# K-stability and Fujita approximation

Chi Li

###### Abstract

This note is a continuation to the paper [26]. We derive a formula for non-
Archimedean Monge-Ampère measures of big models. As applications, we derive a
positive intersection formula for non-Archimedean Mabuchi functional, and
further reduces the (${\rm Aut}(X,L)_{0}$)-uniform Yau-Tian-Donaldson
conjecture for polarized manifolds to a conjecture on the existence of
approximate Zariski decompositions that satisfy some asymptotic vanishing
condition. In an appendix, we also verify this conjecture for some of
Nakayama’s examples that do not admit birational Zariski decompositions.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Positive intersection formula
  4. 4 First Riemann-Roch coefficients of big line bundles and Fujita approximations
  5. A Conjecture 4.7 for Nakayama’s examples without birational Zariski decomposition

##  1 Introduction

Let $(X,L)$ be a polarized projective manifold. The Yau-Tian-Donaldson (YTD)
conjecture predicts that the existence of constant scalar curvature Kähler
(cscK) metrics in the Kähler class $c_{1}(L)$ is equivalent to a K-stability
condition for the pair $(X,L)$. The K-stability condition is usually expressed
as a positivity condition on the Futaki invariants of test configurations. In
a recent work [26], it was proved that the existence of cscK metrics is
equivalent to the uniform positivity of Mabuchi slopes along all maximal
geodesic rays. Here the maximal geodesic rays, as introduced by Berman-
Boucksom-Jonsson [1], are essentially the geodesic rays in the space of
(mildly singular) positive metrics in $c_{1}(L)$ that can be algebraically
approximated by the data of test configurations. It is known that for test
configurations, the Mabuchi slopes (of geodesic rays associated to test
configurations) are the Futaki invariants. So our result is of a Yau-Tian-
Donaldson type. However the approximability of Mabuchi slopes of (maximal)
geodesic rays, is not well-understood yet. In [26], we did a partial
comparison between the Mabuchi slope with non-Archimedean Mabuchi functional
and reduced the ($G$-)uniform version of YTD conjecture to a non-Archimedean
version of entropy regularization conjecture of Boucksom-Jonsson ([12]).

Furthermore we carried out a partial regularization process (based on
Boucksom-Favre-Jonsson’s work on Non-Achimedean Calabi-Yau theorems) and
proved that K-stability for model filtrations is a sufficient (and
conjecturally also a necessary) condition for the existence of cscK metrics.
By a model filtration, we mean a filtration of the section ring
$R(X,L)=\bigoplus_{m=0}^{+\infty}H^{0}(X,mL)$ induced by a model
$({\mathcal{X}},{\mathcal{L}})$ of $(X,L)$. See Definition 2.1 for the
definition of a model, for which the ${\mathbb{Q}}$-line bundle
${\mathcal{L}}$ is not assumed to be semiample compared to a test
configuration in the usual definition of K-stability (see [31, 19]). The main
goal of this paper is to further reduce Boucksom-Jonsson’s non-Archimedean
regularization conjecture and hence the YTD conjecture to some purely algebro-
geometric conjecture about big line bundles (see Conjecture 4.4, or more
generally Conjecture 4.7, for the conjectural statements), which could be
studied even without the background on K-stability or non-Archimedean
geometry.

More specifically, we will first derive a formula for the non-Archimedean
Monge-Ampère measure of big models, which implies a positive intersection
formula for the non-Archimedean Mabuchi functional of model filtrations. We
refer to section 2 for definitions of terms in the following statement of our
main results.

######  Theorem 1.1.

For any normal and big model $({\mathcal{X}},{\mathcal{L}})$ of $(X,L)$, if
$\phi_{({\mathcal{X}},{\mathcal{L}})}$ denotes the associated non-Archimedean
psh metric, then the following statements hold true.

  1. (i)

If the central fibre is given by ${\mathcal{X}}_{0}=\sum_{i=1}^{I}b_{i}E_{i}$,
and $x_{i}=r(b_{i}^{-1}{\rm ord}_{E_{i}})$ is the Shilov point associated to
$E_{i}$, then the non-Archimedean Monge-Ampère measure of
$\phi_{({\mathcal{X}},{\mathcal{L}})}$ is given by the formula:

${\rm MA}^{\rm
NA}(\phi_{({\mathcal{X}},{\mathcal{L}})})=\sum_{i=1}^{I}b_{i}\left(\langle\bar{{\mathcal{L}}}^{n}\rangle\cdot
E_{i}\right)\delta_{x_{i}},$ (1)

where $\langle\bar{{\mathcal{L}}}^{n}\rangle\in H^{n,n}(\bar{{\mathcal{X}}})$
is the positive intersection product of big line bundles (see section 2.2).

  2. (ii)

The non-Archimedean Mabuchi functional of any big model
$({\mathcal{X}},{\mathcal{L}})$ is given by:

${\bf M}^{\rm
NA}({\mathcal{X}},{\mathcal{L}})=\langle\bar{{\mathcal{L}}}^{n}\rangle\cdot\left(K^{\log}_{\bar{{\mathcal{X}}}/{\mathbb{P}}^{1}}+\frac{\underline{S}}{n+1}\bar{{\mathcal{L}}}\right).$
(2)

The non-Archimedean Monge-Ampère measure on Berkovich spaces were introduced
by A. Chambert-Loir [13] and the formula (1) is a generalization of the
formula of non-Archimedean Monge-Ampère measures for smooth semipositive non-
Archimedean metrics. We refer to section 2.2 for the definition of positive
intersection numbers that arise in the study of restricted volumes of big line
bundles.

The formula (2), which was announced in [26], generalizes the intersection
formula for non-Archimedean Mabuchi functional of a test configuration ([10,
32]) which coincides with the CM weight when the central fibre of the test
configuration is reduced (see [34, 29, 27]). As mentioned above, it together
with the work in [26] further reduce the proof of YTD conjecture to some
algebraic conjecture (Conjecture 4.4). Here for the convenience of the reader
we recall the main result from [26], which is the recent progress in the
variational approach to the YTD conjecture (as proposed in [1, 4]) and
incorporates the analytic existence result of Chen-Cheng [14].

######  Definition 1.2.

$(X,L)$ is uniformly K-stable for models if there exists $\gamma>0$ such that
for any model $({\mathcal{X}},{\mathcal{L}})$, we have:

${\bf M}^{\rm NA}({\mathcal{X}},{\mathcal{L}})\geq\gamma\cdot{\bf J}^{\rm
NA}({\mathcal{X}},{\mathcal{L}})$ (3)

where ${\bf M}^{\rm NA}$ and ${\bf J}^{\rm NA}$ are given in (23)-(24).

######  Theorem 1.3 ([26]).

If a polarized manifold $(X,L)$ is uniformly K-stable for models, then $(X,L)$
admits a cscK metric.

We will see that the positive intersection formula (2) implies that it
suffices to test the uniform K-stability for the models with reduced central
fibres in which case
$K^{\log}_{\bar{{\mathcal{X}}}/{\mathbb{P}}^{1}}=K_{{\mathcal{X}}/{\mathbb{P}}^{1}}$
(see Proposition 3.4).

The converse direction of Theorem 1.3 is expected to be true if ${\rm
Aut}(X,L)_{0}$ is discrete. Indeed, it is implied by Conjecture 4.4. Moreover
there is a version in the case when ${\rm Aut}(X,L)_{0}$ is not discrete (see
[26] for details). As observed by Y. Odaka, such results can be applied to get
immediately the G-uniform version of Yau-Tian-Donaldson conjecture for
polarized spherical manifolds (see some beautiful refinement by T. Delcroix
[16, 17] in this case and Remark 4.14).

We end this introduction with the organization of this paper. In section 2.1,
we recall the construction of non-Archimedean psh metrics from models. In
section 2.2 we recall the concepts related to restricted volumes of big line
bundles and positive intersection products, and important results from [6, 20]
about the relation between them. In section 3, we prove Theorem 1.1. In the
section 4, we propose a general conjecture which strengthens the usual Fujita
approximation theorem and (in the ${\mathbb{C}}^{*}$-equivariant case) would
imply the uniform YTD conjecture for cscK metrics. In the appendix, we verify
this algebraic conjecture for some of Nakayama’s examples that do not admits
birational Zariski decompositions. Acknowledgement: The author is partially
supported by NSF (Grant No. DMS-1810867) and an Alfred P. Sloan research
fellowship. Part of this paper was written when the author was working at
Purdue University. I would like to thank members of algebraic geometry group
at Purdue, especially Sai-Kee Yeung, for their interests in this work, and
Linquan Ma for helpful discussions. I am grateful to Sebastien Boucksom and
Mattias Jonsson for their interest in this work and patient discussions about
Conjecture 4.7, especially for suggesting ways to approach it and pointing out
some delicate difficulty (see Remark 4.8).

##  2 Preliminaries

###  2.1 Non-Archimedean metrics associated to models

This paper is a following-up work of [26] and we will mostly follow the
notations from that work.

######  Definition 2.1.

  * •

A model of $(X,L)$ is a flat family of projective varieties
$\pi:{\mathcal{X}}\rightarrow{\mathbb{C}}$ together with a ${\mathbb{Q}}$-line
bundle ${\mathcal{L}}$ satisfying:

    1. (i)

There is a ${\mathbb{C}}^{*}$-action on $({\mathcal{X}},{\mathcal{L}})$ such
that $\pi$ is ${\mathbb{C}}^{*}$-equivariant;

    2. (ii)

There is a ${\mathbb{C}}^{*}$-equivariant isomorphism
$({\mathcal{X}},{\mathcal{L}})\times_{{\mathbb{C}}}{\mathbb{C}}^{*}\cong(X,L)\times{\mathbb{C}}^{*}$.

  * •

The trivial model of $(X,L)$ is given by
$(X\times{\mathbb{C}},L\times{\mathbb{C}})=:(X_{\mathbb{C}},L_{\mathbb{C}})$.

Two models $({\mathcal{X}}_{i},{\mathcal{L}}_{i}),i=1,2$ are called equivalent
if there exists a model $({\mathcal{X}}_{3},{\mathcal{L}}_{3})$ and two
${\mathbb{C}}^{*}$-equivariant birational morphisms
$\mu_{i}:{\mathcal{X}}_{3}\rightarrow{\mathcal{X}}_{i}$ such that
$\mu_{1}^{*}{\mathcal{L}}_{1}=\mu_{2}^{*}{\mathcal{L}}_{2}$.

  * •

If we forget about the data $L$ and ${\mathcal{L}}$, then we say that
${\mathcal{X}}$ is a model of $X$.

If there is a ${\mathbb{C}}^{*}$-equivariant birational morphism
$r_{{\mathcal{X}}_{1},{\mathcal{X}}_{2}}:{\mathcal{X}}_{1}\rightarrow{\mathcal{X}}_{2}$
for two models ${\mathcal{X}}_{i},i=1,2$, then we say that ${\mathcal{X}}_{1}$
dominates ${\mathcal{X}}_{2}$ and write
${\mathcal{X}}_{1}\geq{\mathcal{X}}_{2}$. If ${\mathcal{X}}\geq
X_{\mathbb{C}}$, then we say that ${\mathcal{X}}$ is dominating.

