NA}}.$

To that end, we proceed in a similar manner to [Reb22, Theorem 2.6.1]: we
consider first the following special case, where we assume that $s_{m}$
belongs to an orthogonal basis for $\left\|\cdot\right\|_{m}^{\mathrm{NA}}$,
and $s_{n}$ to an orthogonal basis for
$\left\|\cdot\right\|_{n}^{\mathrm{NA}}$. Let us write
$\alpha=-\log\left\|s_{m}\right\|^{\mathrm{NA}}_{m}$, and
$\beta=-\log\left\|s_{n}\right\|^{\mathrm{NA}}_{n}$, so that $\alpha/m$ and
$\beta/n$ belong to the support of
$\sigma(\left\|\cdot\right\|^{\mathrm{NA}}_{\bullet})$. Since $s_{m}\cdot
s_{n}\in H^{0}(X,(m+n)L)$, there is a decomposition

$s_{m}\cdot s_{n}=\sum a_{j}s_{j,m+n}$

according to the fixed orthogonal basis $(s_{j,m+n})$ for
$\left\|\cdot\right\|^{\mathrm{NA}}_{m+n}$. Let us write

$\gamma:=-\log\left\|s_{m}\cdot s_{n}\right\|_{m+n}^{\mathrm{NA}}.$

Since for all $j$ with $s_{j}\neq 0$ we have

$\left\|s_{j,m+n}\right\|_{m+n}^{\mathrm{NA}}\leq\max_{a_{j}\neq
0}\left\|s_{j,m+n}\right\|_{m+n}^{\mathrm{NA}}=\left\|s_{m}\cdot
s_{n}\right\|_{m+n}^{\mathrm{NA}},$

then for all $\delta_{j}=-\log\left\|s_{j,m+n}\right\|_{m+n}^{\mathrm{NA}}$
with $a_{j}\neq 0$ we have

$\gamma\leq\delta_{j},$

and furthermore since $\left\|\cdot\right\|^{\mathrm{NA}}_{\bullet}$ is
submultiplicative, one has

(2.1) $\alpha+\beta\leq\gamma\leq\delta_{j}.$

Note now that

$\frac{\alpha+\beta}{m+n}=\frac{m}{m+n}\frac{\alpha}{m}+\frac{n}{m+n}\frac{\beta}{n}.$

Convexity of $\mathfrak{f}$ then yields

$\mathfrak{f}\left(\frac{\alpha+\beta}{m+n}\right)\leq\frac{m}{m+n}\mathfrak{f}\left(\frac{\alpha}{m}\right)+\frac{n}{m+n}\mathfrak{f}\left(\frac{\beta}{n}\right),$

so that

(2.2) $(m+n)\mathfrak{f}\left(\frac{\alpha+\beta}{m+n}\right)\leq
m\mathfrak{f}\left(\frac{\alpha}{m}\right)+n\mathfrak{f}\left(\frac{\beta}{n}\right).$

With those results in hand, we can prove submultiplicativity in this first
special case. Indeed,

$\displaystyle\left\|s_{m}\cdot s_{n}\right\|_{m+n}^{\mathrm{NA}}$
$\displaystyle=\max_{a_{j}\neq
0}(e^{-\delta_{j}}e^{(m+n)\mathfrak{f}\left(\frac{\delta_{j}}{m+n}\right)}).$

By (2.1), and the fact that $\mathfrak{f}$ is decreasing, we have in
particular that

$\left\|s_{m}\cdot s_{n}\right\|_{m+n}^{\mathrm{NA}}\leq
e^{-(\alpha+\beta)}e^{(m+n)\mathfrak{f}\left(\frac{\alpha+\beta}{m+n}\right)}.$

Thus, by (2.2),

$\displaystyle\left\|s_{m}\cdot s_{n}\right\|_{m+n}^{\mathrm{NA}}$
$\displaystyle\leq
e^{-\alpha}e^{m\mathfrak{f}\left(\frac{\alpha}{m}\right)}e^{-\beta}e^{n\mathfrak{f}\left(\frac{\beta}{n}\right)}$
$\displaystyle=\left\|s_{m}\right\|_{m,\mathfrak{f}}^{\mathrm{NA}}\left\|s_{n}\right\|_{n,\mathfrak{f}}^{\mathrm{NA}}$

by definition.

