The Gromov Width of Bott-Samelson Varieties

Abstract

We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.

1 Introduction

The Gromov width of a 2n-dimensional symplectic manifold is the largest capacity a for which a ball in $\mathbb {R}^{2n}$ of radius $\sqrt {a/\pi }$, equipped with the standard symplectic form, can be symplectically embedded in this manifold. By Darboux Theorem, this symplectic invariant is a positive number; it originates in the work of Gromov’s while proving his celebrated non-squeezing theorem [15]. There have been many works dedicated to the computation or estimates of the Gromov width of symplectic manifolds (see, e.g., [5, 9, 13, 21, 24, 34] and references therein).

In this paper, we compute the Gromov width of the Bott-Samelson varieties which are natural desingularizations of complex Schubert varieties. In particular, we extend the results on the Gromov width of rational coadjoint orbits of connected simple compact Lie groups obtained in [13]. Gromov widths are closely related to Seshadri constants of line bundles (see [35]). The latter have been busily investigated in algebraic geometry as a measure of local positivity (see, e.g., Chap.5 in [31]). From our computation of the Gromov width, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.

Our computation of the Gromov width is performed in two steps. First, by using Gromov’s J-holomorphic curves techniques applied to minimal curves, we prove that the Gromov width of the Bott-Samelson varieties we consider is bounded above by the symplectic areas of the minimal curves of these varieties (Theorem 1.1). Secondly, to obtain a lower bound, we apply embedding methods: symplectic embeddings of balls into symplectic toric manifolds can be derived from embeddings of simplices into the momentum images of the latter manifolds; this method can be carried out to more general projective manifolds via toric degenerations associated to Newton-Okounkov bodies (see [26, 36]). This approach has been already followed by several authors; see, e.g., [36] for some review. This method applied to concrete simplices included in Newton-Okounkov bodies unimodular to generalized string polytopes (as defined in [14]) enables us to get a lower bound for the Gromov width in case the Bott-Samelson varieties are equipped with a positive integral 2-form (Theorem 1.2). Finally, by making use of Brion–Kannan’s characterization [8] of the minimal curves of Bott-Samelson varieties that are birational to Schubert varieties, we show that the minimum symplectic area among such curves equals the size of one of the simplices alluded above. From this, we can conclude that the lower and upper bounds we obtain do coincide (Corollary 1.5).

To state our results more explicitly, let us set up some notation. Let G be a connected semisimple complex algebraic group of rank n. Fix a maximal torus T and a Borel subgroup B ⊂ G containing T. Let α₁,…,α_n and $\alpha _{1}^{\vee }, \ldots , \alpha _{n}^{\vee }$ be the corresponding simple roots and simple coroots respectively. The latter form a basis of the cocharacter lattice Ξ^∗(T) with dual basis consisting of the fundamental weights ϖ₁,…,ϖ_n. Let W be the Weyl group of (G,T) and s_i ∈ W be the reflection corresponding to the simple root α_i. By P_i we denote the minimal parabolic subgroup of G generated by B and any representative $\dot {s}_{i}$ of s_i.

Given a sequence of simple roots $\textbf {i}= (\alpha _{i_{1}}, \ldots ,\alpha _{i_{r}})$, the corresponding Bott-Samelson varietyZ_i is the quotient

$$ Z_{\textbf{i}}:= (P_{i_{1}}\times {\cdots} \times P_{i_{r}})/B^{r} $$

where B^r acts on $P_{i_{1}}\times {\cdots } \times P_{i_{r}}$ by

$$ (p_{1}, \ldots, p_{r})\cdot (b_{1}, \ldots, b_{r}):=(p_{1}b_{1}, b_{1}^{-1}p_{2}b_{2}, \ldots, b_{r-1}^{-1}p_{r}b_{r}) $$

for $p_{j}\in P_{i_{j}}$ and b_j ∈ B for all 1 ≤ j ≤ r.

Let $w=s_{i_{1}}s_{i_{2}}{\cdots } s_{i_{r}}$. Throughout this paper, we assume that this is a reduced decomposition of w ∈ W. In this case, the corresponding Bott-Samelson variety is a desingularization of the Schubert variety in G/B associated to w.

