# Height filtrations and base loci on flag bundles over a curve

Yangyu Fan  Academy for Multidisciplinary Studies, Beijing National Center for
Applied Mathematics, Capital Normal University, Beijing, 100048, People’s
Republic of China<EMAIL_ADDRESS>,  Wenbin Luo  Beijing International Center
for Mathematical Research, Peking University, Beijing 100871, China
<EMAIL_ADDRESS>and  Binggang Qu  Beijing International Center for
Mathematical Research, Peking University, Beijing 100871, China
<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be an algebraically closed field of characteristic zero. Let $C/k$ be
a projective smooth curve with function field $K=k(C)$ and $G/k$ be a
connected reductive group. Let $F$ be a principal $G$-bundle on $C$. Let
$P\subseteq G$ be a parabolic subgroup and $\lambda:P\longrightarrow G$ be a
strictly anti-dominant character. Then $F/P\longrightarrow C$ is a flag bundle
and $\mathcal{L}_{\lambda}=F\times_{P}k_{\lambda}$ on $F/P$ is a relatively
ample line bundle.

Let $h_{\mathcal{L}_{\lambda}}$ be the height function on $X=(F/P)_{K}$
induced by $\mathcal{L}_{\lambda}$. We compute its height filtration and
successive minima: the height filtration is given by a Bruhat decomposition
and successive minima are given by slopes.

An interesting application is that, in this case, Zhang’s inequality on
successive minima can be enhanced into an equality. This is proved from the
convex geometry point of view.

Let $f\in N^{1}(F/P)$ be the numerical class of a vertical fiber. We also
compute the augmented base loci $\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)$ for
any $t\in\mathbb{R}$ and it turns out that they are almost the same as the
height filtration, which accords with the philosophy in Arakelov geometry that
the positivity of the line bundle $\mathcal{L}_{\lambda}$ is equivalent to the
positivity of its induced height function $h_{\mathcal{L}_{\lambda}}$. As a
corollary, we compute the $k$-th movable cones of flag bundles over curves for
all $k$.

###### Contents

  1. 1 Introduction
  2. 2 Height filtrations and successive minima
  3. 3 Reinforcement of Zhang’s inequality into an equality
  4. 4 Base loci and movable cones

##  1\. Introduction

The main purpose of this paper is to give an explicit description of the
height function and its related topics on a flag bundle over a curve. In this
paper, all fields are of characteristic zero and all curves are projective
smooth. By variety we mean an integral seperated scheme of finite type over a
field.

###  1.1. Height filtration and successive minima

We start by giving a reminder on height functions in Arakelov theory and the
relevant history. Let $K$ be either a number field or $K=k(C)$ where $C$ is a
projective smooth curve over a field $k$. Let $X/K$ be a projective variety of
dimension $d$ and $\overline{L}$ be an adelic line bundle on $X$. These data
induce an Arakelov height function $h_{\overline{L}}$ on $X$ ([27], see also
[25, §9] for a survey). A typical case is the geometric height, which is the
one we concern in this article. Here we give a definition.

If $K=k(C)$, let $\mathcal{X}\longrightarrow C$ be a projective flat morphism
with generic fiber $X\longrightarrow\operatorname{Spec}(K)$ and $\mathcal{L}$
be a line bundle on $\mathcal{X}$ with $\mathcal{L}_{K}\cong L$. The data
$(\mathcal{X},\mathcal{L})$ define an adelic line bundle $\overline{L}$ and
the height function $h_{\overline{L}}$ is given by

$\displaystyle
h_{\overline{L}}:X(\overline{K})\longrightarrow\mathbb{Q},\;x\longmapsto\frac{\mathcal{L}\cdot\overline{\\{x\\}}}{{\mathrm{deg}}(x)}\;\;\text{where
$\overline{\\{x\\}}$ is the closure of $x$ in $\mathcal{X}$}.$

We also denote this by $h_{\mathcal{L}}$ if there is no ambiguity. If $K$ is a
number field, the height function can be defined similarly by arithmetic
intersection theory.

Assume $L$ is ample. Let $Z_{t}(X,h_{\overline{L}})=\text{Zariski closure
of}\;\big{\\{}x\in X(\overline{K}):h_{\overline{L}}(x)<t\big{\\}}$. Note that

  * •

$t\longmapsto Z_{t}(X,h_{\overline{L}})$ is an increasing filtration of
Zariski closed subsets.

  * •

$Z_{t}(X,h_{\overline{L}})=X$ when $t\gg 0$ and
$Z_{t}(X,h_{\overline{L}})=\emptyset$ when $t\ll 0$.

Since Zariski topology is Noetherian, the filtration
$Z_{t}(X,h_{\overline{L}})$ gives a finite filtration $X_{0}=X\supsetneq
X_{1}\supsetneq X_{2}\supsetneq\cdots\supsetneq X_{r}=\emptyset$, the _height
filtration_. Its jumping points
$\zeta_{i}(X,h_{\overline{L}})=\inf\big{\\{}t:Z_{t}(X,h_{\overline{L}})=X_{i-1}\big{\\}}$
are called the _successive minima_.

Note that our definition of successive minima are slightly different with
Zhang [27]: Zhang considers only dimension jumps
$e_{i}(X,h_{\overline{L}})=\inf\big{\\{}t:\dim Z_{t}(X,h_{\overline{L}})\geq
d-i+1\big{\\}}$, $i=1,\dots,d+1$. In the following,
$e_{i}(X,h_{\overline{L}})$ will be called _Zhang’s successive minima_ to
avoid ambiguity.

The first minimum
$\zeta_{1}(X,h_{\overline{L}})=e_{1}(X,h_{\overline{L}})=\inf\big{\\{}t:Z_{t}(X,h_{\overline{L}})=X\big{\\}}$
is of particular interest, it is called the _essential minimum_ and also
denoted by $\zeta_{\text{ess}}(X,h_{\overline{L}})$. This invariant is the
protagonist of the story of the Bogomolov conjecture, which in number field
case says that a closed subvariety $X$ of an abelian variety $A$ has essential
minimum zero with respect to the Néron-Tate height on $A$, if and only if it
is the translation of an abelian subvariety by a torsion point, we refer the
reader to [26, 24] for more details. The height filtration induced by the
Néron-Tate height is $A\supsetneq\emptyset$ with the unique jumping point
$\zeta=0$. This allows the equidistribution of small points to happen, and
consequently implies the Bogomolov conjecture.

The height filtration (of toric height functions) on toric varieties is
completely determined in [10]. An important example is on the projective space
$\mathbb{P}^{n}_{\mathbb{Q}}$. Consider the height function induced by the
universal bundle $\mathcal{O}_{\mathbb{P}^{n}_{\mathbb{Z}}}(1)$ equipped with
the Fubini-Study metric $\|\cdot\|_{\text{FS}}$. The height filtration is
given by successively deleting torus orbits, that is,
$\mathbb{P}^{n}_{\mathbb{Q}}\supsetneq\bigcup_{i}\big{\\{}x_{i}=0\big{\\}}\supsetneq\bigcup_{i,j}\big{\\{}x_{i}=0,x_{j}=0\big{\\}}\supsetneq\cdots\supsetneq\bigcup_{i}\big{\\{}x_{i}\neq
0\text{ and $x_{j}=0$ for all $j\neq i$}\big{\\}}\supsetneq\emptyset$ with
successive minima $\frac{1}{2}\log(k+1)$, $k=0,1,\dots,n$.

In this article, we consider another class of varieties with group actions:
flag varieties. Let $k$ be an algebraic closed field, $G/k$ be a connected
reductive group, $C/k$ be a curve, and $F$ be a principal $G$-bundle on $C$
with canonical reduction $F_{Q}$ to $Q$. This defines a linear functional
${\mathrm{deg}}(F_{Q})$ on $X(T)_{\mathbb{Q}}$. Let $P\subseteq G$ be a
parabolic subgroup and $\lambda:P\longrightarrow G$ be a strictly anti-
dominant character. Then $F/P\longrightarrow C$ is a flag bundle and
$\mathcal{L}_{\lambda}=F\times_{P}k_{\lambda}$ on $F/P$ is a relatively ample
line bundle. Let $X=(F/P)_{K}$ and $h_{\mathcal{L}_{\lambda}}$ be the height
function on $X$ induced by $(F/P,\mathcal{L}_{\lambda})$. We compute its
height filtration and successive minima.

######  Theorem A (Theorem 2.11).

The height filtration of $h_{\mathcal{L}_{\lambda}}$ on $X$ is given by
successively deleting Schubert cells $C_{w}=(F_{Q}\times_{Q}QwP/P)_{K}$ for
$w\in W_{Q}\backslash W/W_{P}$, i.e.

$\displaystyle
Z_{t}=X\Bigg{\backslash}\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq
t}C_{w}=\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle<t}C_{w}.$

In particular, successive minima are
$\zeta_{w}=\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ and Zhang’s
successive minima are $e_{i}=\min\big{\\{}\zeta_{w}:\ell(w)=\dim
G/P-i+1\big{\\}}$ where $\ell(w)=\max\limits_{\sigma\in
W_{Q}wW_{P}}\min\limits_{\tau\in\sigma W_{P}}\ell(\tau)$.

###  1.2. Zhang’s inequality

Let $X/K$ be a projective variety of dimension $d$, and $\overline{L}$ be a
relative semipositive adelic line bundle with the underlying line bundle $L$
being ample. The height of $X$ is defined to be
$h_{\overline{L}}(X)=\frac{\overline{L}^{d+1}}{(d+1)L^{n}}$. In [27], Zhang
gives the following inequality:

$\displaystyle e_{1}\geq
h_{\overline{L}}(X)\geq\frac{1}{d+1}\sum_{i=1}^{d+1}e_{i}.$

In the case of toric varieties, Zhang’s successive minima can be intepreted by
convex geometry as shown in [9, 10]. Moreover, Zhang’s inequality can be
proved directly [10, Theorem C] as an integral inequality of a convex function
on a polytope. This can be partially generalized to arbitrary projective
varieties by the theory of Okounkov bodies [19] and concave transforms [8, 2].

In this article, we carry on this spirit in the setting of flag bundles $F/P$.
Given a strictly anti-dominant character
$\lambda:P\longrightarrow\mathbb{G}_{m}$, representation theorists could
associate to $\lambda$ a _string polytope_ $\Delta_{\lambda}$, a _weight
polytope_ $\mathcal{P}_{\lambda}$ and a _weight map_
$p:\Delta_{\lambda}\longrightarrow\mathcal{P}_{\lambda}$. Let
$M_{\lambda}=G\times_{P}k_{\lambda}$ be the line bundle on $G/P$ induced by
$\lambda$. Then integral points in $\Delta_{\lambda}$ parametrizes a weight
basis of $H^{0}(G/P,M_{\lambda})$ and
$p:\Delta_{\lambda}\longrightarrow\mathcal{P}_{\lambda}$ sends an integral
point to the weight of the vector parametrized by it [18, §2.2].