We now show submultiplicativity in general. Let $s_{m}$, $s_{n}$ be arbitrary
sections, which we decompose in their respective orthogonal bases as

$s_{m}=\sum_{i}c_{i}s_{i,m},\,s_{n}=\sum_{j}c_{j}s_{j,n}.$

We then have that

$\displaystyle\left\|s_{m}\cdot s_{n}\right\|^{NA}_{m+n,mff}$
$\displaystyle\leq\max_{i,c_{i}\neq 0;j,c_{j}\neq 0}\left\|s_{i,m}\cdot
s_{j,n}\right\|^{NA}_{m+n,\mathfrak{f}}$
$\displaystyle\leq\left(\max_{i,c_{i}\neq
0}\left\|s_{i,m}\right\|^{NA}_{m,\mathfrak{f}}\right)\cdot\left(\max_{j,c_{j}\neq
0}\left\|s_{j,n}\right\|^{NA}_{n,\mathfrak{f}}\right)$
$\displaystyle=\left\|s_{m}\right\|^{NA}_{m,\mathfrak{f}}\left\|s_{n}\right\|^{NA}_{n,\mathfrak{f}}.$

The first inequality follows from the non-Archimedean triangle inequality
(with redundancies, since the $s_{i,m}\cdot s_{j,n}$ generate
$H^{0}(X,(m+n)L)$); the second inequality follows from the result proven above
in the case of orthogonal sections and splitting the independent maxima; and
the equality in the last line is definitional. ∎

The flat cone we will embed in the space of positive metrics is therefore the
space of bounded, decreasing convex functions on the support of the
Duistermaat–Heckman measure of a bounded graded norm. It is easily seen to be
infinite-dimensional, and flat since $L^{p}$ with $p>1$ is a strictly convex
norm.

##  3\. Embedding the flat cone.

###  3.1. Rescaling by filtrations, I: the finite-dimensional case.

Let $V$ be a finite-dimensional complex vector space, and $\underline{v}$ be a
fixed basis of $V$. We define the apartment
$\mathcal{A}_{\underline{v}}\subset\mathcal{N}^{H}(V)$ to be the set of all
Hermitian norms admitting ${\underline{v}}$ as an orthogonal basis. For
$p\in[1,\infty]$, this is easily seen to be isometric to
$(\mathbb{R}^{N},L^{p})$ via the following construction: one fixes a Hermitian
norm $\left\|\cdot\right\|\in\mathcal{A}_{\underline{v}}$, and one defines for
${\underline{a}}\in\mathbb{R}^{N}$

$\iota_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}})$

to be the unique Hermitian norm in $\mathcal{A}_{\underline{v}}$ such that,
for all $i$,

$\iota_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}})(v_{i})=\left\|v_{i}\right\|e^{-a_{i}}.$

This defines a rescaling map

$\iota_{\underline{v}}(\cdot,\cdot):\mathcal{A}_{\underline{v}}\times\mathbb{R}^{N}\to\mathcal{A}_{\underline{v}}.$

One would like to extend this definition to non-Hermitian norms. Of course,
one cannot in general pick an arbitrary diagonalising basis (as those may not
exist in general) for this operation. One instead uses an envelope-type
construction for rescaling. To that end, we construct a gauge non-Archimedean
norm as follows. Having fixed a basis $\underline{v}$, and a vector
${\underline{a}}$, let

$\left\|\cdot\right\|^{NA}_{{\underline{v}},\underline{a}}:V\to\mathbb{R}_{>0}$

be the non-Archimedean norm given by

$\left\|\sum\alpha_{i}v_{i}\right\|^{NA}_{\underline{a}}=\max_{i,\alpha_{i}\neq
0}e^{-a_{i}}.$

We then define the rescaling of $\left\|\cdot\right\|$ via

(3.1)
$\iota_{\underline{v}}^{1}(\left\|\cdot\right\|,{\underline{a}})(w):=\inf_{w=\sum
w_{i}}\left\\{\sum_{i}\left\|w_{i}\right\|\left\|w_{i}\right\|^{NA}_{{\underline{v}},\underline{a}}\right\\},$

where the infimum ranges over all decompositions of $w$ as a sum of vectors
$w_{i}\in V$. This gives a rescaling map