In Section 2, the reader can find some material on Bott-Samelson varieties and results of [8] on minimal curves.

In Section 3, we prove that the Gromov width w_G(Z_i,ω) of Z_i equipped with any Kähler 2-form ω is bounded from above by the symplectic area ${\int \limits }_{C} \omega$ of any minimal curve C of Z_i. The latter is denoted by ω([C]) below.

Theorem 1.1

Let $[\omega ]\in H^{2}(Z_{\mathbf {i}},\mathbb {R})$ be Kähler. Then

$$ w_{G}(Z_{\mathbf{i}}, \omega)\leq\min\{\omega([C]): C \text{ minimal curve of } Z_{\textbf{i}}\}, $$

In Section 4, we show the following theorem (and Corollary 1.3) giving a lower bound for the Gromov width.

Theorem 1.2

Let $\mathbf {m}=(m_{1}, \ldots , m_{r})\in \mathbb {Q}_{>0}^{r}$ and ω_m be the associated rational Kähler 2-form of the Bott-Samelson variety Z_i. Then

$$ \begin{array}{@{}rcl@{}} w_{G}(Z_{\mathbf{i}}, \omega_{\mathbf{m}}) &\geq& \min\big\{ m_{j}+\sum\limits_{j<\ell\leq r} m_{\ell}\langle \varpi_{i_{\ell}}, s_{i_{\ell}}{\cdots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle : 1\leq j\leq r \big\} \end{array} $$

(1.1)

$$ \begin{array}{@{}rcl@{}} &\geq& \min\{\omega_{\mathbf{m}}([C]): T\text{-stable curve } C \text{ of }Z_{\mathbf{i}} \text{ containing } z_{0}\} \end{array} $$

(1.2)

$$ \begin{array}{@{}rcl@{}} &\geq& \min\{\omega_{\mathbf{m}}([C]): C \text{ minimal curve of } Z_{\mathbf{i}}\}. \end{array} $$

(1.3)

Theorem 1.2 extends the results obtained by Fang-Littelmann-Pabiniak in [13] concerning the lower bound of the Gromov width of rational coadjoint orbits equipped with the Kirillov-Kostant-Souriau symplectic form. Namely, we recover

Corollary 1.3 (Fang-Littelmann-Pabiniak)

Let K be the connected compact Lie group such that $G=K^{\mathbb {C}}$. Given $\lambda \in {\Xi }^{*}(T)\otimes _{\mathbb {Z}}\mathbb {Q}$ in the Weyl chamber defined by B, let O_λ be the coadjoint orbit Kλ equipped with the Kirillov-Kostant-Souriau form. Then

$$ w_{G}(O_{\lambda})\geq\min \big\{\lambda(\alpha_{j}^{\vee}):\lambda(\alpha_{j}^{\vee})\neq 0, 1\leq j\leq n\big\}. $$

Remark 1.4

It is an open conjecture of Karshon and Tolman that the Gromov width of a coadjoint orbit of a connected compact simple Lie group is precisely the lower bound given in the above corollary; see [1, 20] for the state of the art on this conjecture.

As a straightforward consequence of Theorems 1.1 and 1.2, we get the following:

Corollary 1.5

Let $\mathbf {m}=(m_{1}, \ldots , m_{r})\in \mathbb {Q}_{>0}^{r}$ and ω_m be the associated rational Kähler 2-form of the Bott-Samelson variety Z_i. Then

$$ \begin{array}{@{}rcl@{}} w_{G}(Z_{\mathbf{i}}, \omega_{\mathbf{m}}) &=& \min\big\{ m_{j} + {\sum}_{j<\ell\leq r} m_{\ell}\langle \varpi_{i_{\ell}}, s_{i_{\ell}}{\cdots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle : 1\leq j\leq r \big\} \\ &= & \min \{\omega_{\mathbf{m}}([C]): C \text{ minimal curve of } Z_{\mathbf{i}}\}. \end{array} $$