If ${\mathcal{X}}$ is normal, we say that ${\mathcal{X}}$ is a normal model.
We say a model ${\mathcal{X}}$ is a SNC (i.e. simple normal crossing) if
$({\mathcal{X}},{\mathcal{X}}^{\rm red}_{0})$ is a simple normal crossing
pair.

  * •

Let $(\bar{{\mathcal{X}}},\bar{{\mathcal{L}}})$ be the canonical
${\mathbb{C}}^{*}$-equivariant compactification of
$({\mathcal{X}},{\mathcal{L}})$ over ${\mathbb{P}}^{1}$ by adding the trivial
$(X,L)$ at $\infty\in{\mathbb{P}}^{1}$.

We say that $({\mathcal{X}},{\mathcal{L}})$ is a big model if
$\bar{{\mathcal{L}}}$ is a big ${\mathbb{Q}}$-line bundle over
$\bar{{\mathcal{X}}}$ and the stable base ideal of $m\bar{{\mathcal{L}}}$ is
the same as the $\pi$-base ideal of $m{\mathcal{L}}$ for $m\gg 1$. In
particular, the stable base locus satisfies ${\bf
B}({\mathcal{L}})\subseteq{\mathcal{X}}_{0}=\pi^{-1}(\\{0\\})$. (This
definition is motivated by [12, Lemma A.6].)

In the following for simplicity of notations, if there is no confusion, we
also just write $({\mathcal{X}},{\mathcal{L}})$ for
$(\bar{{\mathcal{X}}},\bar{{\mathcal{L}}})$.

  * •

If ${\mathcal{L}}$ is semiample over ${\mathbb{C}}$, then we call the model
$({\mathcal{X}},{\mathcal{L}})$ to be a test configuration of $(X,L)$.

######  Remark 2.2.

Rigorously speaking, the model of $(X,L)$ should be called the model of
$(X\times{\mathbb{C}},L\times{\mathbb{C}})$. In other words, with the language
of [11], we used the base change from the trivially valued case to the
discrete valued case.

In the original literature of K-stability, which we adopt in this paper, the
line bundle ${\mathcal{L}}$ is assumed to be semi-ample. For us this is the
only difference between the definition of test configurations and models.

We refer to [7, 11] for the definition of Berkovich analytification $(X^{\rm
NA},L^{\rm NA})$ of $(X,L)$ with respect to the trivially valued field
${\mathbb{C}}$ and the definition of non-Archimedean psh metrics $L^{\rm NA}$
which are represented by $\phi_{\rm triv}$-psh functions on $X^{\rm NA}$
(where $\phi_{\rm triv}$ is the metric associated to the trivial test
configuration).

For each model $({\mathcal{X}},{\mathcal{L}})$ of $(X,L)$, we can associate a
non-Archimedean psh metric $\phi_{({\mathcal{X}},{\mathcal{L}})}$ in the
following way. If $\mathfrak{b}_{m}$ denotes the $\pi$-relative base ideal of
$m{\mathcal{L}}$ and $\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ is
the normalized blowup of $\mathfrak{b}_{m}$ with the exceptional divisor
denoted by $\tilde{E}_{m}$, then
$({\mathcal{X}}_{m},{\mathcal{L}}_{m}=\mu_{m}^{*}{\mathcal{L}}-\frac{1}{m}\tilde{E}_{m})$
is a semiample test configuration. $({\mathcal{X}}_{m},{\mathcal{L}}_{m})$
defines a smooth non-Archimedean metric
$\phi_{({\mathcal{X}}_{m},{\mathcal{L}}_{m})}\in{\mathcal{H}}^{\rm NA}(L)$ and
we set

$\phi_{({\mathcal{X}},{\mathcal{L}})}=\lim_{m\rightarrow+\infty}\phi_{({\mathcal{X}}_{m},{\mathcal{L}}_{m})}.$
(4)

If the base variety ${\mathcal{X}}$ is clear, we just write
$\phi_{({\mathcal{X}},{\mathcal{L}})}$ as $\phi_{{\mathcal{L}}}$. It is easy
to see that equivalent models define the same non-Archimedean psh metrics.
Moreover, if ${\mathcal{L}}$ is semiample, then
$\phi_{({\mathcal{X}},{\mathcal{L}})}=\phi_{({\mathcal{X}}_{m},{\mathcal{L}}_{m})}$
for $m$ sufficiently divisible.

By resolution of singularities, we can assume that ${\mathcal{X}}$ is
dominating via a ${\mathbb{C}}^{*}$-equivariant birational morphism
$\rho:{\mathcal{X}}\rightarrow X_{\mathbb{C}}$. Write
${\mathcal{L}}=\rho^{*}L+D$ with $D$ supported on ${\mathcal{X}}_{0}$. Then
${\mathcal{L}}$ defines a model function $f_{\mathcal{L}}$ on $X^{\rm
div}_{\mathbb{Q}}$ (the set of divisorial valuations on $X$) given by:

$f_{\mathcal{L}}(v)=G(v)(D),\quad\forall v\in X^{\rm div}_{\mathbb{Q}}$ (5)

where $G(v):X^{\rm div}_{\mathbb{Q}}\rightarrow(X\times{\mathbb{C}})^{\rm
div}_{\mathbb{Q}}$ is the Gauss extension, i.e. $G(v)$ is a
${\mathbb{C}}^{*}$-invariant valuation on $X\times{\mathbb{C}}$ that extends
$v$ and satisfies $G(v)(t)=1$. Set $\tilde{\phi}_{\mathcal{L}}=\phi_{{\rm
triv}}+f_{\mathcal{L}}$.

The $\phi_{\rm triv}$-psh upper envelope of $f_{\mathcal{L}}$ is defined as:

$P(f_{\mathcal{L}})(v)=\sup\left\\{(\phi-\phi_{{\rm triv}})(v);\phi\in{\rm
PSH}^{\rm NA}(L),\phi-\phi_{\rm triv}\leq f_{\mathcal{L}}\right\\}.$ (6)

By [8, Theorem 8.5] we have the identity
$\phi_{({\mathcal{X}},{\mathcal{L}})}=\phi_{\rm
triv}+P(f_{\mathcal{L}})=:P(\tilde{\phi}_{\mathcal{L}})$. Moreover, by [8,
Theorem 8.3], $P(f_{\mathcal{L}})$ is a continuous $\phi_{{\rm triv}}$-psh
function.

Because $\bar{{\mathcal{L}}}$ is $\bar{\pi}$-big over the compactification
$\bar{{\mathcal{X}}}\stackrel{{\scriptstyle\bar{\pi}}}{{\rightarrow}}{\mathbb{P}}^{1}$,
when $c\gg 1$, the ${\mathbb{Q}}$-line bundle
$\bar{{\mathcal{L}}}_{c}:=\bar{{\mathcal{L}}}+c{\mathcal{X}}_{0}$ is big over
$\bar{{\mathcal{X}}}$. Moreover, by [7, Lemma A.8], when $c\gg 1$, the
$\pi$-relative base ideal of $m\bar{{\mathcal{L}}}$ is the same as the
absolute base ideal of $m\bar{{\mathcal{L}}}$ for all $m$ sufficiently
divisible. In other words we know that $({\mathcal{X}},{\mathcal{L}}_{c})$ is
a big model in the sense in Definition 2.1. Note that we have
$P(\tilde{\phi}_{\mathcal{L}})+c=P(\tilde{\phi}_{{\mathcal{L}}_{c}})=\phi_{\rm
triv}+P(f_{{\mathcal{L}}_{c}})$.

###  2.2 Restricted volumes and positive intersection products

In this section, we (change the notation and) assume that ${\mathcal{X}}$ is a
compact projective manifold and ${\mathcal{L}}$ is a big line bundle over
${\mathcal{X}}$ of dimensional $n+1$. Recall that the volume of
${\mathcal{L}}$ is defined as:

${\rm
vol}_{\mathcal{X}}({\mathcal{L}})=\limsup_{m\rightarrow+\infty}\frac{h^{0}({\mathcal{X}},m{\mathcal{L}})}{m^{n+1}/(n+1)!}.$
(7)

Denote by $N^{1}({\mathcal{X}})={\rm Div}({\mathcal{X}})/\equiv\;$ the Néron-
Severi group. Then the volume functional extends to be a continuous function
on
$N^{1}({\mathcal{X}})_{\mathbb{R}}=N^{1}({\mathcal{X}})\otimes_{\mathbb{Q}}{\mathbb{R}}$.
By Fujita’s approximation theorem, this invariant can be calculated as the
movable intersection number of ${\mathcal{L}}$ (see [18, 25]). In other words,
if we let $\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ be the
normalized blowup of $\mathfrak{b}(|mL|)$ (or its resolution) with exceptional
divisor $\tilde{E}_{m}$ and set
${\mathcal{L}}_{m}=\mu_{m}^{*}{\mathcal{L}}-\frac{1}{m}\tilde{E}_{m}$, then

${\rm
vol}_{{\mathcal{X}}}({\mathcal{L}})=\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n+1}.$
(8)

As a consequence, the limsup in (7) is indeed a limit.

Next we recall the notion of restricted volume ([33, 20, 6]) and the
asymptotic intersection number that calculates the restricted volume.

######  Definition 2.3 ([20]).

For any irreducible ($d$-dimensional) subvariety $Z\subset{\mathcal{X}}$

  * •

The restricted volume of ${\mathcal{L}}$ along $Z$ is defined as

${\rm
vol}_{{\mathcal{X}}|Z}({\mathcal{L}})=\limsup_{m\rightarrow+\infty}\frac{\dim_{{\mathbb{C}}}{\rm
Im}\left(H^{0}({\mathcal{X}},m{\mathcal{L}})\rightarrow
H^{0}(Z,m{\mathcal{L}}|_{Z})\right)}{m^{d}/d!}.$ (9)

  * •

For any $Z\not\subseteq{\bf B}({\mathcal{L}})$ (the stable base locus of
${\mathcal{L}}$), the asymptotic intersection number of ${\mathcal{L}}$ and
$Z$ is defined as:

$\|{\mathcal{L}}^{d}\cdot
Z\|:=\limsup_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{d}\cdot\tilde{Z}_{m},$
(10)

where $\tilde{Z}_{m}$ is the strict transform of $Z$ under the normalized
blowup $\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ of base ideal of
$|m{\mathcal{L}}|$.

######  Remark 2.4.

It is shown in [20] that the limsup in the formula (9) and (10) are actually
limits.

Boucksom-Favre-Jonsson [6] proved that the restricted volume is equal to a
positive intersection product.

######  Definition 2.5 ([6, Definition 2.5]).

Let ${\mathcal{L}}$ be a big ${\mathbb{Q}}$-line bundle. For any effective
divisor $D$, define:

$\langle{\mathcal{L}}^{n}\rangle\cdot
D=\sup_{\mu,E}\;(\mu^{*}L-E)^{n}\cdot\mu^{*}D,$ (11)

where supremum is taken over all birational morphism
$\mu:\tilde{{\mathcal{X}}}\rightarrow{\mathcal{X}}$ and an effective divisor
$E$ such that $\mu^{*}L-E$ is nef. If $D=\sum_{i}b_{i}D_{i}$ with
$b_{i}\in{\mathbb{R}}$ with $D_{i}$ effective, then we extend the definition
(11) linearly:

$\langle{\mathcal{L}}^{n}\rangle\cdot
D=\sum_{i}b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot D_{i}.$

######  Remark 2.6.

In [6], Boucksom-Favre-Jonsson defined positive intersection product
$\langle\xi^{p}\rangle$ for any big class $\xi\in
N^{1}({\mathcal{X}})_{\mathbb{R}}$ and $1\leq p\leq n+1$, by developing an
intersection theory on the Riemann-Zariski space. For example, when $p=n+1$,
$\langle\xi^{n+1}\rangle={\rm vol}(\xi)$; when $p=1$, $\langle\xi\rangle$ is
the collection of positive parts of divisorial Zariski decomposition of
$\pi^{*}\xi$ for all smooth blowups
$\pi:{\mathcal{X}}_{\pi}\rightarrow{\mathcal{X}}$. We refer to [6] for details
on these more general definitions.