$\iota_{\underline{v}}^{1}(\cdot,\cdot):\mathcal{N}(V)\times\mathbb{R}^{N}\to\mathcal{N}(V).$

More generally, we will occasionally denote by

$\iota^{1}(\cdot,\cdot):\mathcal{N}(V)\times\mathcal{N}^{\mathrm{NA}}(V)\to\mathcal{N}(V)$

be the map sending a norm to its envelope by a non-Archimedean norm as defined
above. In this case,

$\iota^{1}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\mathrm{NA}}_{{\underline{v}},{\underline{a}}})=\iota^{1}_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}}).$

Note that, if $\left\|\cdot\right\|$ is Hermitian, then there always exists a
basis $\underline{v}$ which is orthogonal for $\left\|\cdot\right\|$ and
$\left\|\cdot\right\|^{\mathrm{NA}}$. Defining a vector $\underline{a}$ with
components $a_{i}=-\log\left\|v_{i}\right\|^{\mathrm{NA}}$, one then has that

$\iota^{1}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\mathrm{NA}})=\iota^{1}(\left\|\cdot\right\|,\underline{a})=\iota(\left\|\cdot\right\|,\underline{a}).$

Importantly, the vector $\underline{a}$ depends on the choice of such a
jointly orthogonal basis.

The following result, which follows from the case $p=\infty$ proven in [Fin22,
Lemma 2.8], shows that these rescaling operations have a very nicely
controlled metric distortion. In our applications, where we will consider
sequences of vector spaces $V_{k}$ whose dimensions will grow at most as
$k^{n}$ for some fixed $n$, this will imply that this distortion is only
growing as $o(k)$.

######  Lemma 3.1.

Let $\underline{v}$ be a basis of $V$, ${\underline{a}}\in\mathbb{R}^{N}$, and
let $\left\|\cdot\right\|\in\mathcal{A}_{\underline{v}}(V)$,
$\left\|\cdot\right\|^{\prime}\in\mathcal{N}(V)$. Then, for $p\in[1,\infty]$,

$d_{p}(\iota_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}}),\iota_{\underline{v}}^{1}(\left\|\cdot\right\|^{\prime},{\underline{a}}))\leq
d_{\infty}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})+\log\dim V.$

###### Proof.

For $t\in[0,\infty)$, let
$\left\|\cdot\right\|_{t}\in\mathcal{A}_{\underline{v}}(V)$ be given by

$\left\|v_{i}\right\|_{t}:=\left\|v_{i}\right\|e^{-ta_{i}}.$

In particular,
$\left\|\cdot\right\|_{1}=\iota_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}})$.
Applying [Fin22, Lemma 2.8] at $t=1$ then gives

$d_{\infty}(\iota_{\underline{v}}(\left\|\cdot\right\|,{\underline{a}}),\iota_{\underline{v}}^{1}(\left\|\cdot\right\|^{\prime},{\underline{a}}))\leq
d_{\infty}(\left\|\cdot\right\|,\left\|\cdot\right\|^{\prime})+\log\dim V,$

and the result follows since $d_{p}\leq d_{\infty}$. ∎

###  3.2. Rescaling by filtrations, II: sequences of norms.

Consider now $\phi$ a continuous positive metric on an ample line bundle $L$
over $X$. Let $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ be a bounded
graded norm on $L$. Then, it is well-known that for each $k$, one can find a
basis $\underline{s}_{k}:=(s_{i,k})_{i}$ of $V_{k}=H^{0}(X,kL)$ which is both
orthogonal in the Hermitian sense for $\left\|\cdot\right\|_{k\phi}$ and in
the non-Archimedean sense for $\left\|\cdot\right\|_{k}^{\mathrm{NA}}$. Let
$\mathfrak{f}$ be a bounded, decreasing convex function on the support of the
Duistermaat–Heckman measure
$\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}})$, which gives a bounded
graded norm $\left\|\cdot\right\|_{\bullet,\mathfrak{f}}^{\mathrm{NA}}$ as in
Theorem 2.1. We then define the two following modifications:

  1. (1)

$\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f})$ is the sequence of
Hermitian norms given in degree $k$ by

$\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{f}):=\iota_{\underline{s}_{k}}(\left\|\cdot\right\|_{k\phi},\left\|\cdot\right\|_{k,\mathfrak{f}}^{\mathrm{NA}}),$

which can be described as the unique Hermitian norm on $V_{k}$ admitting
$\underline{s}_{k}$ as an orthogonal basis and such that for all $i$,

$\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{f})(s_{i,k})=\left\|s_{i,k}\right\|_{k\phi}e^{k\mathfrak{f}(-k^{-1}\log\left\|s_{i,k}\right\|_{k}^{\mathrm{NA}})};$

  2. (2)

$\iota^{1}(\left\|\cdot\right\|_{\bullet\phi,\infty},\mathfrak{f})$ is the
sequence of norms given in degree $k$ by

$\iota^{1}(\left\|\cdot\right\|_{k\phi,\infty},\mathfrak{f}):=\iota^{1}(\left\|\cdot\right\|_{k\phi,\infty},\left\|\cdot\right\|_{k,\mathfrak{f}}^{\mathrm{NA}}).$

######  Theorem 3.2.

In the notations above,

  1. (1)

$\iota^{1}(\left\|\cdot\right\|_{\bullet\phi,\infty},\mathfrak{f})$ is bounded
and submultiplicative;

  2. (2)

$\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f})$ is Bernstein–Markov
with respect to
$\iota^{1}(\left\|\cdot\right\|_{\bullet\phi,\infty},\mathfrak{f})$.

###### Proof.

Proof of (1). The submultiplicativity statement appears in the second
paragraph [Fin22, Section 5.1]. We give here a proof for the convenience of
the reader. Let $s_{m}\in V_{m}$ and $s_{n}\in V_{n}$. We have by definition

$\displaystyle\iota^{1}(\left\|\cdot\right\|_{m+n},\left\|\cdot\right\|_{m+n}^{\mathrm{NA}})(s_{m}\cdot
s_{n})=\inf_{s_{m}\cdot s_{n}=\sum
t_{p}}\left\|t_{p}\right\|_{m+n}\left\|t_{p}\right\|_{m+n}^{\mathrm{NA}}.$

Now, any decomposition $s_{m}=\sum u_{q}\in V_{m}$, $s_{n}=\sum v_{r}\in
V_{n}$ gives a decomposition $s_{m}\cdot s_{n}=\sum u_{q}\cdot v_{r}\in
V_{m+n}$. This implies that

$\displaystyle\iota^{1}(\left\|\cdot\right\|_{m+n},\left\|\cdot\right\|_{m+n}^{\mathrm{NA}})(s_{m}\cdot
s_{n})$ $\displaystyle\leq\inf_{s_{m}=\sum u_{q},s_{n}=\sum
v_{r}}\left\|u_{q}\cdot v_{r}\right\|_{m+n}\left\|u_{q}\cdot
v_{r}\right\|_{m+n}^{\mathrm{NA}}$ $\displaystyle\leq\inf_{s_{m}=\sum
u_{q},s_{n}=\sum
v_{r}}\left\|u_{q}\right\|_{m}\left\|v_{r}\right\|_{n}\left\|u_{q}\right\|_{m}^{\mathrm{NA}}\left\|v_{r}\right\|_{n}^{\mathrm{NA}}$
$\displaystyle=\inf_{s_{m}=\sum
u_{q}}\left(\left\|u_{q}\right\|_{m}\left\|u_{q}\right\|_{m}^{\mathrm{NA}}\right)\inf_{s_{n}=\sum
v_{r}}\left(\left\|v_{r}\right\|_{n}\left\|v_{r}\right\|_{n}^{\mathrm{NA}}\right)$
$\displaystyle=\iota^{1}(\left\|\cdot\right\|_{m},\left\|\cdot\right\|_{m}^{\mathrm{NA}})(s_{m})\cdot\iota^{1}(\left\|\cdot\right\|_{n},\left\|\cdot\right\|_{n}^{\mathrm{NA}})(s_{n}),$

where in the second inequality we have used submultiplicativity of
$\left\|\cdot\right\|_{\bullet}$ and
$\left\|\cdot\right\|^{\prime}_{\bullet}$. Boundedness easily follows from the
fact that $\mathfrak{f}$ satisfies uniform boundedness.