As previously mentioned, one can estimate Seshadri constants by Gromov widths. Given a projective variety X together with an ample line bundle ${\mathscr{L}}$ on X, the Seshadri constant $\varepsilon (X,{\mathscr{L}},x)$ of ${\mathscr{L}}$ at some point x ∈ X is defined as the infimum of the ratio ${\mathscr{L}}\cdot C/\text {mult}_{x} C$ taken over all irreducible and reduced curves C on X passing through x. Here mult_xC stands for the multiplicity of C at x. As shown in [6, Proposition 2.6.1], in case of an ample line bundle ${\mathscr{L}}$, the Seshadri constant $\varepsilon (X,{\mathscr{L}},x)$ is upper bounded by the Gromov width of X equipped with the Fubini-Study form associated to ${\mathscr{L}}$. Together with Corollary 1.5, this yields¹

Corollary 1.6

Let $\mathbf {m}\in \mathbb {Z}^{\mathbf {r}}_{>\mathbf {0}}$. Then the following inequality holds for the Seshadri constants of the Bott-Samelson variety Z_i at any point x ∈ Z_i

$$ \varepsilon(Z_{\mathbf{i}},\omega_{\mathbf{m}},x)\leq \min \{\omega_{\mathbf{m}}([C]): C \text{ minimal curve of } Z_{\mathbf{i}}\}. $$

Remark 1.7

Lower bounds of Seshadri constants can be derived also by embedding methods (see, e.g., [22]). This will be addressed in a forthcoming work.

We conclude our work by further relating our results on Gromov widths to previous ones. In Section 5, we consider the polarized Bott-Samelson varieties which can be degenerated into polarized Bott (toric) manifolds. These toric degenerations (and their underlying combinatorics) were thoroughly studied in [16, 19, 37]. The Gromov widths of polarized generalized Bott manifolds are computed in [21]². These results combined altogether enable us to recover Corollary 1.5 in this very setting (Corollary 5.12); in particular, we get that the Gromov width of the Bott-Samelson varieties under consideration equals the symplectic area of a line. This method can not be applied further to any Bott-Samelson variety to recover fully Corollary 1.5 as Example 5.13 shows.

2 Bott-Samelson Varieties

For the sake of simplicity, we identify the set of simple roots of (G,B,T) with the set I = {1,2,…,n}. For w ∈ W, an expression i of w is a word (i₁,i₂,…,i_r) ∈ I^r such that $w=s_{i_{1}}{\cdots } s_{i_{r}}$. Recall that an expression i of w is called reduced whenever the number r is minimal.

Below we freely recall a few basic facts about Bott-Samelson varieties; for more details, see [12, 30].

Given a (not necessarily reduced) expression i = (i₁,…,i_r) of w ∈ W, recall the definition of the Bott-Samelson variety Z_i stated in the introduction. The variety Z_i is smooth, projective of dimension r. The left multiplication of $P_{i_{1}}$ on the first factor makes Z_i into a $P_{i_{1}}$-variety equipped with an $P_{i_{1}}$-equivariant morphism

$$ \pi_{r}: Z_{\textbf{i}}\longrightarrow G/B, \quad [(p_{1}, \ldots, p_{r})]\longmapsto p_{1}{\cdots} p_{r}B. $$

We can realize Z_i as an iterated $\mathbb {P}^{1}$-bundle. Specifically, let $w^{\prime }=s_{i_{1}}{\cdots } s_{i_{r-1}}$ and $\textbf {i}^{\boldsymbol {\prime }}=(i_{1}, \ldots , i_{r-1})$. Let $f: G/B \longrightarrow G/P_{i_{r}}$ be the map given by $gB \mapsto gP_{i_{r}}$, and let $p_{\textbf {i}^{\boldsymbol {\prime }}}: Z_{\textbf {i}^{\boldsymbol {\prime }}}\longrightarrow G/P_{i_{r}}$ be the map given by $[(p_{1}, \ldots , p_{r-1})]\mapsto p_{1}{\cdots } p_{r-1}P_{i_{r}}$. We have the following commutative diagram:

For 1 ≤ j ≤ r, let Z_i(j) denote the Bott-Samelson variety associated to the sub-expression (i₁,…,i_j) of i = (i₁,…,i_r).

Consider the natural morphisms

$$ \pi_{j}:Z_{\textbf{i}}(j) \longrightarrow G/B \quad \text{and}\quad f_{j}: Z_{\textbf{i}} \longrightarrow Z_{\textbf{i}}(j). $$

Explicitly, f_j maps [(p₁,…,p_r)] to [(p₁,…,p_j)].