Moreover an analytic definition of the positive intersection product was
defined even earlier in [5, Theorem 3.5] (called movable intersection product
there). For each semipositive class $\alpha\in
H^{1,1}({\mathcal{X}},{\mathbb{R}})$, define:

$\langle{\mathcal{L}}^{n}\rangle\cdot\alpha=\sup_{T,\mu}\;\\{\beta^{n}\cdot\mu^{*}\alpha\\}$
(12)

where $T$ ranges over all Kähler currents in $c_{1}({\mathcal{L}})$ that have
logarithmic poles and $\mu:\tilde{{\mathcal{X}}}\rightarrow{\mathcal{X}}$
ranges over the set of those log resolutions satisfying
$\mu^{*}T=\\{E\\}+\beta$ (with $\\{E\\}$ an effective divisor and $\beta$
smooth and semipositive). By Poincaré duality the class
$\langle{\mathcal{L}}^{n}\rangle$ is uniquely defined as a semipositive class
in $H^{n,n}({\mathcal{X}},{\mathbb{R}})$.

In the above definitions, we see that the left-hand-side of (11) depends only
on the numerical class of ${\mathcal{L}}$ and $D$.

Recall that the augmented base locus of ${\mathcal{L}}$ is defined as (see
[20]):

${\bf B}_{+}({\mathcal{L}})=\bigcap_{{\mathcal{L}}=A+E}{\rm Supp}(E),$ (13)

where the intersection is over all decompositions of ${\mathcal{L}}=A+E$ into
${\mathbb{Q}}$-divisors with $A$ ample and $E$ effective. It is know that the
augmented base locus depends only on the numerical class of ${\mathcal{L}}$
(see [20] and reference therein). We will use the following important results:

######  Theorem 2.7.

If ${\mathcal{L}}\rightarrow{\mathcal{X}}$ is a big line bundle, and $Z\subset
X$ is a prime divisor, then the following statements are true:

  1. 1.

([20, Theorem 2.13, Theorem C]) If $Z\not\subseteq{\bf B}_{+}({\mathcal{L}})$
then ${\rm vol}_{{\mathcal{X}}|Z}({\mathcal{L}})=\|{\mathcal{L}}^{n}\cdot
Z\|$. If $Z\subseteq{\bf B}_{+}({\mathcal{L}})$, then ${\rm
vol}_{{\mathcal{X}}|Z}({\mathcal{L}})=0$.

  2. 2.

([6, Theorem B]) There is an identity ${\rm
vol}_{{\mathcal{X}}|Z}({\mathcal{L}})=\langle{\mathcal{L}}^{n}\rangle\cdot Z$.
As a consequence, ${\rm vol}_{{\mathcal{X}}|Z}({\mathcal{L}})$ depends only on
the numerical class of ${\mathcal{L}}$ and $Z$.

As a consequence of these results, we know that in the definition of positive
intersection number in (11), it suffices to take the supremum along the
sequence $\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ which is the
normalized blowup of $\mathfrak{b}(|mL|)$ (or its resolution) with exceptional
divisor $\tilde{E}_{m}$. In other words, if we set
${\mathcal{L}}_{m}=\mu_{m}^{*}{\mathcal{L}}-\frac{1}{m}\tilde{E}_{m}$, then
for any divisor $D$, we have the identity:

$\langle{\mathcal{L}}^{n}\rangle\cdot
D=\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot\mu^{*}D.$ (14)

##  3 Positive intersection formula

Let $\pi:({\mathcal{X}},{\mathcal{L}})\rightarrow{\mathbb{C}}$ be a big model
of $(X,L)$. By resolution of singularities, we can assume that
$({\mathcal{X}},{\mathcal{X}}^{\rm red}_{0})$ is a dominating and SNC model of
$(X,L)$. From now on, for simplicity of notation, we still denote by
$({\mathcal{X}},{\mathcal{L}})$ its natural compactification over
${\mathbb{P}}^{1}$.

Because ${\mathcal{L}}$ is big over ${\mathcal{X}}(=\bar{{\mathcal{X}}})$ (by
the definition of big model), ${\bf B}_{+}({\mathcal{L}})\neq{\mathcal{X}}$,
there exists a fiber ${\mathcal{X}}_{t}=\pi^{-1}(\\{t\\})$ for some
$t\in{\mathbb{P}}^{1}\setminus\\{0\\}$ such that
${\mathcal{X}}_{t}\not\subseteq{\bf B}_{+}({\mathcal{L}})$. In particular,
${\mathcal{X}}_{t}\not\subseteq{\bf B}({\mathcal{L}})$. We then apply Theorem
2.7 to get

$\langle{\mathcal{L}}^{n}\rangle\cdot{\mathcal{X}}_{t}={\rm
vol}_{{\mathcal{X}}|X}({\mathcal{L}})=\|{\mathcal{L}}^{n}\cdot X\|=V.$ (15)

Because $\langle{\mathcal{L}}^{n}\rangle\cdot{\mathcal{X}}_{t}$ depends only
on numerical classes of ${\mathcal{L}}$ and ${\mathcal{X}}_{t}$ (see
definition 2.5), we can use
${\mathcal{X}}_{t}\equiv{\mathcal{X}}_{0}=\sum_{i=1}^{I}b_{i}E_{i}$ to get:

$V=\langle{\mathcal{L}}^{n}\rangle\cdot{\mathcal{X}}_{t}=\sum_{i=1}^{I}b_{i}\left(\langle{\mathcal{L}}^{n}\rangle\cdot
E_{i}\right).$ (16)

Now we can prove the formula (1) for the non-Archimedean Monge-Ampère measure
of non-Archimedean metrics associated to model filtrations. This result
refines and generalizes [7, Lemma 8.5].

###### Proof of Theorem 1.1.(i).

We will use the notations in section 2.1. Via the resolution of singularity,
we can first replace ${\mathcal{X}}$ by any SNC model ${\mathcal{X}}^{\prime}$
that dominates ${\mathcal{X}}$ via
$\pi^{\prime}:{\mathcal{X}}^{\prime}\rightarrow{\mathcal{X}}$ and replace
${\mathcal{L}}$ by ${\mathcal{L}}^{\prime}=\pi^{\prime*}{\mathcal{L}}$. For
simplicity of notations, we will still use the notation
$({\mathcal{X}},{\mathcal{L}})$ instead of
$({\mathcal{X}}^{\prime},{\mathcal{L}}^{\prime})$.

Because the sequence of continuous metrics
$\phi_{m}:=\phi_{({\mathcal{X}}_{m},{\mathcal{L}}_{m})}\in{\mathcal{H}}^{\rm
NA}$ increases to the continuous metric $\phi_{{\mathcal{L}}}=\phi_{\rm
triv}+P(f_{\mathcal{L}})$, by Dini’s theorem we know that $\phi_{m}$ converges
to $\phi_{\mathcal{L}}$ uniformly. In particular, $\phi_{m}$ converges to
$\phi_{\mathcal{L}}$ in the strong topology and ${\rm MA}^{\rm NA}(\phi_{m})$
converges strongly, and hence also weakly, to ${\rm MA}^{\rm
NA}(\phi_{\mathcal{L}})$.

Set $\nu_{{\mathcal{X}},m}=(r_{{\mathcal{X}}})_{*}({\rm MA}^{\rm
NA}(\phi_{m}))$ and $\nu_{{\mathcal{X}}}=(r_{{\mathcal{X}}})_{*}{\rm MA}^{\rm
NA}(\phi_{\mathcal{L}})$ where
$r_{{\mathcal{X}}}:X\rightarrow\Delta_{{\mathcal{X}}}$ is the natural
retraction to the dual complex of ${\mathcal{X}}$ (see [11]). Then they are
supported on $\Delta_{\mathcal{X}}$ and it is easy to see that
$\nu_{{\mathcal{X}},m}$ converges to $\nu_{{\mathcal{X}}}$ weakly. By
Portmanteau’s theorem for weak convergence of measures (see [2, Theorem 2.1]),
we have:

$\limsup_{m\rightarrow+\infty}\nu_{{\mathcal{X}},m}(\\{x_{i}\\})\leq\nu_{{\mathcal{X}}}(\\{x_{i}\\}).$
(17)

On the other hand, we clearly have

$\displaystyle\nu_{{\mathcal{X}},m}(\\{x_{i}\\})$ $\displaystyle=$
$\displaystyle(r_{{\mathcal{X}}})_{*}{\rm MA}^{\rm NA}(\phi_{m})(\\{x_{i}\\})$
$\displaystyle=$ $\displaystyle{\rm MA}^{\rm
NA}(\phi_{m})((r_{{\mathcal{X}}})^{-1}\\{x_{i}\\})\geq{\rm MA}^{\rm
NA}(\phi_{m})(\\{x_{i}\\}).$

So we combine the above two inequalities to get:

$\limsup_{m\rightarrow+\infty}{\rm MA}^{\rm
NA}(\phi_{m})(\\{x_{i}\\})\leq\nu_{{\mathcal{X}}}(\\{x_{i}\\})=:V_{i}.$ (18)

We consider two cases:

  1. 1.

If $E_{i}\not\subseteq{\bf B}_{+}({\mathcal{L}})$, then $E_{i}$ is not
contained in ${\bf B}({\mathcal{L}})$. By the formula of non-Archimedean
Monge-Ampère measures of test configurations (see [11, section 3.4]) we get
that:

$b_{i}{\mathcal{L}}_{m}^{n}\cdot\tilde{E}_{i}={\rm MA}^{\rm
NA}(\phi_{m})(\\{x_{i}\\}),$ (19)

where $\tilde{E}_{i}$ is the strict transform of $E_{i}$ under $\mu_{m}$. So
by (19) and (10) we get

$\limsup_{m\rightarrow+\infty}{\rm MA}^{\rm
NA}(\phi_{m})(\\{x_{i}\\})=\limsup_{m\rightarrow+\infty}b_{i}{\mathcal{L}}_{m}^{n}\cdot\tilde{E}_{i}=b_{i}\|{\mathcal{L}}^{n}\cdot
E_{i}\|.$ (20)

So by Theorem 2.7 and the inequality (18) we have

$b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot E_{i}=b_{i}\cdot{\rm
vol}_{{\mathcal{X}}|E_{i}}({\mathcal{L}})=b_{i}\|{\mathcal{L}}^{n}\cdot
E_{i}\|\leq V_{i}.$ (21)

  2. 2.

If $E_{i}\subseteq{\bf B}_{+}({\mathcal{L}})$, then
$b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot E_{i}=0\leq V_{i}$.