Proof of (2). Since $\left\|\cdot\right\|_{\bullet\phi}$ is Bernstein–Markov
with respect to $\left\|\cdot\right\|_{\bullet\phi,\infty}$, the result then
follows from applying Lemma 3.1. ∎

######  Corollary 3.3.

The limit
$\mathrm{FS}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f}))=\mathrm{FS}(\iota^{1}(\left\|\cdot\right\|_{\bullet\phi,\infty},\mathfrak{f}))$
is well-defined as a bounded positive metric on $L$.

###### Proof.

This follows from Proposition 1.7. ∎

######  Remark 3.4.

It is then clear from the definitions that
$\mathrm{FS}(\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{f}))$ corresponds
with the metric $\phi_{k}^{\mathfrak{f}}$ from the introduction.

###  3.3. Proof of the main result.

Given $\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ a bounded graded norm,
let $\mathcal{C}_{\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}}$ be the
infinite-dimensional cone of bounded decreasing convex functions on the
support of the Duistermaat–Heckman measure of
$\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$.

######  Theorem 3.5.

For all continuous positive metrics $\phi$ on $L$ and all bounded graded norms
$\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$ on $L$, there exists an
isometric embedding

$\displaystyle\iota:(\mathcal{C}_{\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}},L^{p}(\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}})))$
$\displaystyle\to(L^{\infty}\mathrm{PSH}(X,L),d_{p})$
$\displaystyle\mathfrak{f}$
$\displaystyle\mapsto\mathrm{FS}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f})),$

for $p\in[1,\infty)$, mapping the apex of the cone to $\phi$.

###### Proof.

Let us denote for simplicity by $S$ the closed interval that is the support of
the Duistermaat–Heckman measure of
$\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}$. Let
$\mathfrak{f},\mathfrak{g}\in\mathcal{C}_{\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}}$.
By Corollary 3.3 and Theorem 1.4, for $p\in[1,\infty)$, one has

(3.2)
$d_{p}(\mathrm{FS}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f})),\mathrm{FS}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{g})))=d_{p}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f}),\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{g})).$

To prove the isometry result, it therefore suffices to show that we have

(3.3)
$d_{p}(\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{f}),\iota(\left\|\cdot\right\|_{\bullet\phi},\mathfrak{g}))=\|f-g\|_{L^{p}(\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}))}.$

By definition, one has that

$\displaystyle
k^{-1}d_{p}(\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{f}),\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{g}))^{p}$
$\displaystyle=N_{k}^{-1}\sum_{i=1,\dots,N_{k}}|f(-k^{-1}\log\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k})-g(-k^{-1}\log\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k})|^{p}$
$\displaystyle=\int_{S}|f-g|^{p}\,d\sigma_{k}(\left\|\cdot\right\|^{\mathrm{NA}}_{k}).$

Indeed, both norms in the left-hand side are diagonalisable in the basis
$\underline{s}_{k}$, so that their successive minima are

$\lambda_{i}(\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{f}),\iota(\left\|\cdot\right\|_{k\phi},\mathfrak{g}))=\log\frac{\left\|s_{i,k}\right\|_{k\phi}e^{kg(-k^{-1}\log\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k})}}{\left\|s_{k,i}\right\|_{k\phi}e^{kf(-k^{-1}\log\left\|s_{i,k}\right\|^{\mathrm{NA}}_{k})}}.$

It then easily follows, since $\sigma_{k}\to\sigma$ weakly, that one obtains
in the limit the the desired isometry equality (3.3). Note that the map in the
statement of the theorem is then easily seen to be injective, i.e. defines an
isometric embedding, for if $f\neq g$ while their images define the same
metric, then