Henceforth, we assume the expression i is reduced. Let $X(s_{i_{1}}{\cdots } s_{i_{j}})$ be the Schubert variety associated to the Wel group element $s_{i_{1}}{\cdots } s_{i_{j}}$. Then

$$ \pi_{j}(Z_{\textbf{i}}(j))=X(s_{i_{1}}{\cdots} s_{i_{j}}) $$

and Z_i(j) is a desingularization of $X(s_{i_{1}}{\cdots } s_{i_{j}})$.

The Bott-Samelson variety Z_i is equipped with a base point that is

$$ z_{0}=[(\dot{s}_{i_{1}}, \ldots, \dot{s}_{i_{r}})]. $$

Moreover, we have π_r(z₀) = wx₀ with x₀ being the base point of G/B and π_r yields an isomorphism between the B-orbits

$$ Bz_{0}\simeq Bwx_{0} $$

(2.1)

(see, e.g., [23, §II.13.6] for details). In particular, z₀ is fixed by T.

For 1 ≤ j ≤ r, the line bundle ${\mathscr{L}}_{j}$ on Z_i is defined as

$$ \mathcal{L}_{j}:=f_{j}^{*}\pi_{j}^{*}\mathcal{L}_{G/B}(\varpi_{i_{j}}), $$

where ${\mathscr{L}}_{G/B}(\varpi _{i_{j}})$ denotes the line bundle on G/B associated to $\varpi _{i_{j}}$. For $\textbf {m}=(m_{1}, \ldots , m_{r})\in \mathbb {Z}^{r}$, we set

$$ \mathcal{L}_{\textbf{i, m}}:=\mathcal{L}_{1}^{m_{1}}\otimes {\cdots} \otimes \mathcal{L}_{r}^{m_{r}}. $$

Theorem 2.1

[30, Theorem 3.1]

• (1) The isomorphism classes of ${\mathscr{L}}_{1}, \ldots , {\mathscr{L}}_{r}$ form a basis of Pic(Z_i). In particular, the map $\mathbb {Z}^{r}\longrightarrow \text {Pic}(Z_{\textbf {i}}), \mathbf {m} \mapsto {\mathscr{L}}_{\mathbf {i}, \mathbf {m}}$ is an isomorphism of groups.

• (2) The line bundle ${\mathscr{L}}_{\mathbf {i}, \mathbf {m}}$ is (very) ample if and only if m_j > 0 for all j.

• (3) The line bundle ${\mathscr{L}}_{\mathbf {i}, \mathbf {m}}$ is generated by its global sections if and only if m_j ≥ 0 for all j.

In the remainder of this section, we collect some results on the T-stable curves and the minimal rational curves on Bott-Samelson varieties from [8].

First, let us briefly recall the notion of rational curves on any projective variety X. For more details, we refer to [27, Chapter II.2].

Let RatCurves(X) denote the normalization of the space of rational curves on X. Every irreducible component $\mathcal {K}$ of RatCurves(X) is a (normal) quasi-projective variety equipped with a quasi-finite morphism to the Chow variety of X; the image consists of the Chow points of irreducible, generically reduced rational curves. There exists a universal family $p :\mathcal {U} \to \mathcal {K}$ and a projection $\mu :\mathcal {U} \to X$. For any x ∈ X, let $\mathcal {U}_{x}=\mu ^{-1}(x)$ and $\mathcal {K}_{x}=p(\mathcal {U}_{x}$). If $\mathcal {K}_{x}$ is non-empty and projective for a general point x ∈ X then $\mathcal {K}$ is called a family of minimal rational curves and any member of $\mathcal {K}$ is called a minimal rational curve.

We now state the main properties and the characterization of minimal curves in case of Bott-Samelson varieties associated to a reduced expression.