Combining these with (16), we have:

$\displaystyle V$ $\displaystyle=$
$\displaystyle\sum_{i}b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot
E_{i}\leq\sum_{i}V_{i}=\sum_{i}\nu_{{\mathcal{X}}}(\\{x_{i}\\})\leq V.$

So the inequalities in the above chain are actually equalities. So
$b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot
E_{i}=V_{i}=\nu_{{\mathcal{X}}}(\\{x_{i}\\})$ for $i=1,\dots,I$ and
$\nu_{{\mathcal{X}}}=(r_{{\mathcal{X}}})_{*}{\rm MA}^{\rm
NA}(\phi_{\mathcal{L}})$ is supported on the finite set
$\\{x_{i};i=1,\dots,I\\}$. In other words, we have

$(r_{{\mathcal{X}}})_{*}{\rm MA}^{\rm
NA}(\phi_{\mathcal{L}})=\sum_{i=1}^{N}b_{i}\langle\left({\mathcal{L}}^{n}\rangle\cdot
E_{i}\right)\delta_{x_{i}}.$ (22)

But we have said that $({\mathcal{X}},{\mathcal{L}})$ can be replaced by any
SNC model that dominates ${\mathcal{X}}$. Moreover, the pairs
$\\{(x_{i},V_{i});V_{i}\neq 0\\}$ do not depend on the choice of such SNC
models. By using the homemorphism $X^{\rm
NA}=\varprojlim\Delta_{\mathcal{X}}$, it is then easy to conclude that the
Radon measure ${\rm MA}^{\rm NA}(\phi_{\mathcal{L}})$ is indeed only supported
on the finite set $\\{x_{i};i=1,\dots,I\\}$ and the identity (1) holds true. ∎

######  Remark 3.1.

Although our work on K-stability is the through the study of non-Archimedean
geometry in the trivially valued case (which is base-changed to the discretely
valued case, following [11, 12]), the proof of formula for non-Archimedean
Monge-Ampère measure also holds true for more general non-trivially valued
case.

We recall that the formula for non-Archimedean functionals following the works
in [9, 11, 12] (see also [26]). For any continuous continuous psh metric
$\phi$ on $L^{\rm NA}$, the non-Archimedean Mabuchi functional is given by:

${\bf M}^{\rm NA}(\phi)={\bf H}^{\rm NA}(\phi)+({\bf E}^{K_{X}})^{\rm
NA}(\phi)+\underline{S}\;{\bf E}^{\rm NA}(\phi)$ (23)

where the terms on the right-hand-side are given by the following non-
Archimedean integrals:

$\displaystyle{\bf H}^{\rm NA}(\phi)$ $\displaystyle=$
$\displaystyle\int_{X^{\rm NA}}A_{X}(x){\rm MA}^{\rm NA}(\phi),$
$\displaystyle({\bf E}^{K_{X}})^{\rm NA}(\phi)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\int_{X^{\rm NA}}(\phi-\phi_{\rm triv}){\rm
dd^{c}}\psi\wedge{\rm MA}^{\rm NA}(\phi_{\rm triv}^{[i]},\phi^{[n-1-i]})$
$\displaystyle{\bf E}^{\rm NA}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{n+1}\int_{X^{\rm NA}}(\phi-\phi_{\rm
triv}){\rm MA}^{\rm NA}(\phi_{\rm triv}^{[i]},\phi^{[n-i]}),$

where in the second identity $\psi$ is a Hermitian metric on $K_{X}^{{\rm
NA}}$. We also recall the ${\bf J}^{\rm NA}$-functional:

${\bf J}^{\rm NA}(\phi)=L^{n}\cdot\sup(\phi-\phi_{\rm triv})-{\bf E}^{\rm
NA}(\phi).$ (24)

######  Proposition 3.2.

With the above notation, we have:

${\bf H}^{\rm NA}(\phi_{\mathcal{L}})=\langle{\mathcal{L}}^{n}\rangle\cdot
K^{\log}_{{\mathcal{X}}/X_{{\mathbb{P}}^{1}}}.$ (25)

###### Proof.

Note that we have the identity:

$\displaystyle K^{\log}_{{\mathcal{X}}/X_{{\mathbb{P}}^{1}}}$ $\displaystyle=$
$\displaystyle K_{{\mathcal{X}}}+{\mathcal{X}}^{\rm
red}_{0}-(K_{X_{{\mathbb{P}}^{1}}}+{\mathcal{X}}_{0})=\sum_{i}(A_{X_{{\mathbb{P}}^{1}}}(E_{i})-b_{i})E_{i}$
$\displaystyle=$
$\displaystyle\sum_{i}b_{i}(A_{X_{{\mathbb{P}}^{1}}}(b_{i}^{-1}{\rm
ord}_{E_{i}})-1)E_{i}=\sum_{i}b_{i}A_{X}(x_{i})E_{i}.$

So we can use (1) to get the identity:

$\displaystyle{\bf H}^{\rm NA}(\phi_{\mathcal{L}})$ $\displaystyle=$
$\displaystyle\int_{X^{\rm NA}}A_{X}(x){\rm MA}^{\rm
NA}(\phi_{\mathcal{L}})=\sum_{i}A_{X}(x_{i})b_{i}\langle{\mathcal{L}}^{n}\rangle\cdot
E_{i}$ $\displaystyle=$ $\displaystyle\langle{\mathcal{L}}^{n}\rangle\cdot
K^{\log}_{{\mathcal{X}}/X_{{\mathbb{P}}^{1}}}.$

∎

######  Proposition 3.3.

With the above notation, we have the following identities:

$\displaystyle{\bf R}^{\rm NA}(\phi_{\mathcal{L}})$ $\displaystyle=$
$\displaystyle\langle{\mathcal{L}}^{n}\rangle\cdot\rho^{*}K_{X},$ (26)
$\displaystyle{\bf E}^{\rm NA}(\phi_{\mathcal{L}})$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\langle{\mathcal{L}}^{n+1}\rangle=\frac{1}{n+1}\langle{\mathcal{L}}^{n}\rangle\cdot{\mathcal{L}}.$
(27)

###### Proof.

Because $\phi_{m}$ converges to $\phi_{\mathcal{L}}$ strongly, by [11] we
have:

${\bf R}^{\rm NA}(\phi_{\mathcal{L}})=\lim_{m\rightarrow+\infty}{\bf R}^{\rm
NA}(\phi_{m})=\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot\mu_{m}^{*}\rho^{*}K_{X}.$
(28)

Write $\rho^{*}K_{X}=A_{1}-A_{2}$ with $A_{1},A_{2}$ very ample. Moreover we
can choose $A_{i},i=1,2$ to be sufficient general such that $A_{i},i=1,2$ do
not contain the centers of Rees valuations of $\mathfrak{b}_{m}$ for all $m$.
Then the strict transforms of $A_{i},i=1,2$ under
$\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ are the same as the total
transform of $A_{i},i=1,2$. By using Theorem 2.7 we see that the right-hand-
side of (28) is equal to

$\|{\mathcal{L}}^{n}\cdot A_{1}\|-\|{\mathcal{L}}^{n}\cdot
A_{2}\|=\langle{\mathcal{L}}^{n}\rangle\cdot(A_{1}-A_{2})=\langle{\mathcal{L}}^{n}\rangle\cdot\rho^{*}K_{X}.$
(29)

For the first equality in (27), we can again use
$\phi_{m}=\phi_{({\mathcal{X}}_{m},{\mathcal{L}}_{m})}$ (for which (27) is
known to be true) to approximate and directly apply the Fujita approximation
result in [25, Theorem 11.4.11]. The last equality in (27) follows from the
orthogonality property proved in [5, Corollary 4.5] or [6, Corollary 3.6].

∎

We can complete the proof the formula for the non-Archimedean Mabuchi
functional.

###### Proof of Theorem 1.1.(ii).

The formula (2) follows immediately from the decomposition ${\bf M}^{\rm
NA}={\bf H}^{\rm NA}+{\bf R}^{\rm NA}+\underline{S}{\bf E}^{\rm NA}$ and the
formula for each part in (25), (26) and (27).

∎

As an application of the positive intersection formula, we get:

######  Proposition 3.4.

To check the ($G$-) unform K-stability for models (see Definition 1.2 and
[26]), it suffices to consider models with reduced central fibres.

###### Proof.

Let $({\mathcal{X}},{\mathcal{L}})$ be any big model. We can take a base
change
$({\mathcal{X}}^{(d)},{\mathcal{L}}^{(d)})=({\mathcal{X}},{\mathcal{L}})\times_{{\mathbb{C}},t\mapsto
t^{d}}{\mathbb{C}}$ such that its normalization $\tilde{{\mathcal{X}}}$ has
reduced central fibers. Let $f:\tilde{{\mathcal{X}}}\rightarrow{\mathcal{X}}$
be the natural finite morphism and set
$\tilde{{\mathcal{L}}}=f^{*}{\mathcal{L}}$. Then we have the identity

$K^{\log}_{(\tilde{{\mathcal{X}}},\tilde{{\mathcal{X}}}_{0})}:=K_{\tilde{{\mathcal{X}}}}+\tilde{{\mathcal{X}}}_{0}=f^{*}(K_{{\mathcal{X}}}+{\mathcal{X}}_{0}^{\rm
red})=f^{*}K^{\log}_{({\mathcal{X}},{\mathcal{X}}_{0})}.$

It is known that volumes of big line bundles are multiplicative under generic
finite morphisms (see [21, Lemma 4.3]). So we get the identity

$\langle(\tilde{{\mathcal{L}}}+\epsilon
K^{\log}_{(\tilde{{\mathcal{X}}},\tilde{{\mathcal{X}}}_{0})})^{n+1}\rangle=d\cdot\langle({\mathcal{L}}+\epsilon
K^{\log}_{({\mathcal{X}},{\mathcal{X}}_{0})})^{n+1}\rangle.$ (30)

Taking derivative with respect to $\epsilon$ at $\epsilon=0$, we also get:

$\langle\tilde{{\mathcal{L}}}^{n}\rangle\cdot
K^{\log}_{(\tilde{{\mathcal{X}}},\tilde{{\mathcal{X}}}_{0})}=d\cdot\langle{\mathcal{L}}^{n}\rangle\cdot
K^{\log}_{({\mathcal{X}},{\mathcal{X}}_{0})}.$ (31)

Moreover (30) for $\epsilon=0$ gives ${\bf E}^{\rm
NA}(\phi_{\tilde{{\mathcal{L}}}})=d\cdot{\bf E}^{\rm NA}(\phi_{\mathcal{L}})$.
On the other hand, it is known we have the formula (see [11])

$(\phi_{\tilde{{\mathcal{L}}}}-\phi_{\rm
triv})(x)=d\cdot(\phi_{\mathcal{L}}-\phi_{\rm triv})(d^{-1}x),\quad\text{ for
all }x\in X^{\rm NA}.$ (32)

So we get the identity ${\bf J}^{\rm
NA}(\phi_{\tilde{{\mathcal{L}}}})=d\cdot{\bf J}^{\rm
NA}(\phi_{{\mathcal{L}}})$ by (24). Combining these identities with the
positive intersection formula (2), the statement now follows easily.

∎

##  4 First Riemann-Roch coefficients of big line bundles and Fujita
approximations

In view of the above intersection formula, it seems natural to consider the
following invariant for big line bundles.