Since the weight polytope is the convex hull of the Weyl orbit $W\lambda$, it
follows immediately from Theorem A that successive minima of
$h_{\mathcal{L}_{\lambda}}$ are the value of ${\mathrm{deg}}(F_{Q})$ at
vertices of $\mathcal{P}_{\lambda}$. We will also prove

$\displaystyle
h_{\mathcal{L}_{\lambda}}(X)=\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p\;\mathrm{d}\mu.$

An interesting application is that, Zhang’s inequality in this case can be
enhanced into an equality, in the sense that

######  Theorem B (Theorem 3.1).

Let $q:W/W_{P}\longrightarrow W_{Q}\backslash W/W_{P}$ be the canoniacl
projection and for $w\in W_{Q}\backslash W/W_{P}$ define $m_{w}=\\#q^{-1}(w)$.
Then

$\displaystyle h_{\mathcal{L}_{\lambda}}(X)=\frac{1}{\\#W/W_{P}}\sum_{w\in
W_{Q}\backslash W/W_{P}}m_{w}\zeta_{w}.$

This follows from a simple computation of integration on the string polytopes:
in fact for any linear functional
$f:X(T)_{\mathbb{R}}\longrightarrow\mathbb{R}$, we have

$\displaystyle\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}f\circ
p\;\mathrm{d}\mu=\frac{1}{\\#W/W_{P}}\sum_{w\in W/W_{P}}f(w\lambda).$

We remark that it is certainly not possible to enhance Zhang’s inequality into
an equality in general. Consider the height function on
$\mathbb{P}^{n}_{\mathbb{Q}}$ induced by the the Hermitian line bundle
$\big{(}\mathcal{O}_{\mathbb{P}^{n}_{\mathbb{Z}}}(1),\|\cdot\|_{\text{FS}}\big{)}$.
The variety height is $\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{i}\frac{1}{j}$
[7], which is a rational number while the successive minima are irrational
numbers $\tfrac{1}{2}\log(k+1)$, $k=0,1,\dots,n$.

###  1.3. Base loci

It is a general philosophy in Arakelov geometry that the positivity of
$\overline{L}$ are equivalent to the positivity of its induced height function
$h_{\overline{L}}$. The study of essential minimum and pseudo-effectivity
provides an evidence [10, 2, 22]. Note that if the underlying line bundle $L$
is big, then $\overline{L}$ is big $\Longleftrightarrow$
$\zeta_{\mathrm{ess}}(h_{\overline{L}})>0$.

On the other hand, the theory of augmented base loci $\mathrm{B}_{+}(\cdot)$
is well-developed [13, 14] in the study of positivity in algebraic geometry.
Now back to the case $\mathcal{X}=F/P$, let $f$ be a line bundle given by a
vertical fiber. Recall that it is a classical result that
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)$ is the union of closed subvariety
over which the restricted volume is $0$ (see §4.1). In particular, if
$\mathcal{Y}\subseteq\mathcal{X}$ is a closed subvariety such that
$(\mathcal{L}_{\lambda}-tf)|_{\mathcal{Y}}$ is not big on $\mathcal{Y}$, then
$\mathcal{Y}\subseteq\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)$.

For $w\in W_{Q}\backslash W/W_{P}$ such that
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\leq t$, our calculation of
$h_{\mathcal{L}_{\lambda}}$ gives that the essential minimum
$\zeta_{\mathrm{ess}}(h_{(\mathcal{L}_{\lambda}-tf)|_{\mathcal{X}_{\omega}}})$
on the Schubert variety $\mathcal{X}_{w}=F_{Q}\times_{Q}\overline{QwP}/P$ is
no bigger than $0$, hence $(\mathcal{L}_{\lambda}-tf)|_{\mathcal{X}_{\omega}}$
is not big by the arithmetic bigness criterion mentioned above. We thus obtain
that

$\displaystyle\bigcup_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\leq
t}\mathcal{X}_{w}\subseteq\bigcup_{\mathcal{Y}\subseteq\mathcal{X},(\mathcal{L}_{\lambda}-tf)|_{\mathcal{Y}}\text{
not big}}\mathcal{Y}\subseteq\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)$

where $\mathcal{Y}$ runs over all closed subvariety of $\mathcal{X}$.
Moreover, we prove that both inclusions are actually equality:

######  Theorem C (Theorem 4.6).

The augmented base locus of $\mathcal{L}_{\lambda}-tf$ is given by

$\displaystyle\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)=\mathcal{X}\Bigg{\backslash}\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w}=\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\leq
t}\mathcal{C}_{w}$

where $\mathcal{C}_{w}=F_{Q}\times_{Q}QwP/P$.

In particular, comparing this with Theorem A, we see that
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)_{K}=Z_{t}$ for all $t\in\mathbb{R}$,
except for the critical points $t=\zeta_{w}$ at which
$B_{+}(\mathcal{L}_{\lambda}-\zeta_{w}f)$ is upper continuous while
$Z_{\zeta_{w}}$ is lower continuous.

###  1.4. Movable cones

Recall that for a projective variety $Y$ of dimension $d$, the _k_ -th movable
cones $\operatorname{Mov}^{k}(Y)$, $k=1,2,\dots,d$ are defined as

$\displaystyle\operatorname{Mov}^{k}(Y)=\big{\\{}\alpha\in
N^{1}(Y)_{\mathbb{R}}:\mathrm{B}_{+}(\alpha)\text{ has codimension $\geq
k$}\big{\\}}.$

$\operatorname{Mov}^{k}(Y)$ are open cones in $N^{1}(Y)_{\mathbb{R}}$ and we
have obvious inclusions

$\displaystyle\operatorname{Mov}^{d}\subseteq\operatorname{Mov}^{d-1}\subseteq\cdots\subseteq\operatorname{Mov}^{1}.$

Note that $\operatorname{Mov}^{1}$ is the big cone and
$\operatorname{Mov}^{d}$ is the ample cone.

Let $k$ be an algebraically closed field, $C/k$ be a curve and $E$ be a vector
bundle on $C$. Note that $N^{1}(\mathbb{P}(E))_{\mathbb{R}}$ is two
dimensional, generated by $f$ and $\xi=\mathcal{O}_{\mathbb{P}(E)}(1)$. The
ample cone is given by extremal ray $f$ and $\xi-\mu_{\min(E)}f$, and the big
cone is given by extremal ray $f$ and $\xi-\mu_{\max(E)}f$. The former is a
theorem of Miyaoka [21] and the latter can be found in [15]. Fulger and
Lehmann computed $\operatorname{Mov}^{2}(\mathbb{P}(E))$ in [16]. In [4], the
big cone of the Grassmann bundle of $E$ is computed and in [6], the ample cone
of flag bundles of $E$ is computed.

As a direct consequence of Theorem C, we compute the $k$-th movable cones of
flag bundles $F/P$ over a curve $C$ for all $k$. Since any element in
$N^{1}(F/P)_{\mathbb{R}}$ can be written as $\mathcal{L}_{\lambda}-tf$, we may
define two functions $\langle\alpha^{\vee},\cdot\rangle$ and
$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ on $N^{1}(F/P)_{\mathbb{R}}$ as

  * •

$\langle\alpha^{\vee},\cdot\rangle$ sends $\mathcal{L}_{\lambda}-tf$ to
$\langle\alpha^{\vee},\lambda\rangle$.

  * •

$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ sends $\mathcal{L}_{\lambda}-tf$
to $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle-t$.

These functions are well-defined, and

######  Theorem D (Theorem 4.9).

The $k$-th movable cone $\operatorname{Mov}^{k}(F/P)$ is the cone defined by
$\langle\alpha^{\vee},\cdot\rangle<0$ for any
$\alpha\in\Delta\backslash\Delta_{P}$ and
$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle>0$ for any $w\in W/W_{P}$ with
$\ell(w)\geq n-k+1$.

In particular, we find that $\mathcal{L}_{\lambda}$ is $k$-movable
$\Longleftrightarrow$ the $k$-th minimum of Zhang of
$h_{\mathcal{L}_{\lambda}}$ is positive.

###  1.5. Example: Grassmann bundles

We provide here the computation of essential minimum and big cone on Grassmann
bundles. Let $k$ be an algebraically closed field and $C/k$ be a curve with
function field $K=k(C)$. Let $E$ be a vector bundle of rank $n$ on $C$ with
Harder-Narasimhan filtration $0\subsetneq E_{1}\subsetneq\cdots\subsetneq
E_{r}$. Let $B$ be the subgroup of $\operatorname{GL}_{n}$ of all upper
triangular matrices, and let $Q$ be the parabolic subgroup containing $B$ of
type
$(\operatorname{rank}(E_{1}),\operatorname{rank}(E_{2}),\dots,\operatorname{rank}(E_{r}))$.
Here we say a parabolic subgroup containing $B$ is of type
$(a_{1},\dots,a_{r})$ if it consists of upper block triguluar matrices of
shape

$\left[\begin{array}[]{cccc}A_{1}&*&\cdots&*\\\ &A_{2}&\cdots&*\\\
&&\cdots&*\\\ &&&A_{r}\end{array}\right],\quad
A_{i}\in\operatorname{GL}_{a_{i}}.$

Let $F(E)$ be the frame bundle of $E$, which is a
$\operatorname{GL}_{n}$-principal bundle. Then the canonical reduction of
$F(E)$ is $F_{Q}$ to $Q$ where $F_{Q}$ is the $Q$-bundle of frames respecting
the filtration $0\subsetneq E_{1}\subsetneq\cdots\subsetneq E_{r}$. Let
$\mu_{i}$ ($i=1,\dots,n$) be the $i$-th number in the sequence

$\displaystyle\underbrace{\mu(E_{1}),\dots,\mu(E_{1})}_{\text{$\operatorname{rank}(E_{1})$
times}},\underbrace{\mu(E_{2}/E_{1}),\dots,\mu(E_{2}/E_{1})}_{\text{$\operatorname{rank}(E_{2}/E_{1})$
times}},\dots,\underbrace{\mu(E/E_{r-1}),\dots,\mu(E/E_{r-1})}_{\text{$\operatorname{rank}(E/E_{r-1})$
times}}.$

Note that $X(T)_{\mathbb{Q}}=\bigoplus_{i=1}^{n}\mathbb{Q}\lambda_{i}$ where
$\lambda_{i}:T\longrightarrow\mathbb{G}_{m}$ is the character of taking the
$i$-th entry, and $\langle{\mathrm{deg}}(F_{Q}),\lambda_{i}\rangle=\mu_{i}$.

Let $\operatorname{Gr}_{r}(E)=F(E)/P$ be the Grassmann bundle of $r$-dimension
quotients, where $P$ is the parabolic subgroup containing $B$ of type
$(n-r,r)$. Identify the Weyl group of $G$ with $S_{n}$ and of $P$ with
$S_{n-r}\times S_{r}$ as usual. Then $W/W_{P}=S_{n}\big{/}(S_{n-r}\times
S_{r})$ can be identified with the multi-index set
$\big{\\{}(i_{1},\dots,i_{r}):1\leq i_{1}<i_{2}<\cdots<i_{r}\leq n\big{\\}}$
via

$\displaystyle\left[\begin{matrix}1&2&\cdots&n-r&n-r+1&\cdots&n\\\
*&*&\cdots&*&i_{1}&\cdots&i_{r}\end{matrix}\right]\longleftrightarrow(i_{1},\dots,i_{r}).$

$N^{1}(\operatorname{Gr}_{r}(E))$ is two dimensional with basis
$\mathcal{O}(1)$ (determinant bundle of the universal bundle on
$\operatorname{Gr}_{r}(E)$) and $f$ (a vertical fiber). Let $\det_{2}$ be the
character

$\displaystyle\det\nolimits_{2}:P\longrightarrow\mathbb{G}_{m},\quad\begin{bmatrix}A_{1}&*\\\
0&A_{2}\end{bmatrix}\longmapsto\det(A_{2}).$

One checks that $\mathcal{O}(1)=F(E)\times_{P}k_{\det_{2}}$,
$\det_{2}=\lambda_{n-r+1}+\cdots+\lambda_{n}$ in $X(T)$ and
$I=(i_{1},\dots,i_{r})\in W/W_{P}$ acts on $\det_{2}$ by
$I\det_{2}=\lambda_{i_{1}}+\cdots+\lambda_{i_{r}}$.