Given a reduced word i = (i₁,…,i_r) of w ∈ W, let

$$ \begin{array}{@{}rcl@{}} \beta_{1}:=\alpha_{i_{1}},\quad \beta_{2}:=s_{i_{1}}(\alpha_{i_{2}}), \ldots, \quad \beta_{r}:=s_{i_{1}}{\cdots} s_{i_{r-1}}(\alpha_{i_{r}}). \end{array} $$

(2.2)

Note that

$$ \begin{array}{@{}rcl@{}} R^{+}\cap w(R^{-})=\{\beta_{1}, \ldots, \beta_{r}\} \end{array} $$

(2.3)

where R⁺ and R⁻ denote the sets of positive and negative roots corresponding to B, respectively.

For any 1 ≤ j ≤ r, consider the following orbit closure in Z_i

$$ C_{j}=\overline{U_{\beta_{j}}z_{0}} $$

with $U_{\beta _{j}}$ being the root subgroup of G corresponding to β_j.

Proposition 2.2

[8, Lemma 4.1] Assume Z_i is a Bott-Samelson variety associated to a reduced expression i.

• (1) The T-stable curves in Z_i through z₀ are exactly the curves C_j.

• (2) Every curve C_j is isomorphic to the projective line $\mathbb {P}^{1}$.

• (3) For 1 ≤ j ≤ r and 1 ≤ k ≤ r, we have

  $$ \mathcal{L}_{k}\cdot C_{j}=\begin{cases} 0 & ~if~ j>k\\ \langle \varpi_{i_{k}}, s_{i_{k}}{\cdots} s_{i_{j+1}} (\alpha_{i_{j}}^{\vee}) \rangle & ~if~j\leq k \end{cases}~. $$

• (4) Let $K_{Z_{\mathbf {i}}}$ be the canonical line bundle on Z_i. Then we have

  $$ -K_{Z_{\mathbf{i}}}\cdot C_{j}= \sum\limits_{i=1}^{n} \langle\varpi_{i}, s_{i_{r}}{\cdots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle+1. $$

The following result gives a description of the minimal rational curves in Z_i.

Theorem 2.3

[8, Theorem 4.3] Assume Z_i is a Bott-Samelson variety associated to a reduced expression i.

• (1) Every minimal family $\mathcal {K}$ on Z_i satisfies $\mathcal {K}_{z_{0}}=\{C_{j}\}$ for some 1 ≤ j ≤ r.

• (2) The minimal rational curves in Z_i through z₀ are exactly those C_j such that $s_{i_{r}}{\cdots } s_{i_{j+1}}(\alpha _{i_{j}})$ is a simple root.

Corollary 2.4

Let $(Z_{\mathbf {i}},{\mathscr{L}}_{\mathbf {i}, \mathbf {m}})$ be a Bott-Samelson variety equipped with an ample line bundle. Assume i is a reduced expression. Then

$$ \begin{array}{@{}rcl@{}} &&\min\{\mathcal{L}_{\mathbf{i},\mathbf{m}} \cdot C: C \text{ minimal curve} \}\\ &=& \min\{\mathcal{L}_{\mathbf{i}, \mathbf{m}}\cdot C: T\text{-stable curve } C\ni z_{0}\}\\ &=& \min \{m_{j} + \sum\limits_{j<k\leq r}m_{k}\langle\varpi_{i_{k}},s_{i_{k}}{\ldots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle: 1\leq j\leq r\}. \end{array} $$

Proof

Note that the curves containing the base point z₀ are dominant and recall that the minimal curves of Z_i containing z₀ are all T-stable (Theorem 2.3(1)). Besides, the degrees of all curves in the same minimal family w.r.t. a given polarization are equal and thanks to [27, Theorem IV.2.4], we know that a family of rational curves of minimal degree (among dominant curves) w.r.t. a given polarization is minimal. All this yields the first equality. The second equality follows easily from Proposition 2.2(3). □

3 Upper Bound

Throughout this section, we assume that i is a reduced word and we let X be the Bott-Samelson variety Z_i. We give an upper bound for the Gromov width of X equipped with a Kähler form $\omega \in H^{2}(X,\mathbb {R})$ by using Gromov-Witten invariants.

We thus start by recalling some basics on Gromov-Witten invariants.