######  Definition 4.1.

Let ${\mathcal{L}}$ be a big line bundle over a projective manifold
${\mathcal{X}}$ of dimension $n+1$. The first Riemann-Roch coefficient (1st-RR
coefficient) of ${\mathcal{L}}$ is defined to be:

$\mathfrak{r}_{1}({\mathcal{X}},{\mathcal{L}})=\langle{\mathcal{L}}^{n}\rangle\cdot
K_{{\mathcal{X}}}.$ (33)

If the base manifold ${\mathcal{X}}$ is clear, we just write
$\mathfrak{r}_{1}({\mathcal{X}},{\mathcal{L}})$ as
$\mathfrak{r}_{1}({\mathcal{L}})$.

The zero-th Riemann-Roch coefficient is of course the volume of
${\mathcal{L}}$:

$\mathfrak{r}_{0}({\mathcal{X}},{\mathcal{L}}):={\rm
vol}_{{\mathcal{X}}}({\mathcal{L}})=\langle{\mathcal{L}}^{n+1}\rangle.$ (34)

One would hope that $\mathfrak{r}_{1}({\mathcal{X}},{\mathcal{L}})$ is the
second order coefficients in the expansion of
$h^{0}({\mathcal{X}},m{\mathcal{L}})$. This is true if ${\mathcal{L}}$ is big
and nef by Fujita’s vanishing theorem. But due to the example in [15], this
does not seem to be true for general big line bundles.

######  Lemma 4.2.

If $\mu:{\mathcal{Y}}\rightarrow{\mathcal{X}}$ is a birational morphism
between smooth projective manifold, which is a composition of blowups along
smooth subvarieties. Then we have:

$\mathfrak{r}_{1}({\mathcal{L}})=\mathfrak{r}_{1}(\mu^{*}{\mathcal{L}}).$ (35)

###### Proof.

Write $K_{{\mathcal{X}}}$ as the difference of very ample divisors
$A_{1}-A_{2}$ and arguing as in the proof of Proposition 3.3, we see that:

$\langle{\mathcal{L}}^{n}\rangle\cdot
K_{{\mathcal{X}}}=\langle\mu^{*}{\mathcal{L}}^{n}\rangle\cdot\mu^{*}K_{{\mathcal{X}}}.$
(36)

Let $E_{i}$ be the exceptional divisor of $\mu$. We just need to show that
$\langle\mu^{*}{\mathcal{L}}^{n}\rangle\cdot E_{i}={\rm
vol}_{{\mathcal{Y}}|E_{i}}(\mu^{*}{\mathcal{L}}^{n})=0$. This can be seen by
the inclusion:

${\rm Im}\left(H^{0}({\mathcal{Y}},m\mu^{*}{\mathcal{L}})\rightarrow
H^{0}(E_{i},m\mu^{*}{\mathcal{L}}|_{E_{i}})\right)\subseteq
H^{0}(E_{i},m\mu^{*}{\mathcal{L}}|_{E_{i}})=H^{0}(\mu_{*}(E_{i}),m{\mathcal{L}}|_{\mu_{*}(E_{i})})$
(37)

and using the fact that the right-hand-side is equal to $o(m^{n})$ because
$\dim(\mu_{*}(E_{i}))<n$. ∎

If we consider ${\mathcal{L}}$ as a Cartier $b$-divisor in the sense of
Shokurov, then because of identity (35), $\mathfrak{r}_{1}({\mathcal{L}})$ is
an invariant of the Cartier $b$-divisor ${\mathcal{L}}$.

The following lemma follows immediately from the results in [28, section 3.1].

######  Lemma 4.3.

Let ${\mathcal{L}}=\mathcal{P}+\mathcal{N}$ be the divisorial Zariski
decomposition of ${\mathcal{L}}$. Then we have:

$\mathfrak{r}_{1}({\mathcal{L}})=\mathfrak{r}_{1}(\mathcal{P}).$ (38)

Moreover if ${\mathcal{L}}$ admits a Zariski decomposition, i.e. if
$\mathcal{P}$ is nef, then we have:

$\mathfrak{r}_{1}({\mathcal{L}})=\mathcal{P}^{n}\cdot K_{{\mathcal{X}}}.$ (39)

We propose the following main conjecture.

######  Conjecture 4.4.

Let $({\mathcal{X}},{\mathcal{L}})$ be a big model of $(X,L)$. Then there
exists a sequence of blowups
$\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ along
${\mathbb{C}}^{*}$-equivariant ideal sheaves cosupported on
${\mathcal{X}}_{0}$ and decompositions into ${\mathbb{Q}}$-divisors
$\mu_{m}^{*}{\mathcal{L}}={\mathcal{L}}_{m}+E_{m}$ with ${\mathcal{L}}_{m}$
semiample and $E_{m}$ effective supported on the exceptional divisor of
$\mu_{m}$ such that:

$\lim_{m\rightarrow+\infty}{\rm
vol}_{{\mathcal{X}}_{m}}(\bar{{\mathcal{L}}}_{m})={\rm
vol}_{{\mathcal{X}}}(\bar{{\mathcal{L}}})\quad\text{ and
}\quad\lim_{m\rightarrow+\infty}\mathfrak{r}_{1}(\bar{{\mathcal{L}}}_{m})=\mathfrak{r}_{1}(\bar{{\mathcal{L}}}).$
(40)

Because of the positive intersection formula in (2) and the reduction in [26],
this indeed implies Boucksom-Jonsson’s regularization conjecture. Moreover by
the following lemma and the work in [26], it would complete the solution of
Yau-Tian-Donaldson conjecture for cscK metrics.

######  Lemma 4.5.

For any big model $({\mathcal{X}},{\mathcal{L}})$, Conjecture 4.4 implies that
there exists $\phi_{m}\in{\mathcal{H}}^{\rm NA}$ such that $\phi_{m}$
converges to $\phi_{({\mathcal{X}},{\mathcal{L}})}$ in the strong topology and
${\bf M}^{\rm NA}(\phi_{m})\rightarrow{\bf M}^{\rm
NA}(\phi_{({\mathcal{X}},{\mathcal{L}})})$.

###### Proof.

By the same base change construction as in the proof of Proposition 3.4, we
can assume that ${\mathcal{X}}$ has a reduced central fibre.

For simplicity of notations, we denote by
$\phi=\phi_{({\mathcal{X}},{\mathcal{L}})}$ (resp. $\phi_{m}$) the non-
Archimedean metrics associated to ${\mathcal{L}}$ (resp. ${\mathcal{L}}_{m}$).
Then because $E_{m}$ is effective, we have $\phi\geq\phi_{m}$. We claim that
$\phi_{m}\rightarrow\phi$ strongly. Indeed, by [12, Proposition 6.26], it
suffices to show the following non-negative quantity converges to 0 as
$m\rightarrow+\infty$:

${\bf J}^{\rm NA}_{\phi}(\phi_{m})=\int_{X^{\rm NA}}(\phi_{m}-\phi){\rm
MA}^{\rm NA}(\phi_{m})-{\bf E}^{\rm NA}(\phi_{m})+{\bf E}^{\rm NA}(\phi).$
(41)

This follows immediately from $\phi_{m}\leq\phi$ and (27):

$\displaystyle 0\leq{\bf J}^{\rm NA}_{\phi}(\phi_{m})\leq-{\bf E}^{\rm
NA}(\phi_{m})+{\bf E}^{\rm NA}(\phi)=\frac{{\rm
vol}_{{\mathcal{X}}}({\mathcal{L}})}{n+1}-\frac{{\rm
vol}_{{\mathcal{X}}_{m}}({\mathcal{L}}_{m})}{n+1}.$

By the positive intersection formula (2) the second identity in (40) implies
${\bf M}^{\rm NA}(\phi_{m})\rightarrow{\bf M}^{\rm NA}(\phi)$. ∎

We hope the conjecture 4.4 can be studied by using the geometric tools
introduced in the study of Fujita’s approximation theorem. We recall the
following definition

######  Definition 4.6 (see [25, Definition 11.4.3]).

Let ${\mathcal{L}}$ be a big line bundle. A Fujita approximation of
${\mathcal{L}}$ consists of a projective birational morphism
$\mu:{\mathcal{X}}^{\prime}\rightarrow{\mathcal{X}}$ with
${\mathcal{X}}^{\prime}$ irreducible together with a decomposition
$\mu^{*}{\mathcal{L}}=A+E$ in $N^{1}({\mathcal{X}})_{\mathbb{Q}}$ such that
$A$ is big and semiample and $E$ is effective.

In the more general context of big line bundles, we conjecture the following
result:

######  Conjecture 4.7.

Let ${\mathcal{L}}$ be a big line bundle over a smooth projective manifold
${\mathcal{X}}$. Then there exists a sequence of Fujita approximations ([25,
Definition 11.4.3]), i.e. birational morphisms
$\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ and decompositions into
${\mathbb{Q}}$-divisors $\mu_{m}^{*}{\mathcal{L}}={\mathcal{L}}_{m}+E_{m}$
with ${\mathcal{L}}_{m}$ ample and $E_{m}$ effective, such that:

$\lim_{m\rightarrow+\infty}{\rm
vol}_{{\mathcal{X}}_{m}}({\mathcal{L}}_{m})={\rm
vol}_{{\mathcal{X}}}({\mathcal{L}})\quad\text{ and
}\quad\lim_{m\rightarrow+\infty}\mathfrak{r}_{1}({\mathcal{L}}_{m})=\mathfrak{r}_{1}({\mathcal{L}}).$
(42)

######  Remark 4.8.

Sebastien Boucksom pointed out to me that this conjecture could be formulated
using the language of b-divisors. Such a formulation has some consequences and
(hopefully) might be useful for studying this problem.

Let’s recall an orthogonality estimate by Boucksom-Demailly-Pǎun-Peternell
(see also [25, Theorem 11.4.21]):

######  Theorem 4.9 ([5, Theorem 4.1]).

Fix any ample line bundle $H$ on ${\mathcal{X}}$. There exists a constant
$C=C({\mathcal{X}},H)>0$ such that any Fujita decomposition
$(\mu:{\mathcal{X}}^{\prime}\rightarrow{\mathcal{X}},\mu^{*}{\mathcal{L}}=A+E)$
satisfies the estimate:

$(A^{n}\cdot E)^{2}\leq C\cdot({\rm vol}_{{\mathcal{X}}}({\mathcal{L}})-{\rm
vol}_{{\mathcal{X}}^{\prime}}(A)).$ (43)

We observe an immediate consequence of this estimate.

######  Lemma 4.10.

Let $\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ be a sequence of
birational morphisms such $\mu_{m}^{*}{\mathcal{L}}={\mathcal{L}}_{m}+E_{m}$
where ${\mathcal{L}}_{m}$ is ample and $E_{m}$ is effective. Assume that the
following conditions are satisfied:

  1. 1.

$\lim_{m\rightarrow+\infty}{\rm
vol}_{{\mathcal{X}}_{m}}({\mathcal{L}}_{m})={\rm
vol}_{\mathcal{X}}({\mathcal{L}})$.

  2. 2.

$\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot\mu_{m}^{*}K_{{\mathcal{X}}}=\langle{\mathcal{L}}^{n}\rangle\cdot
K_{\mathcal{X}}$.