Let $h_{\mathcal{O}(1)}$ be the height function on
$\operatorname{Gr}_{r}(E_{K})$ induced by the model
$(\operatorname{Gr}_{r}(E),\mathcal{O}(1))$. By Theorem A, the successive
minima are $\langle{\mathrm{deg}}(F_{Q}),I\det_{2}\rangle=\sum_{i\in
I}\mu_{i}$. (Theorem A says the successive minima are
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ for $w\in W_{Q}\backslash
W/W_{P}$, but these numbers are really just
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ for $w\in W/W_{P}$.) In
particular, the longest element $I_{0}=(1,2,\dots,r)$ gives the essential
minimum

$\displaystyle\zeta_{\text{ess}}=\langle{\mathrm{deg}}(F_{Q}),I_{0}\det\nolimits_{2}\rangle=\mu_{1}+\cdots+\mu_{r}.$

The only element in $\Delta\backslash\Delta_{P}$ is
$\alpha=\lambda_{n-r}-\lambda_{n-r+1}$ and the function
$\langle\alpha^{\vee},\cdot\rangle$ on $N^{1}(\operatorname{Gr}_{r}(E))$ is
given by $\mathcal{O}(1)\longmapsto\langle\alpha^{\vee},\det_{2}\rangle<0$ and
$f\longmapsto 0$. Let $I_{0}=(1,2,\dots,r)\in W/W_{P}$ be the longest element.
Then the function $\langle{\mathrm{deg}}(F_{Q}),I_{0}\cdot\rangle$ on
$N^{1}(\operatorname{Gr}_{r}(E))$ is given by
$\mathcal{O}(1)\longmapsto\mu_{1}+\cdots+\mu_{r}$ and $f\longmapsto 1$. By
Theorem C, the big cone is given by $\langle\alpha^{\vee},\cdot\rangle<0$ and
$\langle{\mathrm{deg}}(F_{Q}),I_{0}\cdot\rangle>0$, and one sees that it is
the cone given by extremal rays $f$ and
$\mathcal{O}(1)-(\mu_{1}+\dots+\mu_{r})f$.

###  1.6. Notations and conventions

In the following, let $k$ be an algebraically closed field of characteristic
zero and $C/k$ be a projective smooth curve with function field $K=k(C)$. Let
$G/k$ be a connected reductive group and denote the identity component of its
center by $Z(G)$. Fix a Borel subgroup $B\subseteq G$ and a maximal torus
$T\subseteq B$. Let $W$ be the Weyl group and $\Delta$ be the set of simple
roots with respect to $(G,B,T)$. For any $\alpha\in\Delta$, we denote by
$\alpha^{\vee}$ the corresponding simple coroot.

We shall consider only parabolic subgroup $P\subseteq G$ containing $B$. For
any parabolic subgroup $P$, let $W_{P}\subseteq W$ be the Weyl group
$W(L_{P})$ of the Levi factor $L_{P}\subseteq P$ and
$\Delta_{P}\subseteq\Delta$ be the simple roots of $L_{P}$. For parabolic
subgroups $P,Q\subseteq G$ and any double coset $w\in W_{Q}\backslash
W/W_{P}$, set $\ell(w)=\max_{\sigma\in W_{Q}wW_{P}}\min_{\tau\in\sigma
W_{P}}\ell(\tau)$, which equals to $\operatorname{dim}QwP/P$. Here by abuse of
notations, $\ell(\tau)$ is the usual length of the Weyl element $\tau$.

For any linear algebraic group $\Gamma/k$, let $X(\Gamma)$ denote the
character group $\mathrm{Hom}(\Gamma,\mathbb{G}_{m})$. For any
$\mathbb{Z}$-algebra $A$, denote by $X(\Gamma)_{A}$ the tensor product
$X(\Gamma)\otimes_{\mathbb{Z}}A$. For any linear functional $f$ on
$X(\Gamma)$, we shall usually denote $f(\alpha)$ by $\langle f,\alpha\rangle$
for $\alpha\in X(\Gamma)$. For any $\alpha\in X(\Gamma)$, denote by
$k_{\alpha}$ the one-dimensional algebraic representation on the vector space
$k$ with $\Gamma$ acting by $\alpha$.

###  1.7. Organization of the paper

Let $F$ be a principal $G$-bundle on $C$ and
$\lambda:P\longrightarrow\mathbb{G}_{m}$ be a strictly anti-dominant
character. Then $F/P$ is a flag bundle over $C$ and
$\mathcal{L}_{\lambda}=F\times_{P}k_{\lambda}$ on $F/P$ is a relatively ample
line bundle. Let $X=(F/P)_{K}$ and $h_{\mathcal{L}_{\lambda}}$ be the height
function on $X$ induced by $(F/P,\mathcal{L}_{\lambda})$.

In Chapter $2$, we compute the height filtration and successive minima of
$h_{\mathcal{L}_{\lambda}}$. In Chapter $3$, we enhance Zhang’s inequality
into an equality by convex geometry. In Chapter $4$, we compute the augmented
base loci of $\mathcal{L}_{\lambda}-tf$ on $F/P$. As a corollary we get the
movable cones $\operatorname{Mov}^{k}(F/P)$.

##  2\. Height filtrations and successive minima

###  2.1. Vector bundles, slope stability and Harder-Narasimhan filtration

Let $E$ be a vector bundle on $C$. The degree of $E$ is
${\mathrm{deg}}(E)={\mathrm{deg}}(\det(E))$ and the slope of $E$ is
$\mu(E)={\mathrm{deg}}(E)/\operatorname{rk}(E)$. It is called slope semistable
if for every subbundle $F\subseteq E$, $\mu(F)\leq\mu(E)$. This is equivalent
to $\mu(Q)\geq\mu(E)$ for every quotient bundle $Q$ of $E$.

In general, there exists uniquely a filtration $0=E_{0}\subseteq
E_{1}\subseteq\cdots\subseteq E_{r}=E$ such that

  * •

$E_{i}/E_{i-1}$ is semistable;

  * •

$\mu(E_{i}/E_{i-1})>\mu(E_{i+1}/E_{i})$.

This filtration is called the _Harder-Narasimhan filtration_ of $E$ and we
denote $\mu_{i}(E)=\mu(E_{i}/E_{i-1})$. The first and the last term are of
particular interests. The first one $\mu_{1}(E)=\mu(E_{1})$ is called _maximal
slope_ of $E$ and is also denoted by $\mu_{\max}(E)$. The last one
$\mu_{r}(E)=\mu(E/E_{r-1})$ is called _minimal slope_ of $E$ and is also
denoted by $\mu_{\min}(E)$.

###  2.2. Principal bundles

For any linear algebraic group $\Gamma/k$, a _principal $\Gamma$-bundle_ on
$C$ is a variety $F$ equipped with a right action of $\Gamma$ and a
$\Gamma$-equivariant smooth morphism $F\longrightarrow C$ such that the map

$\displaystyle F\times_{C}(C\times\Gamma)\longrightarrow
F\times_{C}F,\quad(f,(x,g))\longmapsto(f,fg)$

is an isomorphism.

Attached to any principal $\Gamma$-bundle $F$, one has the _degree map_

$\displaystyle{\mathrm{deg}}(F):X(\Gamma)\longrightarrow\mathbb{Z},\quad\lambda\longmapsto\langle{\mathrm{deg}}(F),\lambda\rangle={\mathrm{deg}}(F\times_{\Gamma}k_{\lambda}).$

Here ${\mathrm{deg}}(F\times_{\Gamma}k_{\lambda})$ is the degree of the line
bundle $F\times_{\Gamma}k_{\lambda}$ on the curve $C$.

Let $H$ be a closed subgroup of a linear algebraic group $\Gamma/k$. A
_reduction of structure group_ of $F$ to $H$ is a pair $(F_{H},\varphi)$ where
$F_{H}$ is a principal $H$-bundle and $\varphi:F_{H}\times_{H}\Gamma\cong F$
is an isomorphism. By the universal property of the quotient $F/H$, the
assignment to any section $\sigma:C\longrightarrow F/H$ the reduction
$\sigma^{*}F$ of $F$ to $H$ is a one-one correspondence between reductions of
structure group of $F$ to $H$ and sections of $F/H\longrightarrow C$.

Let $F$ be a principal $G$-bundle on $C$ for the connected reductive group
$G$. Note that for any parabolic subgroup $P\subseteq G$ with Levi subgroup
$L_{P}$, the natural inclusion

$\displaystyle X(P)\longrightarrow X(L_{P})\longrightarrow X(Z(L_{P}))$

becomes an isomorphism after tensoring with $\mathbb{Q}$. Thus we have

$\displaystyle X(T)_{\mathbb{Q}}\longrightarrow
X(Z(L_{P}))_{\mathbb{Q}}=X(P)_{\mathbb{Q}}$

and by taking duals, we get the so-called _slope map_
$X(P)^{\vee}_{\mathbb{Q}}\longrightarrow X(T)^{\vee}_{\mathbb{Q}}$ introduced
in [23, §2.1.3]. Let $F_{P}$ be a reduction of $F$ to $P$. By applying the
slope map to ${\mathrm{deg}}(F_{P})$ we can define
$\langle{\mathrm{deg}}(F_{P}),\lambda\rangle$ for any $\lambda\in X(T)$.

###  2.3. Semistability and canonical reduction

We mainly follow [5].

######  Definition 2.1 (semistable).

A principal $G$-bundle $F$ on $C$ is called _semistable_ if for any parabolic
subgroup $P$, any reduction $F_{P}$ of $F$ to $P$ and any dominant character
$\lambda$ of $P$ which is trivial on $Z(G)$, we have
$\langle{\mathrm{deg}}(F_{P}),\lambda\rangle\leq 0$.

######  Definition 2.2 (canonical reduction).

Let $F$ be a principal $G$-bundle on $C$. A reduction $F_{P}$ of $F$ to a
parabolic subgroup $P$ is called _canonical_ if the following two conditions
hold:

  1. (1)

The the principal $L_{P}$ bundle $F_{P}\times_{P}L_{P}$ is semistable.

  2. (2)

For any non-trivial character $\lambda$ of $P$ which is non-negative linear
combination of simple roots, $\langle{\mathrm{deg}}(F_{P}),\lambda\rangle>0$.

Behrend in [3] proved canonical reduction exists uniquely. When
$G=\operatorname{GL}_{n}$, the above definition of semistability recovers
Mumford’s slope stability of vector bundles and the above deinition of
canonical reduction recovers the Harder-Narasimhan filtration of vector
bundles.

###  2.4. Height function induced by $(F/P,\mathcal{L}_{\lambda}$)

Let $F$ be a principal $G$-bundle with canonical reduction $F_{Q}$ to some
parabolic subgroup $Q\subseteq G$. Let $P\subseteq G$ be a parabolic subgroup.
Set $\mathcal{X}=F/P$ and $X=\mathcal{X}_{K}$.