Given $A\in H_{2}(X,\mathbb {Z})$, consider the moduli space $\overline {{\mathscr{M}}^{X}_{0,k}}(A)$ of stable maps of genus 0 into X of class A and with k marked points. This space carries a virtual fundamental class $[\overline {{\mathscr{M}}]}^{\text {vir}}$ in the rational Čech homology group $H_{d}(\overline {{\mathscr{M}}_{0,k}}(A),\mathbb {Q})$ where d denotes the expected dimension of $\overline {{\mathscr{M}}^{X}_{0,k}}(A)$, that is

$$ d={\dim}X+c_{1}(A)+k-3 $$

where c₁ denotes the first Chern class of the tangent bundle T_X of X.

Proposition 3.1

Let A be the class of a T-stable curve of X through the generic point z₀ of X. Then the moduli space $\overline {{\mathscr{M}}^{X}_{0,k}}(A)$ is smooth and has the expected dimension. Moreover,

$$ {[\overline{\mathcal{M}^{X}_{0,k}}(A)]} = [\overline{\mathcal{M}]}^{\text{vir}}. $$

Proof

For any T-stable curve C of X going through z₀, we have: $H^{1}(C, {T_{X}}_{|_{C}})=0$ by [8, Lemma 2.5(i)]. Therefore, the moduli space $\overline {{\mathscr{M}}^{X}_{0,k}}(A)$ is unobstructed. The proposition follows; see Section 2 in [33]. □

Let

$$ ev^{k}:\overline{\mathcal{M}^{X}_{0,k}}(A)\longrightarrow X^{k} $$

be the evaluation map sending a stable map to the k-tuple of its values at the k marked points.

Corollary 3.2

Let A be the class of a minimal curve of X through the generic point z₀ of X. Then the evaluation map $ev^{1}:\overline {{\mathscr{M}}^{X}_{0,1}}(A)\rightarrow X$ is an isomorphism.

Proof

By Proposition 2.3, A = [C_j] for some j such that $s_{i_{r}}{\cdots } s_{i_{j+1}}(\alpha _{i_{j}})$ is a simple root. For such a j, we have in particular, that $s_{i_{1}}...s_{i_{j-1}}s_{i_{j+1}}...s_{i_{r}}$ is reduced hence the natural morphism $X\rightarrow Z_{(i_{1},....i_{j-1},i_{j+1},...i_{r})}$ is a $\mathbb {P}^{1}$-fibration with fiber over $[\dot {s}_{i_{1}},\ldots ,\dot {s}_{i_{j-1}},\dot {s}_{i_{j+1}},\ldots ,\dot {s}_{i_{}j-1}]$ the curve C_j itself. It follows that the evaluation map ev¹ is bijective. Thanks to Proposition 3.1, we can conclude the proof. □

For $\alpha _{i}\in H^{*}(X,\mathbb {Q})$ with i = 1,...,k, the Gromov-Witten invariant is defined to be the rational number

$$ GW^{X}_{A,k}(\alpha_{1},...,\alpha_{k}):={\int}_{[\overline{\mathcal{M}}]^{\text{vir}}} (ev^{k})^{*}(\alpha_{1}\times ...\times\alpha_{k}), $$

whenever the degrees of α₁,…,α_k sum up to the expected dimension d; otherwise it is 0.

Proposition 3.3

Let A be the class of a minimal curve C of X through z₀. Then

$$ GW^{X}_{A,1}(\text{PD}[pt])\neq 0. $$

Proof

Note that c₁(A) = 2 by Proposition 2.2(4). It follows that PD[pt] satisfies the codimension condition that is, its degree equals the expected dimension which is $d={\dim }X+2-2$. By Proposition 3.1 along with Corollary 3.2, we have the equality

$$ GW^{X}_{A,1}(\text{PD}[pt])={\int}_{[X]} \text{PD}[pt]. $$

The right hand side being obviously non-equal to 0, the proposition follows. □

The variety X being projective and smooth and ω being Kähler by assumption, the moduli space $\overline {{\mathscr{M}}^{X}_{0,k}}(A)$ is homeomorphic to the moduli space of stable J-holomorphic maps of genus 0 into X of class A and with k-marked points. Here J stands for the complex structure of X. Moreover, the algebraic and symplectic virtual fundamental classes as constructed in [4] and [38] resp. coincide; see [38]. As a consequence, Theorem 4.1 in [21] (for peculiar cocycles) reads as follows.