Then
$\lim_{m\rightarrow+\infty}\mathfrak{r}_{1}({\mathcal{L}}_{m})=\mathfrak{r}_{1}({\mathcal{L}})$
if and only if

$\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot
K_{{\mathcal{X}}_{m}/{\mathcal{X}}}=0.$ (44)

In particular, if there exists a constant $C>0$ independent of $m$ such that
for any irreducible component $F$ of $E_{m}$ we have ${\rm
ord}_{F}(K_{{\mathcal{X}}_{m}/{\mathcal{X}}})\leq C\cdot{\rm ord}_{F}(E_{m})$,
then we have the convergence:
$\lim_{m\rightarrow+\infty}\mathfrak{r}_{1}({\mathcal{L}}_{m})=\mathfrak{r}_{1}({\mathcal{L}})$.

######  Remark 4.11.

The above lemma suggests that the techniques from birational algebraic
geometry might be useful for achieving (44). Indeed, our hope is that the MMP
techniques (based on the work of Birkar-Casini-Hacon-McKernan) could be used
to extract suitable exceptional divisors satisfying the conditions in the
above lemma. Note that such type of techniques has prove to be very powerful
in the study of K-stability for Fano varieties (see for example [3]).

By the works in [18, 25] and [20, 6], the sequence $\\{{\mathcal{L}}_{m}\\}$
that satisfy the first two conditions can be obtained by blowing up base
ideals. Moreover one can also get ${\mathcal{L}}_{m}$ by blowing up
appropriate asymptotic multiplier ideals, which satisfy the important Nadel-
vanishing and global generation properties. We review the construction in [25,
11.4.B, Proof of Theorem 11.4] for the reader’s convenience. Fix a very ample
bundle $H$ on ${\mathcal{X}}$ such that $G:=K_{\mathcal{X}}+(n+2)H$ is very
ample. For $m\geq 0$, set $M_{m}=m{\mathcal{L}}-G$. Given $\epsilon>0$ there
exists $m\gg 1$ such that ${\rm vol}(M_{m})\geq m^{n+1}({\rm
vol}({\mathcal{L}})-\epsilon)$. Set
$\mathcal{J}=\mathcal{J}({\mathcal{X}},\|M_{m}\|)$, let
$\mu_{m}:{\mathcal{X}}_{m}\rightarrow{\mathcal{X}}$ be a common resolution of
$\mathcal{J}$ such that
$\mu_{m}^{*}\mathcal{J}={\mathcal{O}}(-\tilde{E}_{m})$. Then
${\mathcal{L}}_{m}:={\mathcal{L}}-\frac{1}{m}\tilde{E}_{m}$ is semiample and
${\mathcal{L}}_{m}^{n+1}\geq{\rm vol}({\mathcal{L}})-\epsilon$ (see [25,
11.4.B] for more details). By letting $\epsilon\rightarrow 0$, we see that the
first condition in Lemma 4.10 is thus satisfied.

Now we claim that in this construction, the second condition in Lemma 4.10 can
also be satisfied. This fact will be used in the calculations of appendix A.
To see this, we use some similar argument as in [20, Proof of Theorem 2.13].
Choose $m_{0}\gg 1$ such that $m_{0}{\mathcal{L}}-G=N^{\prime}$ is effective.
Fix a very ample divisor $H$ such that $N:=N^{\prime}+H$ is ample. Then we
have the inclusion

$\displaystyle\mathfrak{b}(|(m-m_{0}){\mathcal{L}}|){\mathcal{O}}_{\mathcal{X}}(-N)$
$\displaystyle\subseteq$
$\displaystyle\mathfrak{b}(|(m-m_{0}){\mathcal{L}}){\mathcal{O}}(-N^{\prime})$
$\displaystyle\subseteq$
$\displaystyle\mathfrak{b}(|m{\mathcal{L}}-G|)=\mathfrak{b}(|M_{m}|)\subseteq\mathcal{J}(\|M_{m}\|).$

We can also assume that $\mu_{m}$ is both resolutions of
$\mathfrak{b}(|M_{m}|)$ and $\mathfrak{b}(|(m-m_{0}){\mathcal{L}}|)$
satisfying the identities
$\mu_{m}^{*}\mathfrak{b}(|m{\mathcal{L}}-G|)={\mathcal{O}}_{{\mathcal{X}}_{m}}(-\tilde{F}_{m})$
and
$\mu_{m}^{*}\mathfrak{b}(|(m-m_{0}){\mathcal{L}}|)={\mathcal{O}}_{{\mathcal{X}}_{m}}(-\tilde{Q}_{m})$.
Set
${\mathcal{L}}^{\prime}_{m}=\mu_{m}^{*}({\mathcal{L}}-\frac{G}{m})-\frac{1}{m}\tilde{F}_{m}$
and
${\mathcal{L}}^{\prime\prime}_{m}=\mu_{m}^{*}{\mathcal{L}}-\frac{1}{m-m_{0}}\tilde{Q}_{m}$.

Fix any effective divisor $D$ on ${\mathcal{X}}$. Let $\tilde{D}_{m}$ be the
strict transform of $D$ under $\mu_{m}$. Then the above inclusion implies:

${\rm vol}(({\mathcal{L}}_{m}+\frac{N}{m})|_{\tilde{D}_{m}})\geq{\rm
vol}(({\mathcal{L}}^{\prime}_{m}+\frac{N}{m})|_{\tilde{D}_{m}})\geq{\rm
vol}({\mathcal{L}}^{\prime\prime}_{m}|_{\tilde{D}_{m}}).$

Because ${\mathcal{L}}_{m}$ and ${\mathcal{L}}^{\prime\prime}_{m}$ are both
semiample, this implies

$({\mathcal{L}}_{m}+\frac{N}{m})^{n}\cdot\tilde{D}_{m}\geq{\mathcal{L}}^{\prime\prime
n}_{m}\cdot\tilde{D}_{m}.$ (45)

Now we can fix a very ample line bundle $H$ such that
$\mu_{m}^{*}(mH)-{\mathcal{L}}_{m}=m\mu^{*}(H-{\mathcal{L}})+\tilde{E}_{m}$ is
effective. Then for any $1\leq i\leq n$, we have:

$\limsup_{m\rightarrow+\infty}\frac{1}{m^{n}}(m{\mathcal{L}}_{m}^{n-i})\cdot
N^{i}\cdot\tilde{D}_{m}\leq\limsup_{m\rightarrow+\infty}\frac{1}{m^{n}}(mH)^{n-i}\cdot
N^{i}\cdot\tilde{D}_{m}=0.$

By expanding the left-hand-side of (45), this implies

$\langle{\mathcal{L}}^{n}\rangle\cdot
D\geq\limsup_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot\tilde{D}_{m}\geq\limsup_{m\rightarrow+\infty}{\mathcal{L}}^{\prime\prime
n}_{m}\cdot\tilde{D}_{m}=\|{\mathcal{L}}^{n}\cdot D\|$

which, by using Theorem 2.7, implies the equality

$\lim_{m\rightarrow+\infty}{\mathcal{L}}_{m}^{n}\cdot\mu_{m}^{*}D=\|{\mathcal{L}}^{n}\cdot
D\|=\langle{\mathcal{L}}^{n}\rangle\cdot D.$

Writing $-K_{{\mathcal{X}}}=D_{1}-D_{2}$ with $D_{1},D_{2}$ effective, we then
see that the second condition of Lemma 4.10 is satisfied too.

Finally we point out that Conjecture 4.7 holds true any for any big line
bundle that admits a birational Zariski decomposition (in the sense of
Cutkosky-Kawamata-Moriwaki). Unfortunately not all big line bundles admit such
birational Zariski decomposition by the counterexamples of Nakayama [30]. On
the other hand, we verify in the appendix that conjecture 4.7 indeed holds for
some of Nakayama’s examples. Indeed, we will show that in these examples the
bound of discrepancies in the above lemma is indeed satisfied. So it seems to
be very interesting to know whether (44) can be achieved in general.

######  Definition 4.12.

We say a big line bundle ${\mathcal{L}}$ admits a birational Zariski
decomposition if there is a modification
$\mu:\tilde{{\mathcal{X}}}\rightarrow{\mathcal{X}}$, a nef
${\mathbb{R}}$-divisor $\mathcal{P}$ and an ${\mathbb{R}}$-effective divisor
$\mathcal{N}$ on $\tilde{{\mathcal{X}}}$ with the following properties:

  * •

$\mu^{*}{\mathcal{L}}=\mathcal{P}+\mathcal{N}$.

  * •

For any positive integer $m>0$, the map

$H^{0}(\tilde{{\mathcal{X}}},{\mathcal{O}}_{\tilde{{\mathcal{X}}}}(\lfloor
mP\rfloor))\rightarrow
H^{0}(\tilde{{\mathcal{X}}},{\mathcal{O}}_{\tilde{{\mathcal{X}}}}(m{\mathcal{L}}))$
(46)

induced by the section $e_{m}$ is an isomorphism, where $e_{m}$ is the
canonical section of $\lceil m\mathcal{N}\rceil$.

######  Lemma 4.13.

If a big line bundle ${\mathcal{L}}$ admits a birational Zariski
decomposition, then the conjecture 4.7 for ${\mathcal{L}}$ is true.

###### Proof.

By Lemma 4.2 and (39), we have
$\mathfrak{r}_{1}({\mathcal{L}})=\mathfrak{r}_{1}(\mu^{*}{\mathcal{L}})=\mathfrak{r}_{1}(\mathcal{P})=\mathcal{P}^{n}\cdot
K_{\tilde{{\mathcal{X}}}}$. Choose any ample divisor $A$ on
$\tilde{{\mathcal{X}}}$. Because $\mathcal{P}$ is big and nef, we know that
for $k\gg 1$, $k\mathcal{P}-A=\Delta_{k}$ is effective. So we get:

$(m+k)\mathcal{P}=m\mathcal{P}+A+E_{k},$ (47)

which implies the decomposition over $\tilde{{\mathcal{X}}}$:

$\mu^{*}{\mathcal{L}}=\mathcal{P}+\mathcal{N}=\frac{1}{m+k}(m\mathcal{P}+A)+\frac{1}{m+k}\Delta_{k}+\mathcal{N}.$
(48)

By perturbing the coefficients of $A$, we can assume that $m\mathcal{P}+A$ is
a ${\mathbb{Q}}$-divisor. Set
${\mathcal{L}}_{m}=\frac{1}{m+k}(m\mathcal{P}+A)$. Then it is easy to see that
(42) holds true.

∎

######  Remark 4.14.

If $(X,L)$ is a polarized spherical manifold, it is known that its models in
the sense of Definition 2.1 is a Mori dream space (see the appendix A by Y.
Odaka to [16]). Since Zariski decomposition of big lines bundles always exist
on Mori dream spaces, the above lemma in the ${\mathbb{C}}^{*}$-equivariant
setting gives an explanation why the Yau-Tian-Donaldson conjecture holds for
polarized spherical manifolds. See [16, Appendix A] for a slightly different
proof of this fact (again based on Theorem 1.3).