A character $\lambda:P\longrightarrow\mathbb{G}_{m}$ is called strictly anti-
dominant if the natural pairing $\langle\alpha^{\vee},\lambda\rangle<0$ for
any $\alpha\in\Delta\backslash\Delta_{P}$. Let
$\lambda:P\longrightarrow\mathbb{G}$ be a stricly anti-dominant character.
Then the line bundle $M_{\lambda}=G\times_{P}k_{\lambda}$ on $G/P$ is ample.
Therefore $\mathcal{L}_{\lambda}=F\times_{G}M_{\lambda}$ is a relatively ample
line bundle on $\mathcal{X}=F\times_{G}G/P$ and induces a height function

$\displaystyle\quad\quad\quad h_{\mathcal{L}_{\lambda}}:X(\overline{K})$
$\displaystyle\longrightarrow\mathbb{Q},\quad
x\longmapsto\frac{\mathcal{L}_{\lambda}\cdot\overline{\\{x\\}}}{{\mathrm{deg}}(x)}$

where $\overline{\\{x\\}}$ is the Zariski closure of $x$ in $\mathcal{X}$.

For $x\in X(K)$, let $\sigma_{x}:C\longrightarrow\mathcal{X}$ be the section
induced by $x:\operatorname{Spec}(K)\longrightarrow X$ via valuative
criterion, and let $F_{P,x}=\sigma_{x}^{*}F$ be the corresponding reduction to
$P$.

######  Lemma 2.3.

For $x\in X(K)$,
$h_{\mathcal{L}_{\lambda}}(x)=\langle{\mathrm{deg}}(F_{P,x}),\lambda\rangle$.

###### Proof.

By definition, $h_{\mathcal{L}_{\lambda}}(x)$ is the degree of
$\sigma_{x}^{*}(\mathcal{L}_{\lambda})$ and
$\langle{\mathrm{deg}}(F_{P,x}),\lambda\rangle$ is the degree of
$F_{P,x}\times_{P}k_{\lambda}$. We have equality of line bundles

$\displaystyle\sigma_{x}^{*}(\mathcal{L}_{\lambda})=\sigma_{x}^{*}(F\times_{P}k_{\lambda})=(\sigma_{x}^{*}F)\times_{P}k_{\lambda}=F_{P,x}\times_{P}k_{\lambda}$

and the lemma follows by taking degree. ∎

###  2.5. A height lower bound in Schubert cells

For $w\in W_{Q}\backslash W/W_{P}$, write
$\mathcal{C}_{w}=F_{Q}\times_{Q}QwP/P$,
$\mathcal{X}_{w}=F_{Q}\times_{Q}\overline{QwP}/P$, $C_{w}=\mathcal{C}_{w,K}$
and $X_{w}=\mathcal{X}_{w,K}$. We shall show that

######  Proposition 2.4.

For any $x\in C_{w}(\overline{K})$,
$h_{\mathcal{L}_{\lambda}}(x)\geq\langle{\mathrm{deg}}F_{Q},w\lambda\rangle$.

Note that for any $w^{\prime}\in W_{Q}$ and $\lambda\in X(T)$, we have
$w^{\prime}\lambda-\lambda\in\mathbb{Z}[\Delta_{Q}]$ consequently
$\langle{\mathrm{deg}}(F_{Q}),w^{\prime}\lambda\rangle=\langle{\mathrm{deg}}(F_{Q}),\lambda\rangle$.
Note also that for any $\lambda\in X(P)$ and $w\in W_{P}$, $w\lambda=\lambda$.
So the number $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ is well-defined
for any $\lambda\in X(P)$ and $w\in W_{Q}\backslash W/W_{P}$.

######  Definition 2.5 (relative position).

A reduction $F_{P}$ of $F$ to $P$ is called _in relative position $w\in
W_{Q}\backslash W/W_{P}$ with respect to_ $F_{Q}$ if the image of the natural
map

$\displaystyle F_{Q}\times_{C}F_{P}\longrightarrow G_{C},\quad(a,b)\longmapsto
a^{-1}b$

lies in $QwP_{C}$. Here we implicitly apply the injections

$\displaystyle F_{P}\hookrightarrow F_{P}\times_{P}G\cong F,\quad
F_{Q}\hookrightarrow F_{Q}\times_{Q}G\cong F.$

Note that by the universal property of the quotient stacks $[G//Q\times P]$
and $[QwP//Q\times P]$, this definition coincides with the one in [23, §4.1].

######  Lemma 2.6.

When $x\in C_{w}(K)$, $F_{P,x}$ is of relative position $w$ with respect to
$F_{Q}$.

###### Proof.

Note that the injection $F_{P,x}=\sigma_{x}^{*}F\longrightarrow F$ has image
in $F_{Q}\times_{Q}QwP$ by the commutativity of the following diagram

Now the lemma follows since $F_{Q}\times_{C}F_{P,x}\longrightarrow C\times G$
factors as

$\displaystyle F_{Q}\times_{C}F_{P,x}\longrightarrow
F_{Q}\times_{C}\Big{(}F_{Q}\times_{Q}QwP\Big{)}\longrightarrow$
$\displaystyle\Big{(}F_{Q}\times_{C}F_{Q}\Big{)}\times_{Q}QwP\longrightarrow(C\times
Q)\times_{Q}QwP=C\times QwP.$

∎

######  Proposition 2.7 ([23], Proposition $4.6$).

If $F_{P}$ is in relative position $w$ with respect to $F_{Q}$, then
$\langle{\mathrm{deg}}(F_{P}),\lambda\rangle\geq\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$
for any anti-dominant character $\lambda$.

Now we can prove Proposition 2.4.

###### Proof of _Proposition_ 2.4.

Let $L/K$ be any finite field extension. Assume that $x\in C_{w}(L)$. Consider
the following commutative diagram

---
$\textstyle{X_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{C_{L}}/P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F/P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spec}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{x}}$$\scriptstyle{x}$$\textstyle{C_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{\widetilde{x}}}$$\scriptstyle{\sigma_{x}}$$\textstyle{\operatorname{Spec}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C.}$

where $C_{L}$ is the normalization of $C$ in $L$ and
$F_{C_{L}}=F\times_{C}C_{L}$. Let $\mathcal{L}_{C_{L},\lambda}$ be the
pullback of $\mathcal{L}_{\lambda}$ via the map $F_{C_{L}}/P\longrightarrow
F/P$, which is also the line bundle associated to the character $\lambda$ of
$P$ via the principal bundle $F_{C_{L}}$ on $C_{L}$. Then by Lemma 2.3,

$\displaystyle
h_{\mathcal{L}_{\lambda}}(x)=\tfrac{1}{[L:K]}{\mathrm{deg}}(\sigma_{x}^{*}\mathcal{L}_{\lambda})=\tfrac{1}{[L:K]}{\mathrm{deg}}(\sigma_{\widetilde{x}}^{*}\mathcal{L}_{C_{L},\lambda})=\tfrac{1}{[L:K]}\langle{\mathrm{deg}}(F_{C_{L},P,\widetilde{x}}),\lambda\rangle$

where $F_{C_{L},P,\widetilde{x}}$ is the reduction of $F_{C_{L}}$ to $P$
corresponding to the rational point $\widetilde{x}\in X_{L}(L)$.

Note that the canonical reduction of $F_{C_{L}}$ is
$F_{Q,C_{L}}=F_{Q}\times_{C}C_{L}$ and
${\mathrm{deg}}(F_{Q,C_{L}})=[L:K]{\mathrm{deg}}(F_{Q})$ in
$X(T)_{\mathbb{Q}}^{\vee}$. By Lemma 2.6, $F_{Q,C_{L}}$ is of relative
position $w$ with respect to $F_{C_{L},P,\widetilde{x}}$. We conclude by
applying Proposition 2.7 to $F_{C_{L}}$. ∎

###  2.6. Height filtration and successive minima of
$h_{\mathcal{L}_{\lambda}}$

In this subsection, we employ Ballaÿ’s theorem [2, Theorem 1.2] to compute
essential minimum. We start by finding the maximal slope of
$\pi_{*}\mathcal{L}_{\lambda}|_{\mathcal{X}_{w}}$.

######  Lemma 2.8 ([17], Part I, §5.18).

On $\mathcal{X}_{w}=F_{Q}\times_{Q}\overline{QwP}/P\subseteq\mathcal{X}$, we
have
$\pi_{*}\big{(}\mathcal{L}_{\lambda}|_{\mathcal{X}_{w}}\big{)}=F_{Q}\times_{Q}H^{0}(\overline{QwP}/P,M_{\lambda})$.

Schieder computed the Harder-Narasimhan filtration of
$F_{Q}\times_{Q}H^{0}(\overline{QwP}/P,M_{\lambda})$ in [23, §5.1]. Let $V/k$
be a finite dimensional $Q$-representation with weight decomposition
$V=\bigoplus_{\nu\in X(T)}V[\nu]$. The $Q$-filtration
$V_{\bullet,{\mathrm{deg}}(F_{Q})}$ of $V$ is defined by

$\displaystyle
V_{t,{\mathrm{deg}}(F_{Q})}=\bigoplus_{\langle{\mathrm{deg}}(F_{Q}),\nu\rangle\geq
t}V[\nu].$

######  Proposition 2.9.

The Harder-Narasimhan filtration of
$F_{Q}\times_{Q}H^{0}(\overline{QwP}/P,M_{\lambda})$ is
$F_{Q}\times_{Q}H^{0}(\overline{QwP}/P,M_{\lambda})_{\bullet,{\mathrm{deg}}(F_{Q})}$.
Moreover, the maximal slope is $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$.

###### Proof.

The first assertion is a slight generalization of [23, Proposition 5.1], whose
original texts only consider $G$-representations restricted to $Q$, but the
proof actually works for arbitrary $Q$-representations. For the second
assertion, we only need to show the highest weights in
$H^{0}(\overline{QwP}/P,M_{\lambda})$ belong to
$w\lambda+\mathbb{Z}[\Delta_{Q}]$.

When $Q=B$, $H^{0}(\overline{BwP}/P,M_{\lambda})$ has a single highest weight
$w\lambda$ by the theory of Demazure modules [17, Chapter 14]. For general
$Q$, let $\pi:W/W_{P}\longrightarrow W_{Q}\backslash W/W_{P}$ be the canoniacl
projection. Note that

$\displaystyle\overline{QwP}/P=\bigcup_{w^{\prime}\in\pi^{-1}(w)}\overline{Bw^{\prime}P}/P.$

Since the restriction map

$\displaystyle H^{0}(\overline{QwP}/P,M_{\lambda})\longrightarrow
H^{0}(\overline{Bw^{\prime}P}/P,M_{\lambda})$

is surjective for each $w^{\prime}\in\pi^{-1}(w)$ and the diagonal map

$\displaystyle
H^{0}(\overline{QwP}/P,M_{\lambda})\longrightarrow\prod_{w^{\prime}\in\pi^{-1}(w)}H^{0}(\overline{Bw^{\prime}P}/P,M_{\lambda})$

is injective, the highest weights in $H^{0}(\overline{QwP}/P,M_{\lambda})$
belong to $w\lambda+\mathbb{Z}[\Delta_{Q}]$. ∎

######  Corollary 2.10.

The essential minimum $\zeta_{1}(h_{\mathcal{L}_{\lambda}},X_{w})$ of
$h_{\mathcal{L}_{\lambda}}$ on $X_{w}$ is
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$.