$\|f-g\|_{L^{p}(\sigma(\left\|\cdot\right\|_{\bullet}^{\mathrm{NA}}))}=0.$

∎

###  3.4. Discussion and open questions.

Relation with Okounkov bodies. The Okounkov body $\Delta(X,L)$ [Oko03, LM09,
KK12] is a convex body in $\mathbb{R}^{n}$ associated to the algebra of
sections of $L$, which captures the volume of $L$. Its definition requires a
flag, in particular a hypersurface $Y_{1}\subset X$. Such a hypersurface
induces a bounded graded filtration $\mathcal{F}_{1}$ (via its order of
vanishing valuation). In [Nys14], the second author defines an operation
sending a bounded positive metric $\phi$ on $L$ to a convex function on
$\Delta(X,L)$, its Chebyshev transform $c[\phi]$. By our main result, one can
show that a decreasing convex function $\mathfrak{f}$ on the support of the
measure associated to $\mathcal{F}_{1}$ induces a modification
$\phi^{\mathfrak{f}}$ satisfying
$c[\phi^{\mathfrak{f}}]=c[\phi]+\mathfrak{f}$. In particular, geodesics in
this cone are characterised by their Chebyshev transform being affine.

In [RN17, Theorem 8.1], the authors show that there exists a partial moment
map from the space of positive metrics to the first coordinate of the Okounkov
body. Namely, if $H_{1}$ is the exhaustion function in the sense of [RN17,
Section 8] associated to the first divisor $Y_{1}$ in the flag used to
construct the Okounkov body, then for all continuous positive metrics on $L$,
one has

$(H_{1})_{*}(dd^{c}\phi)^{n}=(p_{1})_{*}d\lambda,$

where $d\lambda$ is the Lebesgue measure, and $p_{1}$ is the projection onto
the first variable. Exhaustion functions can be constructed using Legendre
transforms of test curves; such test curves also admit Chebyshev transforms
which give functions in $\mathcal{C}$ under certain singularity assumptions.
Finally, $H_{1}$ can also be obtained via a related, but different,
quantisation procedure to ours [RN17, Theorem 8.3]. It would therefore be
interesting to understand better the connection between our results and those
of [RN17].

The building structure on the space of Kähler metrics. The apartments we
defined in the space of Hermitian norms on a vector space give it the
structure of an Euclidean building, with the key property that any two norms
share an apartment: in other words, any two Hermitian norms can be
diagonalised by the same basis, and they (and the geodesic joining them) lie
in an entire flat isometric to $\mathbb{R}^{n}$. Our results hint at the
existence of such an apartment structure in the space of positive metrics,
where the apartments are given by sequences of bases diagonalising a
filtration, and the flat model is the space bounded decreasing convex
functions on the support of the associated Duistermaat–Heckman measure. This
suggests the following, seemingly very difficult open queestion:

Question. Let $\phi_{0},\phi_{1}$ be two continuous positive metrics on $L$.
Then, there exists a choice of filtration such that they lie in the image of a
same embedding in the sense of our main theorem.

This result would then completely realise the space of positive metrics as an
asymptotic building, suggesting that the results and techniques from the
theory of Euclidean buildings can also be applied to it.

Generalisations. Much recent work has been devoted to extending pluripotential
theory and quantisation procedures both to big or prescribed singularity
classes, and to the Kähler setting [DX21, DDNL23, DRN+23]. Notions of test
configurations (hence possibly filtrations) for Kähler manifolds, more or less
algebraic, exist in the literature [DR17, SD18, DXZ23]. It is expected that an
adequate formulation of our results also hold for big or prescribed
singularity classes, although many necessary tools, such as Finski’s results
and a satisfactory theory of Duistermaat–Heckman measures, are as of yet
missing.

Toric degenerations. In the toric setting, one can pick a filtration
$\mathcal{F}$ that is not only submultiplicative but multiplicative. Thus, the
proof of Theorem 2.1 shows that one can embed the entire space of convex
functions on the support of the Duistermaat–Heckman measure in the space of
positive metrics; not just the decreasing ones. Assuming that $(X,L)$ is not
toric, but admits a (metrised) toric degeneration, it would be interesting to
understand how the phenomenon of obtaining the entire cone of convex functions
arises at the limit.

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