##  Appendix A Conjecture 4.7 for Nakayama’s examples without birational
Zariski decomposition

In this appendix, we will use Lemma 4.10 to show that conjecture 4.7 is indeed
true for some examples of big line bundles that do not have a birational
Zariski decomposition. Such examples were first discovered by Nakayama [30].
Here we will do a case study based on the construction of Fujita approximation
in [25, Theorem 11.4.4] and the calculation of asymptotic multiplier ideals
for Nakayama’s examples in the work of Koike [23].

We first write down some notations. Set $S=E\times E$ for an elliptic curve
$E$ without complex multiplication. Then the pseudoeffective cone ${\rm
PE}(S)$ coincides with the nef cone ${\rm Nef}(S)$. Fix a point $p\in E$ and
consider in $N^{1}(S)_{\mathbb{R}}$ three classes:

$f_{1}=[\\{P\\}\times E]=:[F_{1}],\quad
f_{2}=[E\times\\{P\\}]=:[F_{2}],\quad\delta=[\Delta]$

where $\Delta\subset E\times E$ is the diagonal. Then $N^{1}(S)_{\mathbb{R}}$
is spanned by $\\{f_{1},f_{2},\delta\\}$ and the description of the nef cone
is known (see [24, Lemm 1.5.4]): $\alpha=x\cdot f_{1}+y\cdot
f_{2}+z\cdot\delta\in N^{1}(S)_{\mathbb{R}}$ is nef if and only if

$xy+xz+yz\geq 0,\quad x+y+z\geq 0.$ (49)

By standard linear algebra, we can use the following linear transformation to
diagonalize the above relation:

$\displaystyle l_{1}=\frac{1}{6}(f_{1}+f_{2}-2\delta),\quad
l_{2}=\frac{1}{6}(-\sqrt{3}f_{1}+\sqrt{3}f_{2}),\quad\frac{1}{6}(f_{1}+f_{2}+\delta)$
$\displaystyle a=x+y-2z,\quad b=-\sqrt{3}x+\sqrt{3}y,\quad c=2(x+y+z).$

such that $\alpha=al_{1}+bl_{2}+cl_{3}\in N^{1}(S)_{\mathbb{R}}$ is nef if and
only if

$c^{2}\geq a^{2}+b^{2},\quad c\geq 0.$ (50)

Let $L_{i},i=0,1,2$ be three line bundles over $S$. Set

$X={\mathbb{P}}(\mathcal{O}_{S}\oplus(L_{1}-L_{0})\oplus(L_{2}-L_{0}))\cong{\mathbb{P}}(L_{0}\oplus
L_{1}\oplus L_{2}).$ (51)

Denote by $H=\mathcal{O}_{X}(1)$ the tautological line bundle.

We use the description of $X$ as a toric bundle over $S$ as in [30]. Let
$\Sigma$ denote the standard fan of ${\mathbb{P}}^{2}$, i.e. the fan generated
by three cones:

$\sigma_{1}={\rm Cone}\\{e_{1},e_{2}\\},\quad\sigma_{2}={\rm
Cone}\\{e_{2},-(e_{1}+e_{2})\\},\quad\sigma_{3}={\rm
Cone}\\{-(e_{1}+e_{2}),e_{1}\\}.$

Let $h:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ be the piecewise linear
function on $\Sigma$ satisfying $h(e_{1})=h(e_{2})=0$ and
$h(-(e_{1}+e_{2}))=-1$. Then $X$ is the toric bundle associated to $\Sigma$
and $h$ determines the line bundle $H$. Set $\mathbb{L}=\pi^{*}L_{0}+D_{h}$.
By a result of Cutkosky (see [23, Lemma 6.1]), $\mathbb{L}$ is a big line
bundle if and only if there exists $(k_{0},k_{1},k_{2})\in{\mathbb{N}}^{3}$
such that $L_{0}^{k_{0}}\otimes L_{1}^{k_{1}}\otimes L_{2}^{k_{3}}$ is an
ample line bundle over $S$. Moreover it is well-known that the canonical line
bundle of the projective bundle $X$ is given by:

$K_{X}=\pi^{*}(K_{S}+(L_{1}-L_{0})+(L_{2}-L_{0}))-3H=L_{1}+L_{2}-2L_{0}-3H.$
(52)

The last identity uses the triviality of $K_{S}$.

We will consider the example in [23, Example 6.5]. Set
$L_{0}=4F_{1}+4F_{2}+\Delta$, $L_{1}={\mathcal{O}}_{V}$,
$L_{2}={\mathcal{O}}_{V}(-F_{1}+9F_{2}+\Delta)$. Then

$c_{1}(L_{0})=6(l_{1}+3l_{3}),\quad c_{1}(L_{1})=0,\quad
c_{1}(L_{2})=6l_{1}+10\sqrt{3}l_{2}+18l_{3}.$ (53)

Because $c_{1}(L_{0})$ is in the interior of the nef cone, $L_{0}$ is ample.
Note that $H$ is relatively ample. So it is easy to see that there exist
$a,b\in{\mathbb{Z}}_{>0}$ such that $\frac{aL_{0}+bH-K_{X}}{n+1}$ is a very
ample line bundle. Set $G=aL_{0}+bH$ and

$\displaystyle M_{p}$ $\displaystyle=$ $\displaystyle
p\,\mathbb{L}-G=p(L_{0}+H)-(aL_{0}+bH)=(p-a)L_{0}+(p-b)H$ $\displaystyle=$
$\displaystyle(p-b)\left(\frac{p-a}{p-b}L_{0}+H\right)=:(p-b)(Q_{p}+H).$

Set $\mathcal{J}_{p}:=\mathcal{J}(X,\|M_{p}\|)$. Let $\mu_{p}:Y_{p}\rightarrow
X$ be the normalized blowup of $\mathcal{J}_{p}$ with
$E_{p}:=\mu_{p}^{*}\mathcal{J}_{p}={\mathcal{O}}_{Y_{p}}(-\sum_{i}c_{p,i}E_{p,i})$.
Set $A_{p}=\mu_{p}^{*}(\mathbb{L})-\frac{1}{p}E_{p}$.

By the discussion after Lemma 4.10, we known that $(Y_{p},A_{p})$ satisfies
the first two conditions of Lemma 4.10. So, by Lemma 4.10, it suffices to show
that there exists $C>0$ such that $A_{X}(E_{i,p})\leq Cp^{-1}c_{p,i}$ for any
$i,p$.

For any ${\mathbb{Q}}$-line bundle $L$ on $S$ and with $h$ as above, define a
compact convex set following [30, §2.b] (we identify line bundles with their
Chern classes):

$\Box(L,h)=\\{(x,y)\in{\mathbb{R}}_{\geq 0}^{2};x+y\leq 1\text{ and
}L+x(L_{1}-L_{0})+y(L_{2}-L_{0})\in{\rm PE}(S)\\}.$ (54)

Then it is straight-forward to use (50) and (53) to get:

$\displaystyle\Box(Q_{p},h)=\left\\{(x,y)\in{\mathbb{R}}^{2}_{\geq
0};\;\;x+\frac{5}{2\sqrt{6}}\frac{p-b}{p-a}y\leq 1\right\\}.$

Let $\varphi_{p,\min}$ be the metric of minimal singularity on $M_{p}$. Then
it is known that
$\mathcal{J}(\varphi_{p,\min})=\mathcal{J}(\|M_{p}\|)=\mathcal{J}_{p}$. For
each fan $\sigma_{i},i=0,1,2$, there exists an open set $U_{i}\cong
S\times{\mathbb{C}}^{2}$ which is an affine toric bundle over $S$. Applying
the result in [23, 5.2], $\mathcal{J}_{p}$ is trivial on $U_{0},U_{1}$, and
over $U_{2}$ we can choose the canonical affine coordinate $(z_{1},z_{2})$ on
${\mathbb{C}}^{2}$ such that the multiplier ideal is generated by monomials:

$\mathcal{J}_{p}(U_{2})=\left\langle
z_{1}^{m_{1}}z_{2}^{m_{2}};\;\;(m_{1}+1,m_{2}+1)\in{\rm
Int}((p-b)S_{p})\cap{\mathbb{Z}}^{2}\right\rangle$ (55)

where

$S_{p}=\left\\{(x,y)\in{\mathbb{R}}^{2}_{\geq
0};\;\;x+\frac{y}{1-\frac{2\sqrt{6}(p-a)}{5(p-b)}}\geq 1\right\\},$ (56)

is generated by the exponents
$(0,1),(1,0),(0,1-\frac{2\sqrt{6}(p-a)}{5(p-b)})$, which are the images of
$(0,0)$, $(1,0)$, $(0,\frac{2\sqrt{6}(p-a)}{5(p-b)})$ under the linear map
$(a,b)\mapsto(\langle(a,b)-m_{\sigma_{3}},v_{i}\rangle)_{i=1,2}=(a,1-a-b)$
with $m_{\sigma_{3}}=(0,1),v_{1}=(1,0),v_{2}=(-1,-1)$ (see [23, Definition
4.1]). In particular, the multiplier ideal sheaf is co-supported on
${\mathbb{P}}(L_{2})\subset{\mathbb{P}}(L_{0}\oplus L_{1}\oplus L_{2})$.
Moreover, as pointed out in [23], by using [30, Theorem 2.10], we know that
the line bundle $\mathbb{L}$ does not admit birational Zariski decomposition.

Equivalently, we have:

$\mathcal{J}_{p}(U_{2})=\left\langle
z_{1}^{m_{1}}z_{2}^{m_{2}};\;\;(m_{1},m_{2})\in{\mathbb{Z}}_{\geq
0},\frac{m_{1}}{\alpha_{p}}+\frac{m_{2}}{\beta_{p}}>1\right\rangle.$ (57)

where
$\alpha_{p}=\frac{p-b}{d_{p}},\quad\beta_{p}=\frac{(1-\frac{2\sqrt{6}}{5})p+\frac{2\sqrt{6}}{5}a-b}{d_{p}},\quad
d_{p}=1-\frac{1}{p-b}-\frac{1}{(1-\frac{2\sqrt{6}}{5})p+\frac{2\sqrt{6}}{5}a-b}$.
We only need to know that there exists $C=\frac{4.9-2\sqrt{6}}{5}>0$ such that
$\alpha_{p}\geq Cp,\beta_{p}\geq Cp$ for $p\gg 1$ since $d_{p}=1+O(p^{-1})$.

Because the multiplier ideal is monomial, we can use the result about Rees
valuations of monomial ideals to see that the blow-up of $\mathcal{J}_{p}$
corresponds to the sides of the Newton-polygons of $\mathcal{J}_{p}$ (see [22,
15.4]). Indeed, such blowup also corresponds to a subdivision of the cone
$\sigma_{2}$.