###### Proof.

By Ballaÿ’s theorem [2, Theorem 1.2],

$\displaystyle\quad\quad\quad\zeta_{1}(h_{\mathcal{L}_{\lambda}},X_{w})$
$\displaystyle=\displaystyle\lim_{n\rightarrow\infty}\frac{\mu_{\max}(\pi_{*}\mathcal{L}_{n\lambda}|_{\mathcal{X}_{w}})}{n}$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{n\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle}{n}=\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle.$

∎

######  Theorem 2.11.

The height filtration of $h_{\mathcal{L}_{\lambda}}$ on $X$ is given by
successively deleting Schubert cells $C_{w}$ for $w\in W_{Q}\backslash
W/W_{P}$, i.e.

$\displaystyle
Z_{t}=X\Bigg{\backslash}\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq
t}C_{w}=\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle<t}C_{w}.$

In particular, successive minima are
$\zeta_{w}=\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ and Zhang’s
successive minima are $e_{i}=\min\big{\\{}\zeta_{w}:\ell(w)=d-i+1\big{\\}}$.

###### Proof.

On $X_{w}$, $\zeta_{1}(h_{\mathcal{L}_{\lambda}},X_{w})=\zeta_{w}$ and for any
$x\in C_{w}$, $h(x)\geq\zeta_{w}$. Thus $Z_{\zeta_{w}}(X_{w})\subseteq
X_{w}\backslash C_{w}=\bigcup_{w^{\prime}<w}X_{w^{\prime}}$. For a closed
subvariety $Y$ of $X$, let $Z_{t}(Y)$ be the Zariski closure in $Y$ of
$\big{\\{}y\in Y:h(y)<t\big{\\}}$. We claim that
$Z_{\zeta_{w}}(X_{w})=X_{w}\backslash
C_{w}=\bigcup_{w^{\prime}<w}X_{w^{\prime}}$.

In fact, suppose conversely that $Z_{\zeta_{w}}(X_{w})\subsetneq
X_{w}\backslash C_{w}$. Then there exists $X_{w^{\prime}}\subsetneq X_{w}$
such that $Z_{\zeta_{w}}(X_{w})\cap X_{w^{\prime}}\subsetneq X_{w^{\prime}}$.
This forces $Z_{\zeta_{w}}(X_{w^{\prime}})\subseteq Z_{\zeta_{w}}(X_{w})\cap
X_{w^{\prime}}\subsetneq X_{w^{\prime}}$. But on $X_{w^{\prime}}$, the
essential minimum is $\zeta_{w^{\prime}}<\zeta_{w}$, so
$Z_{\zeta_{w}}(X_{w^{\prime}})=X_{w^{\prime}}$. We get contradiction and we
have thus proved that on $X_{w}$, the set $Z_{t}(X_{w})=X_{w}$ when
$t>\zeta_{w}$ and $Z_{t}(X_{w})=X_{w}\backslash C_{w}$ when $t\leq\zeta_{w}$.

Let $w_{0}\in W_{Q}\backslash W/W_{P}$ be the longest element. On
$X=X_{w_{0}}$, when $t>\zeta_{w_{0}}$, $Z_{t}=X_{w_{0}}$ and when
$t\leq\zeta_{w_{0}}$, $Z_{t}=X_{w_{0}}\backslash
C_{w_{0}}=\bigcup_{w^{\prime}}X_{w^{\prime}}$ with $w^{\prime}<w_{0}$ and
$\ell(w^{\prime})=\ell(w_{0})-1$. Repeating this procedure on each smaller
$X_{w^{\prime}}$, we can complete the proof by induction. ∎

##  3\. Reinforcement of Zhang’s inequality into an equality

In this section, we shall establish the following Zhang-type equality on flag
varieties over function fields: the height of the variety in fact equals to a
weighted average of the successive minima.

######  Theorem 3.1.

Let $q:W/W_{P}\longrightarrow W_{Q}\backslash W/W_{P}$ be the canoniacl
projection. For $w\in W_{Q}\backslash W/W_{P}$, define $m_{w}=\\#q^{-1}(w)$.
Then

$\displaystyle h_{\mathcal{L}_{\lambda}}(X)=\frac{1}{\\#W/W_{P}}\sum_{w\in
W_{Q}\backslash W/W_{P}}m_{w}\zeta_{w}.$

We shall establish this theorem by convex geometry. Let $E$ be a vector bundle
on $C$ with HN filtration $0=E_{0}\subseteq E_{1}\subseteq\cdots\subseteq
E_{r}=E$ with successive slopes $\mu_{i}=\mu(E_{i}/E_{i-1})$. Chen defines a
measure $\nu_{E}$ on $\mathbb{R}$ by

$\displaystyle\nu_{E}=\frac{1}{\operatorname{rank}(E)}\sum_{i=1}^{r}\operatorname{rank}(E_{i}/E_{i-1})\delta_{\mu_{i}},$

where $\delta_{a}$ is the Dirac measure on $\mathbb{R}$ supported at
$a\in\mathbb{R}$. For any $\varepsilon>0$ and any measure $\mu$, let
$T_{\varepsilon}\mu$ be the unique measure such that
$\displaystyle\int_{\mathbb{R}}f(x)\mathrm{d}T_{\varepsilon}\mu(x)=\int_{\mathbb{R}}f(\varepsilon
x)\mathrm{d}\mu(x)$ for any compactly-supported continuous function
$f:\mathbb{R}\longrightarrow\mathbb{R}$.

######  Theorem 3.2 (Chen; [11], Theorem 1.1).

Let $\pi:\mathcal{X}\longrightarrow C$ be a projective flat morphism and
$\mathcal{L}$ be a line bundle on $\mathcal{X}$. Assume $\mathcal{L}$ is
generically big. Then

  1. (1)

$\big{\\{}T_{\frac{1}{n}}\nu_{\pi_{*}\mathcal{L}^{n}}\big{\\}}_{n}$ converges
weakly. We denote the limit measure by $\nu^{\pi}_{\mathcal{L}}$.

  2. (2)

$\displaystyle\operatorname{Vol}(\mathcal{L})=\dim\mathcal{X}\cdot\operatorname{Vol}(\mathcal{L}_{K})\cdot\int_{\mathbb{R}}x_{+}\;\mathrm{d}\nu^{\pi}_{\mathcal{L}}$,
where $x_{+}=\max\\{0,x\\}$.

Let $w_{0}\in W$ be the longest element and fix a reduced decomposition
$w_{0}=s_{1}\cdot s_{2}\cdots s_{{\ell(w_{0})}}$. Recall that
$\lambda:P\longrightarrow\mathbb{G}_{m}$ is a strictly anti-dominant character
and $N=\dim G/P$. Let
$\Delta_{\lambda}=\Delta_{\underline{w}_{0},\lambda}\subseteq\mathbb{R}^{N}$
be the string polytope associated to $\lambda$ with respect to
$\underline{w}_{0}=(s_{1},\cdots,s_{\ell(w_{0})})$. It is equipped with a
weight map $p:\Delta_{\lambda}\longrightarrow\mathcal{P}_{\lambda}$ where
$\mathcal{P}_{\lambda}\subseteq X(T)_{\mathbb{R}}$ is the weight polytope
associated to $\lambda$ (see [1, §1] for details). The following properties
will be useful:

  1. (1)

The volume $\operatorname{Vol}(\Delta_{\lambda})$ of the convex polytope
equals to the volume $\operatorname{Vol}(M_{\lambda})$ of the line bundle.

  2. (2)

Integral points $\Delta_{\lambda}\cap\mathbb{Z}^{N}$ parametrize a basis of
$H^{0}(G/P,M_{\lambda})$. In particular,
$\\#\Delta_{\lambda}\cap\mathbb{Z}^{N}=\dim H^{0}(G/P,M_{\lambda})$.

  3. (3)

Lattice points $\mathcal{P}_{\lambda}\cap X(T)$ parameterize weights occuring
in $H^{0}(G/P,M_{\lambda})$ and $p$ sends an integral point to the underlying
weight of the corresponding basis vector.

  4. (4)

$\Delta_{\lambda_{1}}+\Delta_{\lambda_{2}}=\Delta_{\lambda_{1}+\lambda_{2}}$
and $p_{\lambda_{1}}+p_{\lambda_{2}}=p_{\lambda_{1}+\lambda_{2}}$. Here the
sum is the Minkowski sum. In particular, the diagram

$\textstyle{\Delta_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\lambda}}$$\scriptstyle{\cdot
n}$$\textstyle{\Delta_{n\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n\lambda}}$$\textstyle{\mathcal{P}_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
n}$$\textstyle{\mathcal{P}_{n\lambda}}$

is commutative for any $n\in\mathbb{N}$, where $\cdot n$ is the scalar
multiplication by $n$.

Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{N}$ normalized with respect
to $\mathbb{Z}^{N}$.

######  Lemma 3.3.

The following equalities hold:

$\displaystyle\nu_{\pi_{*}\mathcal{L}^{n}_{\lambda}}=\frac{1}{\\#(\Delta_{n\lambda}\cap\mathbb{Z}^{N})}\sum_{x\in\Delta_{n\lambda}\cap\mathbb{Z}^{N}}\delta_{{\mathrm{deg}}F_{Q}\circ
p(x)},$ $\displaystyle
T_{\frac{1}{n}}\nu_{\pi_{*}\mathcal{L}_{\lambda}^{n}}=\frac{1}{\\#(\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N})}\sum_{x\in\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N}}\delta_{{\mathrm{deg}}F_{Q}\circ
p(x)},$
$\displaystyle\nu^{\pi}_{\mathcal{L}_{\lambda}}=\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\cdot({\mathrm{deg}}F_{Q}\circ
p)_{*}\mu.$

###### Proof.

Bearing Lemma 2.8, Proposition 2.9 and the aforementioned properties of string
polytopes at hand, we can carry over the argument in [11, Proposition 3.5] to
show the first equality. Appling $T_{1/n}$ and taking limits, we obtain the
second and third equalities. ∎

######  Theorem 3.4.

The variety height $h_{\mathcal{L}_{\lambda}}(X)$ of $X$ is
$\displaystyle\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p\;\mathrm{d}\mu$.

###### Proof.

We may consider $\mathcal{L}_{\lambda}+tf$ for large $t$ so that
$\mathcal{L}_{\lambda}+tf$ is ample, where $f$ is the line bundle given by a
vertical fiber. The measure $\mu^{\pi}_{\mathcal{L}_{\lambda}+tf}$ is
supported on $\mathbb{R}_{+}$ and is a right shift of
$\mu^{\pi}_{\mathcal{L}_{\lambda}}$ by $t$, so Theorem 3.2 says

$\displaystyle\quad\quad\quad(\mathcal{L}_{\lambda}+tf)^{N+1}$
$\displaystyle=(N+1)\operatorname{Vol}(\mathcal{L}_{\lambda,K})\int_{\mathbb{R}}x\;\mathrm{d}\nu^{\pi}_{\mathcal{L}_{\lambda}+tf}$
$\displaystyle=(N+1)\int_{\Delta_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p+t\;\mathrm{d}\mu$

where the second equality is due to Lemma 3.3 and
$\operatorname{Vol}(\mathcal{L}_{\lambda,K})=\operatorname{Vol}(\Delta_{\lambda})$.
Thus

$\displaystyle
h_{\mathcal{L}_{\lambda}+tf}(X)=\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p\;\mathrm{d}\mu+t.$

Note that $h_{\mathcal{L}_{\lambda}+tf}(X)=h_{\mathcal{L}_{\lambda}}(X)+t$. So
substracting $t$, we get

$\displaystyle
h_{\mathcal{L}_{\lambda}}(X)=\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p\;\mathrm{d}\mu.$

∎

To prove Theorem 3.1, we need the following general result.

######  Proposition 3.5.

For any linear function $f:X(T)_{\mathbb{R}}\longrightarrow\mathbb{R}$,

$\displaystyle\frac{1}{\operatorname{Vol}(\Delta_{\lambda})}\int_{\Delta_{\lambda}}f\circ
p\;\mathrm{d}\mu=\frac{1}{\\#W/W_{P}}\sum_{w\in W/W_{P}}f(w\lambda).$

We start with two lemmata.

######  Lemma 3.6.

The pushforward measure $p_{*}\mu$ on $\mathcal{P}_{\lambda}$ is
$W$-invariant.

###### Proof.

Note that the Lebesgue measure $\mu$ on $\Delta_{\lambda}$ is the limit of the
Dirac measure $\delta_{\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N}}$, i.e.
$\mu=\lim_{n}\delta_{\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N}}$. Consider
the map $\Delta_{n\lambda}\longrightarrow\mathcal{P}_{n\lambda}$ and note that
it sends integral points to its corresponding weights. Dividing by n, we see
that

$\displaystyle
p_{*}\delta_{\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N}}=\operatorname{mult}_{n}\delta_{\mathcal{P}_{\lambda}\cap\frac{1}{n}X(T)},$

where $\operatorname{mult}_{n}$ is the multiplicity function

$\displaystyle\operatorname{mult}_{n}:\mathcal{P}_{\lambda}\cap\tfrac{1}{n}X(T)\longrightarrow\mathbb{N},\quad\mu\longmapsto\dim
H^{0}(G/P,L_{n\lambda})[n\mu].$

Since both $\operatorname{mult}_{n}$ and
$\delta_{\mathcal{P}_{\lambda}\cap\tfrac{1}{n}X(T)}$ are $W$-invariant, we
deduce that

$\displaystyle
p_{*}\mu=\lim_{n}p_{*}\delta_{\Delta_{\lambda}\cap\frac{1}{n}\mathbb{Z}^{N}}$

is also $W$-invariant. ∎

######  Lemma 3.7.

The map $\mathcal{P}_{\lambda}\longrightarrow\mathcal{P}_{\lambda}$,
$\mu\longmapsto\sum_{w\in W}w\mu$ is constant.

###### Proof.

We claim that for any $\mu_{1},\mu_{2}\in X(T)$ such that
$\mu_{1}|_{Z(G)}=\mu_{2}|_{Z(G)}$,

$\displaystyle\sum_{w\in W}w\mu_{1}=\sum_{w\in W}w\mu_{2}.$

In fact, for any simply connected cover $G^{sc}\longrightarrow G$, the
preimage $T^{sc}$ of $T$ is a maximal torus in $G^{sc}$ and
$W=W(G,T)=W(G^{sc},T^{sc})$. Via the natural map $X(T)\longrightarrow
X(T^{sc})$, we are reduced to the case $G$ is simply connected. In this case,
$T^{W}=Z(G)$ and the desired equality follows directly.

$H^{0}(G/P,L_{\lambda})$ is a simple $G$-module so all weights $\mu$ occuring
in $H^{0}(G/P,L_{\lambda})$ sharing the same restriction to $Z(G)$. So by the
above claim, the lemma is true for lattice points $\mathcal{P}_{\lambda}\cap
X(T)$. Note that the map $\mu\longmapsto\sum_{w\in W}w\mu$ is linear, we are
done. ∎

###### Proof of _Proposition 3.5_.

Let $\mathcal{P}_{\lambda,+}$ be the intersection of $\mathcal{P}_{\lambda}$
with the anti-dominant cone
$\big{\\{}\langle\alpha^{\vee},\lambda\rangle<0:\alpha\in\Delta\backslash\Delta_{P}\big{\\}}$.
Note that the subgroup $W_{P}\subseteq W$ acts trivially on
$\mathcal{P}_{\lambda}$. By Lemma 3.6, one has

$\displaystyle\int_{\Delta_{\lambda}}f\circ
p\;\mathrm{d}\mu=\int_{\mathcal{P}_{\lambda}}f\;\mathrm{d}p_{*}\mu=\int_{\mathcal{P}_{\lambda,+}}\Big{(}\sum_{w\in
W/W_{P}}f(wx)\Big{)}\;\mathrm{d}p_{*}\mu.$

By the linearity of $f$ and Lemma 3.7, it moreover equals to

$\displaystyle\int_{\mathcal{P}_{\lambda,+}}f\Big{(}\sum_{w\in
W/W_{P}}wx\Big{)}\;\mathrm{d}p_{*}\mu=\Big{(}\sum_{w\in
W/W_{P}}f(w\lambda)\Big{)}\int_{\mathcal{P}_{\lambda,+}}1\;\mathrm{d}p_{*}\mu.$

On the other hand,

$\displaystyle\operatorname{Vol}(\Delta_{\lambda})=\int_{\mathcal{P}_{\lambda}}1\;\mathrm{d}p_{*}\mu=\\#{W/W_{P}}\int_{\mathcal{P}_{\lambda,+}}1\;\mathrm{d}p_{*}\mu.$

Taking ratios, we obtain the desired equality. ∎

###### Proof of _Theorem 3.1_.

Take $f={\mathrm{deg}}(F_{Q})$ in Proposition 3.5 and apply Theorem 3.4. ∎

######  Remark 3.8.

The fact that the variety height and successive minima are given by

$\displaystyle\displaystyle
h_{\mathcal{L}_{\lambda}}(X)=\frac{1}{\operatorname{Vol(\Delta_{\lambda})}}\int_{\mathcal{P}_{\lambda}}{\mathrm{deg}}(F_{Q})\;\mathrm{d}p_{*}\mu,\quad\zeta_{w}=\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$

shows that the Arakelov geometry of $(F/P,\mathcal{L}_{\lambda})$ depends only
on the function ${\mathrm{deg}}(F_{Q})$ on $X(T)_{\mathbb{Q}}$, which might be
surprising at the first glance.

This is in fact because the Arakelov geometry of $(F/P,\mathcal{L}_{\lambda})$
is determined by the Harder-Narasimhan filtration of
$\pi_{*}\mathcal{L}_{n\lambda}=F_{Q}\times_{Q}H^{0}(G/P,M_{n\lambda})$ (Lemma
2.8) for all $n\geq 1$, and the unipotent part of $Q$ acts trivially [23,
Lemma 5.1].

######  Example 3.9.

Let $G=\operatorname{GL}_{4}$, $E$ be a vector bundle on $C$ of rank $4$ and
consider the $\operatorname{Gr}(2,4)$-bundle $\operatorname{Gr}_{2}(E)$. In
this case the string polytopes (with respect to a certain reduced
decomposition) is just the Gelfand-Zetlin polytope [18, Remark 2.4]. Identify
$X(T)_{\mathbb{Q}}$ with $\bigoplus_{i}\mathbb{Q}\lambda_{i}$ as before. The
line bundle $\mathcal{O}(1)$ correspond to the character $\lambda=(0,0,1,1)$.

The Gelfand-Zetlin polytope $\operatorname{GZ}_{\lambda}$ is defined by $0\leq
x_{2}\leq x_{1}\leq x_{3}\leq 1$ and $x_{2}\leq x_{4}\leq x_{3}$ in
$\mathbb{R}^{4}$, with vertices $(0,0,0,0)$, $(0,0,1,0)$, $(0,0,1,1)$,
$(1,0,1,0)$, $(1,0,1,1)$ and $(1,1,1,1)$. The weight polytope
$\mathcal{P}_{\lambda}$ is the convex hull of $(0,0,1,1)$, $(0,1,0,1)$,
$(0,1,1,0)$, $(1,1,0,0)$, $(1,0,1,0)$ and $(1,0,0,1)$, the weight map
$p:\operatorname{GZ}_{\lambda}\longrightarrow\mathcal{P}_{\lambda}$ is

$\displaystyle(x_{1},x_{2},x_{3},x_{4})\longmapsto(x_{1},-x_{1}+x_{2}+x_{3},-x_{2}-x_{3}+x_{4},-x_{4})+(0,0,1,1)$

and ${\mathrm{deg}}(F_{Q})$ is the function on $X(T)_{\mathbb{Q}}$ given by
$\lambda_{i}\longmapsto\mu_{i}$ as in §1.5.

A direct computation shows that

$\displaystyle\operatorname{Vol}(\operatorname{GZ}_{\lambda})=\frac{1}{12},\quad\int_{\operatorname{GZ}_{\lambda}}{\mathrm{deg}}(F_{Q})\circ
p\;\mathrm{d}\mu=\frac{1}{24}(\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}).$

Thus $h(X)=\frac{1}{2}(\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4})$.

On the other hand, $W/W_{P}=S_{4}\big{/}(S_{2}\times S_{2})$, $\\#W/W_{P}=6$
and

$\displaystyle\sum_{w\in
W/W_{P}}\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle=\sum_{1\leq i<j\leq
4}\mu_{i}+\mu_{j}.$

We see that $\frac{1}{\\#W/W_{P}}\sum_{w\in
W/W_{P}}\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle$ is also
$\frac{1}{2}(\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4})$.

##  4\. Base loci and movable cones

###  4.1. Preliminaries on algebraic geometry

We briefly recall some general theory of base loci. Let $\mathbf{k}$ be a
field. In our case, it would be either the base field $k$ or the function
field $K=K(C)$. Let $Y\rightarrow\operatorname{Spec}\mathbf{k}$ be a normal
projective variety over $\mathbf{k}$, and $L$ be a line bundle over $X$. Let
$V$ be a $\mathbf{k}$-subspace of $H^{0}(Y,L)$. Consider the evaluation map

$\displaystyle
H^{0}(Y,L)\otimes_{\mathbf{k}}L^{\vee}\rightarrow\mathcal{O}_{Y}.$

The base locus $\mathrm{Bs}(V)$ of $V$ is the reduced scheme associated to the
closed subscheme defined by the ideal sheaf given by the image of
$\pi^{*}V\otimes{L^{\vee}}\rightarrow\mathcal{O}_{Y}$.

######  Definition 4.1.

The stable base locus of $L$ is defined as $\mathrm{B}(L)=\bigcap_{n\geq
1}\mathrm{Bs}(H^{0}(Y,nL))$. Take any ample line bundle $A$. The augmented
base locus of $L$ is defined as $\mathrm{B}_{+}(L)=\bigcap_{n\geq
1}\mathrm{B}(nL-A)$, which is independent of the choice of $A$.

The following characterization of augmented base loci in terms of restricted
volumes is well known, which we include here for the reader’s convenience.

######  Theorem 4.2 ([14], Theorem C).

Let $Z$ be a closed subvariety of $Y$. The restricted volume is defined as

$\displaystyle\operatorname{Vol}(Y|Z,L)=\limsup_{n\rightarrow+\infty}\frac{\dim_{\mathbf{k}}\mathrm{Im}(H^{0}(Y,nL)\rightarrow
H^{0}(Z,nL|_{Z}))}{n^{\dim Z}/(\dim Z)!}$

where $\mathrm{Im}(\cdot)$ denotes the image. Then
$\mathrm{B}_{+}(L)=\bigcup_{\operatorname{Vol}(Y|Z,L)=0}Z$.