Now denote the sides of the Newton polygon be given by
$\overline{P_{i-1}P_{i}},1\leq i\leq r$ with $P_{i}=(x_{i},y_{i})$. Then it is
easy to see that $P_{0}=(0,\lfloor\beta\rfloor+1)$ and
$P_{r}=(\lfloor\alpha\rfloor+1,0)$. Note that
$(a_{i},b_{i})=(y_{i-1}-y_{i},x_{i}-x_{i-1})\in{\mathbb{R}}_{>0}^{2}$ is a
normal vector of $\overline{P_{i-1}P_{i}}$. The monomial valuation ${\rm
ord}_{E_{i}}$ that corresponds to the side $\overline{P_{i}P_{i+1}}$ can be
chosen to be given by the weighted blowup with weights $(a_{i},b_{i})$. Set
$\tau_{i}=b_{i}/a_{i}>0$. It is easy to see that:

$\displaystyle w(E_{i}):=\frac{A(E_{i})}{{\rm ord}_{E_{i}}(\mathcal{J}_{p})}$
$\displaystyle=$
$\displaystyle\frac{a_{i}+b_{i}}{a_{i}x_{i-1}+b_{i}y_{i-1}}=\frac{1+\tau_{i}}{x_{i-1}+y_{i-1}\tau_{i}}$
$\displaystyle=$
$\displaystyle\frac{a_{i}+b_{i}}{a_{i}x_{i}+b_{i}y_{i}}=\frac{1+\tau_{i}}{x_{i}+y_{i}\tau_{i}}.$

As a consequence, we have:

$w(E_{i})-w(E_{i+1})=\frac{1+\tau_{i}}{x_{i}+y_{i}\tau_{i}}-\frac{1+\tau_{i+1}}{x_{i}+y_{i}\tau_{i+1}}=\frac{(y_{i}-x_{i})(\tau_{i+1}-\tau_{i})}{(x_{i}+y_{i}\tau_{i})(x_{i}+y_{i}\tau_{i+1})}.$
(58)

From this identity, we easily see that $\max\\{w(E_{i});1\leq i\leq
r\\}=\max\\{w(E_{1}),w(E_{r})\\}$. Now note that $\tau_{1}^{-1}$ is at most
the absolute value of the slope of the line $\overline{P_{0}P^{\prime}}$ where
$P^{\prime}$ is the point $(1,-\frac{\beta}{\alpha}+\beta)$ one the line
connecting $(\alpha,0)$ and $(0,\beta)$, which gives the inequality:

$\displaystyle w(E_{1})$ $\displaystyle=$
$\displaystyle\frac{1+\tau_{1}}{(\lfloor\beta\rfloor+1)\tau_{1}}=\frac{1}{\lfloor\beta\rfloor+1}+\frac{1}{(\lfloor\beta\rfloor+1)\tau_{1}}\leq\frac{1}{\beta}+\frac{1}{\lfloor\beta\rfloor+1}(\lfloor\beta\rfloor+1-\beta+\frac{\beta}{\alpha})$
$\displaystyle=$
$\displaystyle\frac{1}{\beta}+\frac{\lfloor\beta\rfloor+1-\beta}{\lfloor\beta\rfloor+1}+\frac{\beta}{\lfloor\beta\rfloor+1}\frac{1}{\alpha}=O(p^{-1}).$

By the same argument (or just by symmetry), we also get $w(E_{r})=O(p^{-1})$.
According to the previous discussion, the verification of 3rd condition in
Lemma 4.10 is complete.

######  Remark A.1.

It is easy to see that the above arguments, which reduce the problem to the
estimates for Rees valuations of monomial ideals, works for many more examples
of Nakayama type.

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Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067.

Current address: Department of Mathematics, Rutgers University, Piscataway, NJ
08854-8019.

E-mail address<EMAIL_ADDRESS>


# HEAVY-QUARK PRODUCTION AT NNLO

SIMONE DEVOTO  Dipartimento di Fisica Aldo Pontremoli, Università di Milano
and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

###### Abstract

We present the calculation of QCD radiative corrections to heavy-quark
production at NNLO. The calculation is implemented in an updated version of
Matrix, which allows us to obtain multidifferential predictions for stable top
quarks. We also report on the NNLO calculation for $b{\bar{b}}$ production and
comment on the inclusion of top-quark decays and of the extension of our
results to $t\bar{t}H$.

##  1 Introduction

Top quark ($t$) production is a process of key importance for the LHC physics
programme. The mass of the top quark represents one of the fundamental
parameters of the Standard Model: its high value leads to a strong coupling
with the Higgs boson and thus to an important role in electroweak-symmetry
breaking. Furthermore, the top quark is a possible window on new physics
scenarios, and its production constitutes an important background for both
precision Standard Model measurements and Beyond the Standard Model searches.
The main source of top events at the LHC is the production of a $t\bar{t}$
pair, whose measurement is now reaching an impressive experimental precision.
During the last decades, there have been many efforts from the theoretical
community to provide predictions that can match such experimental accuracy.

In the following, we will focus on our computation of NNLO QCD corrections to
top-quark pair production[1, 2, 3] that, through its implementation within the
Matrix[4] framework, provided a first public NNLO generator for the process.
We will briefly discuss further developments of this work, including bottom-
quark (b) pair production[5], $t{\bar{t}}H$ production and $t{\bar{t}}$
production including top-quark decays.

##  2 Top-quark pair production: inclusion in MATRIX 2.1.0

Our computation of $t\bar{t}$ production relies on the $q_{T}$-subtraction
formalism[6] as a method to handle and cancel infrared divergences. Originally
developed for the production of colourless final states, with the inclusion of
extra contributions it can be extended to massive, colourful final states.

Within the $q_{T}$-subtraction formalism, the cross section for the production
of a general final state $F$ can be written as follows:

$d\sigma^{F}_{NNLO}=\mathcal{H}^{F}_{NNLO}\otimes
d\sigma_{LO}^{F}+\left[d\sigma_{NLO}^{F+{jet}}-d\sigma^{CT}_{NNLO}\right]\;.$
(1)

The term in the square brackets represents the contribution to the cross
section with $q_{T}\neq 0$, $q_{T}$ being the transverse momentum of the
system $F$. Since the final state $F$ has $q_{T}=0$ at Born level kinematics,
this contribution at NNLO is effectively captured by the NLO cross section for
the final state $F+$jet that can thus be computed with known NLO subtraction
techniques such as, for instance, the dipole subtraction method[7]. There are
some extra singularities of pure NNLO type associated to the limit $q_{T}\to
0$, which are subtracted by the counterterm $d\sigma^{CT}_{NNLO}$. If $F$ is
colourless, $d\sigma^{CT}_{NNLO}$ is built by using the knowledge of the
small-$q_{T}$ behaviour of the cross section as given by all-order
resummation[8]. The coefficient $\mathcal{H}^{F}_{NNLO}$ provides the
contribution to the process with $q_{T}=0$, and thus contains the information
on the virtual corrections.

The extension of eq.1 to a massive colourful final state $F$ requires
additional soft contributions, affecting both the counterterm
$d\sigma^{CT}_{NNLO}$ and the coefficient ${\cal H}_{NNLO}^{F}$. The
contribution to $d\sigma^{CT}_{NNLO}$ is known from Ref.[ [9]], while the
contribution to ${\cal H}_{NNLO}^{F}$ has been recently evaluated by us [10]
(see also Ref.[ [11]]).

With the inclusion of those final ingredients, we were able to apply
$q_{T}$-subtraction to the production of a colourful final state and to carry
out several phenomenological studies on $t\bar{t}$ production at NNLO. We
considered the inclusive cross section[1], differential distributions and
comparisons with experimental data[2], and predictions in the $\overline{MS}$
renormalisation scheme for the top mass[3].

Our calculation has been implemented in Matrix, a computational framework that
allows the user to evaluate fully differential cross sections for a wide class
of process at hadron colliders at NNLO QCD and NLO EW. The core of Matrix is
the Monte Carlo integrator Munich, that contains an implementation of the
Catani-Seymour subtraction method[7, 12] for NLO QCD computations. The
required amplitudes are obtained either from OpenLoops[13] or, in the case of
the two-loop amplitudes, in the form of numerical grids or analytic
expressions. Matrix then implements then the $q_{T}$-subtraction formalism and
provides the user with the desired differential distribution for the chosen
infrared-safe observable.

A preliminary version of the code for the computation of multi-differential
distributions for $t\bar{t}$ production has been made available to
experimental collaborations, and has already been used for data-theory
comparisons[14]. In the latest release of Matrix, version 2.1.0, the inclusion
of $t\bar{t}$ production has become publicly available111The latest version of
Matrix can be downloaded at: https://matrix.hepforge.org.

Matrix 2.1.0 contains several improvements with respect to the previous
version. The number of supported processes increases: not only $t\bar{t}$
production is included, but also tri-photon production[15]. Furthermore, the
user has now access to the option to compute double-differential distributions
for arbitrary infrared-safe observable. Finally, on a more technical side, the
inclusion of power corrections[16] and of the bin-wise extrapolation in the
$r_{\rm cut}\to 0$ limit of the slicing parameter improve the accuracy of the
predictions for the distributions.

##  3 Further Developments: bottom-quark pair production

Figure 1: Bottom-quark pair production cross section as measured by the Alice,
Phenix, CDF and UA1 collaborations. The green band represent the NNLO
prediction computed with Matrix. Plot taken from Ref.[ [17]].

The completion of the computation of $t\bar{t}$ production within the
$q_{T}$-subtraction formalism opens the door to several possible extensions.

A first development was the computation of NNLO QCD corrections to $b\bar{b}$
production[5]. This has been implemented in the Matrix framework as an
extension of $t\bar{t}$ production, by changing the heavy-quark mass and by
allowing an arbitrary number of light flavours (from $n_{f}=5$ to $n_{f}=4$),
which corresponds to a computation with a massive bottom quark and a top quark
decoupled from the process. We obtained results for the inclusive cross
section and first predictions for differential distributions. The computation
is fully implemented in a (at the moment) private release of Matrix.
Predictions are provided upon request, and have already been used by the Alice
collaboration in their analysis.

In Figure 1 we show the comparison, presented by the Alice collaboration[17],
between different experimental measurements and theoretical predictions for
$b\bar{b}$ production. The Alice analysis corresponds to a centre of mass
energy $\sqrt{s}=5.02$ TeV, and the cross section for $b\bar{b}$ production is
extracted from D-meson measurements. Figure 1 additionally shows results at
other collider energies as obtained by several other experiments (Phenix, CDF,
UA1). The data are compared to our NNLO predictions (green band) and to the
FONLL predictions[18] (blue band), which are obtained by combining NLO QCD
corrections and resummed computations. The inclusion of NNLO QCD corrections
leads to a better agrement with the data at all the considered collider
energies, and reduces the theoretical uncertainties, as obtained by using the
customary 7-points scale variation.

##  4 Conclusions and Outlook

The $q_{T}$-subtraction formalism has been extended to the production of
heavy-quark pairs in hadron collisions, leading to several phenomenological
application to $t\bar{t}$ and $b\bar{b}$ production. This computation has been
implemented in the Matrix framework, and $t\bar{t}$ production is now
supported as a process in the latest version of the public release of Matrix.

Our work can be extended in several directions. We can consider the production
of a heavy-quark pair with additional colourless particles. First steps in
this direction have been carried out with the computation of NNLO corrections
for $t{\bar{t}}H$ production in the flavour off-diagonal channels [19]. To
complete the computation in the other channels some ingredients are still
missing. In particular, the two-loop amplitudes are not yet available, and the
contribution of soft final state emission has to be properly evaluated.

We can also consider the inclusion of top quark decays and off-shell effects.
In this case, the process is actually $pp\to
e^{+}\nu_{e}b\mu^{-}{\bar{\nu}}_{\mu}{\bar{b}}+X$ and higher-order QCD
corrections can be computed by considering the production of the $b{\bar{b}}$
pair recoiling against the colourless $2l2\nu$ system. Work in these
directions is ongoing.

## References

## References

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