Let $\mathrm{Pic}(Y)$ be the picard group of $Y$. Since
$\mathrm{B}(L)=\mathrm{B}(mL)$ for any $m>1$, we can extend the definitions of
$\mathrm{B}(\cdot)$ and $\mathrm{B}_{+}(\cdot)$ to
$\mathrm{Pic}(Y)_{\mathbb{Q}}:=\mathrm{Pic}(Y)\otimes\mathbb{Q}$. Moreover,
let $N^{1}(Y):=\mathrm{Div}(X)/\equiv_{num}$ be the numerical group of
divisors. It is proved in [13] that the augmented base loci are well defined
on $N^{1}(Y)_{\mathbb{R}}:=N^{1}(Y)\otimes_{\mathbb{Z}}\mathbb{R}$.

Let $\mathcal{Y}$ be a projective variety with a surjective morphism
$\mathcal{Y}\rightarrow C$, and $\mathcal{L}$ be a line bundle over
$\mathcal{Y}$. For any scheme-theoretical point $p\in C$, we denote by
$\kappa(p)$ the residue field of $p$. Let $\mathcal{Z}$ be a subvariety of
$\mathcal{Y}$. The fiber $\mathcal{Z}\times_{C}\mathrm{Spec}(\kappa(p))$ of
$\mathcal{Z}$ over $p$, is denoted as $\mathcal{Z}_{p}$. We also denote by
$\mathcal{L}_{p}$ the restriction $\mathcal{L}|_{\mathcal{Y}_{p}}$ of
$\mathcal{L}$ to $\mathcal{Y}_{p}$.

We have the following lemma, which will be useful for our computation of
augmented base loci in the next subsection.

######  Lemma 4.3.

The restriction of $\mathrm{B}(\mathcal{L})$ to $\mathcal{Y}_{p}$ can be
computed by

$\displaystyle B(\mathcal{L})_{p}=\bigcap\nolimits_{n\geq
1}\mathrm{Bs}(\mathrm{Im}(H^{0}(\mathcal{X},n\mathcal{L})\otimes_{k}\kappa(p)\rightarrow
H^{0}(\mathcal{Y}_{p},n\mathcal{L}_{p}))).$

###  4.2. Augmented base loci on the flag bundle $\mathcal{X}$

Let $\lambda\in X(P)$ be a strictly anti-dominant character. We fix a closed
point $p_{0}\in C$, and set $f=\pi^{*}(\mathcal{O}_{C}(p_{0}))$. In this
subsection, we are going to compute the augmented base locus
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)$. Remind that we have the
decomposition

$\displaystyle H^{0}(G/P,M_{\lambda})=\bigoplus_{\nu\in
X(T)}H^{0}(G/P,M_{\lambda})[\nu]$

and the filtration

$\displaystyle
H^{0}(G/P,M_{\lambda})_{t,{\mathrm{deg}}(F_{Q})}=\bigoplus_{\langle{\mathrm{deg}}(F_{Q}),\nu\rangle\geq
t}H^{0}(G/P,M_{\lambda})[\nu].$

We start with a lemma describing the vanishing behaviour of sections in
$H^{0}(G/P,M_{\lambda})[\nu]$.

######  Lemma 4.4.

Take $w\in W/W_{P}$. Then

  1. (1)

$s\neq 0\in H^{0}(G/P,M_{\lambda})[w\lambda]$ $\Longrightarrow$ $s(x)\neq 0$
for any $x\in BwP/P$.

  2. (2)

$s\in H^{0}(G/P,M_{\lambda})[\nu]$ for $\nu\not\leq w\lambda$
$\Longrightarrow$ $s(x)=0$ for any $x\in\overline{BwP}/P$.

###### Proof.

Consider the restriction map

$\displaystyle\mathrm{Res}:\ H^{0}(G/P,M_{\lambda})\longrightarrow
H^{0}(\overline{BwP}/P,M_{\lambda}),$

which is $B$-equivariant and surjective. Since
$H^{0}(G/P,M_{\lambda})[w\lambda]$ and
$H^{0}(\overline{BwP}/P,M_{\lambda})[w\lambda]$ are both one-dimensional, one
has $\mathrm{Res}(s)\neq 0$ for any $s\neq 0\in
H^{0}(G/P,M_{\lambda})[w\lambda]$.

By the closedness of the vanishing locus of $\mathrm{Res}(s)$, there exists
$x\in BwP/P$ such that $\mathrm{Res}(s)(x)\neq 0$. Note that by the highest
weight theory, $B$ acts on $\mathrm{Res}(s)$ via $w\lambda$ and $BwP/P$ is a
single $B$-orbit under translation. Consequently, $\mathrm{Res}(s)(x)\neq 0$
for any $x\in BwP/P$.

As for $(2)$, for any $\nu\not\leq w\lambda$,
$H^{0}(\overline{BwP}/P,M_{\lambda})[\nu]=0$. Consequently $\mathrm{Res}(s)=0$
for any $s\in H^{0}(G/P,M_{\lambda})[\nu]$.

∎

######  Lemma 4.5.

Let $p$ be a scheme-theoretical point of $C$. Consider the filtration
$H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{\bullet,{\mathrm{deg}}(F_{Q})}$
of $H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})$ given by

$\displaystyle
H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t,{\mathrm{deg}}(F_{Q})}=F_{Q,\kappa(p)}\times_{Q}H^{0}(G/P,M_{\lambda})_{t,{\mathrm{deg}}(F_{Q})}$

for $t\in\mathbb{R}$, we have
$\operatorname{Bs}\Big{(}H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t,{\mathrm{deg}}(F_{Q})}\Big{)}=\mathcal{X}_{p}\backslash\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq
t}\mathcal{C}_{w,p}$.

###### Proof.

Let $F_{T,\kappa(p)}$ be a reduction of $F_{Q,\kappa(p)}$ to $T$. For any
$\nu\in X(T)$, consider the $\kappa(p)$-subspace
$H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})[\nu]=F_{T,\kappa(p)}\times_{T}H^{0}(G/P,M_{\lambda})[\nu]$
of
$H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})=F_{T,\kappa(p)}\times_{T}H^{0}(G/P,M_{\lambda})$.
For $w^{\prime}\in W/W_{P}$, set
$C_{w^{\prime}}=F_{T,\kappa(p)}\times_{T}Bw^{\prime}P/P$ and
$X_{w^{\prime}}=F_{T,\kappa(p)}\times_{T}\overline{Bw^{\prime}P}/P$. Then
$\mathcal{C}_{w,p}=\coprod_{\pi(w^{\prime})=w}C_{w^{\prime}}$ and
$\mathcal{X}_{w,p}=\bigcup_{\pi(w^{\prime})=w}X_{w^{\prime}}$ where
$\pi:W/W_{P}\longrightarrow W_{Q}\backslash W/W_{P}$ is the natural
projection.

Since the principal $T$-bundle $F_{T,\kappa(p)}$ is trivial, we deduce that
$s(x)\neq 0$ for any $x\in C_{w^{\prime}}$, any $w^{\prime}\in\pi^{-1}(w)$,
and any $s\in
H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})[w^{\prime}\lambda]$ from Lemma
4.4(1). Thus if $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq t$,
$H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})[w^{\prime}\lambda]\subseteq
H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t,{\mathrm{deg}}(F_{Q})}$ for
any $w^{\prime}\in\pi^{-1}(w)$ and consequently,
$x\not\in\operatorname{Bs}(H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{\geq
t})$ for any $x\in C_{w^{\prime}}$. This proves
$\operatorname{Bs}(H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t,{\mathrm{deg}}(F_{Q})})\subseteq\mathcal{X}_{p}\backslash\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq
t}\mathcal{C}_{w,p}$.

On the other hand, we claim that if
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle<t$, then for any $s\in
H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t,{\mathrm{deg}}(F_{Q})}$, the
restriction of $s$ to $X_{w}$ is zero. In fact, we may assume $s\in
H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})[\nu]$ for some $\nu$ and
$\langle{\mathrm{deg}}(F_{Q}),\nu\rangle\geq t$ implies that $\nu\not\leq
w^{\prime}\lambda$ for any $w^{\prime}\in\pi^{-1}(w)$ (because if $\nu\leq
w^{\prime}\lambda$ for some $w^{\prime}$, then
$\langle{\mathrm{deg}}(F_{Q}),\nu\rangle\leq\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle<t$).
By Lemma 4.4(2), the restriction of $s$ to $X_{w^{\prime}}$ is zero for each
$w^{\prime}$. We win. ∎

Now we are ready to compute the augmented base locus of
$\mathcal{L}_{\lambda}-tf$.

######  Theorem 4.6.

The augmented base locus of $\mathcal{L}_{\lambda}-tf$ is given by

$\displaystyle\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)=\mathcal{X}\Bigg{\backslash}\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w}=\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\leq
t}\mathcal{C}_{w}.$

###### Proof.

We may assume that $t\in\mathbb{N}$ by taking a sufficiently large tensor
power of $\mathcal{L}_{\lambda}-tf$. Let
$t_{0}=\min\\{\langle{\mathrm{deg}}(F_{Q}),\nu\rangle:H^{0}(G/P,M_{\lambda})[\nu]\not=0\text{
and }\langle{\mathrm{deg}}(F_{Q}),\nu\rangle>t\\}$. Take
$E=F_{Q}\times_{Q}H^{0}(G/P,M_{\lambda})_{t_{0},{\mathrm{deg}}(F_{Q})}\otimes\mathcal{O}_{C}(-tx)$,
which is a subbundle of $\pi_{*}(\mathcal{L}_{\lambda}-tf).$ Moreover,
$\mu_{\min}(E)=t_{0}-t>0$, i.e. $E$ is ample.

For each scheme-theoretical point $p\in C$, we have
$E(p):=E\otimes_{C}\kappa(p)=F_{Q,\kappa(p)}\times
H^{0}(G/P,M_{\lambda})_{t_{0},{\mathrm{deg}}(F_{Q})}.$ The ampleness of $E$
gives that

$\displaystyle
H^{0}(C,\mathrm{Sym}^{m}E)\otimes_{k}\kappa(p)\longrightarrow\mathrm{Sym}^{m}(E(p))$

is surjective for every $m\gg 0$. Since
$\pi_{*}(\mathcal{L})\otimes_{C}\kappa(p)=H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})$,
we have the commutative diagram

which implies that $V:=\mathrm{Im}(\mathrm{Sym}^{m}E(p)\rightarrow
H^{0}(\mathcal{X}_{p},m\mathcal{L}_{\lambda,p}))$ is contained in the image
$\mathrm{Im}(H^{0}(C,\pi_{*}(m\mathcal{L}_{\lambda}-tmf))\otimes_{k}\kappa(p)\rightarrow
H^{0}(\mathcal{X}_{p},m\mathcal{L}_{\lambda,p}))=\mathrm{Im}(H^{0}(\mathcal{X},m\mathcal{L}_{\lambda}-tmf)\otimes_{k}\kappa(p)\rightarrow
H^{0}(\mathcal{X},m\mathcal{L}_{\lambda,p})).$ Hence

$\displaystyle\mathrm{B}(\mathcal{L}_{\lambda}-tf)_{p}\subseteq\mathrm{Bs}(V)=\mathrm{Bs}(E(p))=\mathcal{X}_{p}\backslash\coprod\nolimits_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w,p}$

by applying Lemma 4.5 to
$H^{0}(\mathcal{X}_{p},\mathcal{L}_{\lambda,p})_{t_{0},{\mathrm{deg}}(F_{Q})}$.
Consequently,

(4.1)
$\displaystyle\mathrm{B}(\mathcal{L}_{\lambda}-tf)\subseteq\mathcal{X}\backslash\coprod\nolimits_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w}.$

Now we take an ample line bundle $\mathcal{A}$ over $\mathcal{X}$ of form
$\mathcal{L}_{\lambda^{\prime}}-t^{\prime}f$ where $\lambda^{\prime}\in X(T)$
is a strictly anti-dominant character and
$\langle{\mathrm{deg}}(F_{Q}),\nu\rangle-t^{\prime}>0$ for any $\nu\in X(T)$
such that $H^{0}(G/P,M_{\lambda^{\prime}})[\nu]\not=0.$

Take sufficiently large $n$ such that
$\langle{\mathrm{deg}}(F_{Q}),w\lambda^{\prime}\rangle-t^{\prime}<n(t_{0}-t)$
for any $w\in W_{Q}\backslash W/W_{P}$. Thus
$\langle{\mathrm{deg}}(F_{Q}),w(n\lambda-\lambda^{\prime})\rangle>nt-t^{\prime}$
if and only if $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\geq t_{0}.$
Similarly as in (4.1), we obtain that

$\displaystyle\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)\subset\mathrm{B}(n(\mathcal{L}_{\lambda}-tf)-\mathcal{L}_{\lambda^{\prime}}-t^{\prime}f)\subseteq\mathcal{X}\Bigg{\backslash}\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w}.$

On the other hand, for any $w\in W_{Q}\backslash W/W_{P}$ such that
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle\leq t$, since

$\displaystyle\limsup_{m\rightarrow+\infty}\frac{\mu_{\max}(\pi_{*}(m(\mathcal{L}_{\lambda}-tf)|_{\mathcal{X}_{w}})}{n}=\zeta_{1}(\mathcal{L}_{\lambda}-tf|_{\mathcal{X}_{w}})=\zeta_{w}-t\leq
0,$

we have $\operatorname{Vol}(\mathcal{L}_{\lambda}-tf|_{\mathcal{X}_{w}})=0$ by
a comparison result [12, Theorem 2.4] deduced from the Riemann-Roch theorem.
Therefore
$\operatorname{Vol}(\mathcal{X}|\mathcal{X}_{w},\mathcal{L}_{\lambda}-tf)=0$,
which implies that $\mathcal{X}_{w}\subseteq B_{+}(\mathcal{L}_{\lambda}-tf)$
due to Theorem 4.2. In conclusion,
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)=\mathcal{X}\backslash\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle>t}\mathcal{C}_{w}.$
∎

######  Remark 4.7.

Comparing Theorem 2.11 and Theorem 4.6, we find that
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)_{K}$ agrees with the height
filtration $Z_{t}$ for every $t\not=\zeta_{w}$. At critical points,
$\mathrm{B}_{+}$ is upper continuous while $Z_{t}$ is lower continuous.

###  4.3. Positive cones of flag bundles over curves

We first give a reminder on the definitions of various positive cones of a
projective variety. Let $Y$ be a smooth projective variety over a field
$\mathbf{k}$ (of characteristic zero). We recall the following:

  1. (1)

The big cone $\mathrm{Big}(Y)\subseteq N^{1}(Y)_{\mathbb{R}}$ is the cone of
numerical classes of big divisors. Its closure is called _Pseudo-effective
cone_ , denoted as $\mathrm{Psef}(Y)$.

  2. (2)

The ample cone $\mathrm{Amp}(Y)\subseteq N^{1}(Y)_{\mathbb{R}}$ is the cone of
numerical classes of ample divisors. Its closure is called nef cone, denoted
as $\mathrm{Nef}(Y)$.

  3. (3)

For each $k=1,\cdots,\dim Y$, the $k$-th movable cone is defined as
$\operatorname{Mov}^{k}(Y)=\big{\\{}D\in
N^{1}(Y)_{\mathbb{R}}:\mathrm{codim}(\mathrm{B}_{+}(D))\geq k\big{\\}}$ which
is open in $N^{1}(Y)_{\mathbb{R}}$. They are open cones in
$N^{1}(Y)_{\mathbb{R}}$ and we have obvious inclusions
$\operatorname{Mov}^{d}\subseteq\cdots\subseteq\operatorname{Mov}^{1}$. Note
that $\operatorname{Mov}^{1}$ is the big cone and $\operatorname{Mov}^{d}$ is
the ample cone.

Now we come back to the case of flag varieties. Denote by

$\displaystyle c:X(P)\longrightarrow\mathrm{Pic}(G/P),\quad\lambda\longmapsto
M_{\lambda}$

the character map. We have an exact sequence [20, Theorem 18.32]:

$\displaystyle 0\longrightarrow X(G)\longrightarrow
X(P)\longrightarrow\mathrm{Pic}(G/P)\longrightarrow\mathrm{Pic}(G)\longrightarrow\mathrm{Pic}(P)\longrightarrow
0.$

Since $G$ and $P$ are connected linear algebraic groups over a characteristic
zero field, both $\mathrm{Pic}(G)$ and $\operatorname{Pic}(P)$ are finite.
Notice that the numerical equivalence and the linear equivalence coincide on
$G/P$ since $G/P$ is smooth fano. Therefore the linear map
$X(P)_{\mathbb{R}}\rightarrow N^{1}(G/P)_{\mathbb{R}}$ is surjective. It is
well-known that $M_{\lambda}$ is ample $\Longleftrightarrow$ $M_{\lambda}$ is
big $\Longleftrightarrow$ $\lambda$ is strictly anti-dominant, thus

$\displaystyle\mathrm{Amp}(G/P)=\mathrm{Big}(G/P)=\big{\\{}c(\lambda):\langle\alpha^{\vee},\lambda\rangle<0\text{
for all $\alpha\in\Delta\backslash\Delta_{P}$}\big{\\}}$

where the pairing $\langle\cdot,\cdot\rangle$ is extended from $X(P)$ to
$X(P)_{\mathbb{R}}$ by linearity. This implies
$\mathrm{Mov}^{i}(G/P)=\mathrm{Amp}(G/P)$ for any $i=1,\cdots,\dim(G/P)$,
which is the image of the strictly anti-dominant cone in $X(P)$ under
$c:X(P)_{\mathbb{R}}\longrightarrow N^{1}(G/P)_{\mathbb{R}}$.

We now compute $\mathrm{Mov}^{i}(F/P)$.

######  Lemma 4.8.

Let $f\in N^{1}(F/P)$ be the class of a vertical fiber. Then every element in
$N^{1}(F/P)$ can be writen as $\mathcal{L}_{\lambda}-tf$.

###### Proof.

Apply [20, Theorem 18.32] to the $G$-bundle $F\longrightarrow C$. We get an
exact sequence

$\displaystyle
X(P)\longrightarrow\operatorname{Pic}(F/P)\longrightarrow\operatorname{Pic}(F)\longrightarrow\operatorname{Pic}(P).$

Apply it to the $P$-bundle $F\longrightarrow F/P$. We get another exact
sequence

$\displaystyle
X(G)\longrightarrow\operatorname{Pic}(C)\longrightarrow\operatorname{Pic}(F)\longrightarrow\operatorname{Pic}(G).$

Note that after tensoring $\mathbb{R}$,
$\operatorname{Pic}(P)_{\mathbb{R}}=\operatorname{Pic}(G)_{\mathbb{R}}=0$.

Consider the diagram

where $g$, $\phi$ and $\psi$ are just pullbacks. Note that $g$ and $\psi$ are
surjective. Let $x\in\operatorname{Pic}(F/P)$ such that $g(x)=\psi(a)$ for
some $a\in\operatorname{Pic}(C)_{\mathbb{R}}$. Then $g(x-\phi(a))=0$ and the
first line is exact at the middle, so $x-\phi(a)=f(b)$. ∎

By the above description, we define two functions
$\langle\alpha^{\vee},\cdot\rangle$ and
$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ on $N^{1}(F/P)_{\mathbb{R}}$ as

  1. (1)

$\langle\alpha^{\vee},\cdot\rangle$ sends $\mathcal{L}_{\lambda}-tf$ to
$\langle\alpha^{\vee},\lambda\rangle$.

  2. (2)

$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ sends $\mathcal{L}_{\lambda}-tf$
to $\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle-t$.

To see they are well-defined, it suffices to deal with the case
$\mathcal{L}_{\lambda}$ equals to a multiple of $f$. In this case, we see that
$M_{\lambda}$ is a trivial line bundle on $G/P$ by restricting
$\mathcal{L}_{\lambda}$ to a fiber. This implies $\lambda\in X(G)$ by the
exact sequence $0\longrightarrow X(G)\longrightarrow
X(P)\longrightarrow\operatorname{Pic}(G/P)$. Then
$\mathcal{L}_{\lambda}=F\times_{P}k_{\lambda}=\pi^{*}(F\times_{G}k_{\lambda})$
where $\pi$ is the structure map $F/P\longrightarrow C$. In particular,
$\mathcal{L}_{\lambda}=\langle{\mathrm{deg}}(F),\lambda\rangle f$ in
$N^{1}(F/P)_{\mathbb{R}}$.

  1. (1)

$\langle\alpha^{\vee},\cdot\rangle$ sends $f$ to zero and sends $\lambda\in
X(G)$ also to zero. This proves $\langle\alpha^{\vee},\cdot\rangle$ is well-
defined.

  2. (2)

$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ sends
$\langle{\mathrm{deg}}(F),\lambda\rangle f$ to
$\langle{\mathrm{deg}}(F),\lambda\rangle$ since $\lambda\in X(G)$ is invariant
under Weyl group action and sends $\lambda\in X(G)$ also to
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle=\langle{\mathrm{deg}}(F_{Q}),\lambda\rangle=\langle{\mathrm{deg}}(F),\lambda\rangle$
since
$F\times_{G}k_{\lambda}=F_{Q}\times_{Q}G\times_{G}k_{\lambda}=F_{Q}\times_{Q}k_{\lambda}$.
This proves $\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle$ is well-defined.

######  Theorem 4.9.

The $k$-th movable cone $\operatorname{Mov}^{k}(F/P)$ is the cone defined by

  1. (1)

$\langle\alpha^{\vee},\cdot\rangle<0$ for any
$\alpha\in\Delta\backslash\Delta_{P}$, and

  2. (2)

$\langle{\mathrm{deg}}(F_{Q}),w\cdot\rangle>0$ for any $w\in W/W_{P}$ with
$\ell(w)\geq n-k+1$.

###### Proof.

The sufficiency follows immediately after Theorem 4.6. For the necessity, let
$\mathcal{L}_{\lambda}-tf$ be a line bundle on $F/P$ such that
$\mathrm{codim}(\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf))\geq k$. Then in
particular $\mathcal{L}_{\lambda}-tf$ is big. This impies $M_{\lambda}$ is big
on $G/P$, which is equivalent to $\langle\alpha^{\vee},\lambda\rangle<0$ for
all $\alpha\in\Delta\backslash\Delta_{P}$. Then $\lambda$ is strictly anti-
dominant and Theorem 4.6 applies, saying
$\mathrm{B}_{+}(\mathcal{L}_{\lambda}-tf)=\mathcal{X}\backslash\coprod_{\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle-t>0}\mathcal{C}_{w}$.
It has $\operatorname{codim}\geq k$ $\Longleftrightarrow$
$\langle{\mathrm{deg}}(F_{Q}),w\lambda\rangle-t>0$ for all $w$ with
$\ell(w)\geq n-k+1$. ∎

######  Corollary 4.10.

$\mathcal{L}_{\lambda}-tf\in\operatorname{Mov}^{k}(F/P)$ $\Longleftrightarrow$
$e_{k}(h_{\mathcal{L}_{\lambda}})>t$ where $e_{k}$ is the $k$-th minimum of
Zhang.

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