Theorem 3.4 (Gromov)

Let (X,ω) be Kähler and $A \in H_{2}(X, \mathbb {Z})\setminus \{0\}$ be a second homology class. If $GW^{X}_{A, k}(\text {PD}[pt], \alpha _{2}, \ldots , \alpha _{k})\neq 0$ for some k and $\alpha _{i}\in H^{*}(X, \mathbb {Q})$ (i = 2,…,k), then the inequality w_G(X,ω) ≤ ω(A) holds.

Corollary 3.5

Let $[\omega ]\in H^{2}(Z_{\mathbf {i}},\mathbb {R})$ be a Kähler form and A be the class of a minimal curve of Z_i. If the expression i is reduced then

$$ w_{G}(Z_{\mathbf{i}}, \omega)\leq \omega(A). $$

Proof

The corollary follows readily from Theorem 3.4 and Proposition 3.3 whenever A = [C_j] for some minimal curve C_j of Z_i. Note that ω(A) = ω([C]) for every curve C in the minimal family containing C_j. This along with Theorem 2.3 yields the inequality for any minimal curve. □

4 Lower Bound

In this section, we prove Theorem 1.2: we give a lower bound for the Gromov width. The bound we obtain is derived from a result of Kaveh’s involving Newton-Okounkov bodies.

Newton-Okounkov bodies of projective algebraic varieties are convex bodies generalizing momentum polytopes of symplectic toric manifolds; they were introduced about the same time in [25] and [32]. In the literature, one can find several non-equivalent definitions of Newton-Okkounkov bodies for Bott-Samelson varieties. Here, we are concerned with one construction which parallels one of Fujita’s definitions.

4.1 Definition of the Newton-Okounkov body

Let us thus start by defining the Newton-Okounkov body of interest to us.

Fix a reduced expression i = (i₁,…,i_r) of a given w ∈ W. Recall the definition of the roots β_i given in (2.2) and that $U_{\beta _{i}}$ denotes the root subgroup of G corresponding to β_i.

We shall regard $U_{\beta _{1}}\times {\cdots } \times U_{\beta _{r}}$ as an affine (open) neighborhood of the base point z₀ of Z_i via the natural isomorphisms

$$ \begin{array}{@{}rcl@{}} U_{\beta_{1}} \times {\cdots} \times U_{\beta_{r}} \simeq Bwx_{0}\simeq Bz_{0}. \end{array} $$

(4.1)

The first isomorphism is well-known; it follows mainly from the characterization of the roots β_i recalled in (2.3) (see, e.g., [23, §II.13.3] for details). The second one is the isomorphism (2.1); recall that x₀ denotes the base point of G/B.

We further identify the function field $\mathbb {C} (Z_{\textbf {i}}) = \mathbb {C}(U_{\beta _{1}}\times {\cdots } \times U_{\beta _{r}})$ with the rational function field $\mathbb {C}(t_{1}, \ldots , t_{r})$ via the isomorphism of algebraic varieties $\mathbb {C}^{r} \simeq U_{\beta _{1}}\times {\cdots } \times U_{\beta _{r}}$ given by

$$ (t_{1},\ldots,t_{r}) \longmapsto (\exp(t_{1} E_{\beta_{1}}),\dots, \exp(t_{r} E_{\beta_{r}})) $$

where $E_{\beta _{i}}$ denotes the root vector associated to the (positive) root β_i.

The lexicographic order ≤ on $\mathbb {Z}^{r}$ induces a total order on the set of monomials in the variables t₁,…,t_r (also denoted by ≤) by setting

$$ t_{1}^{a_{1}}{\cdots} t_{r}^{a_{r}} \leq t_{1}^{a^{\prime}_{1}}{\cdots} t_{r}^{a^{\prime}_{r}} \text{ if and only if } (a_{1}, \ldots, a_{r})\leq (a^{\prime}_{1}, \ldots, a^{\prime}_{r}) \text{ in } \mathbb{Z}^{r}. $$

Given $f(t_{1}, \ldots , t_{r}) = {\sum }_{j=(j_{1}, \ldots , j_{r})}c_{j}t_{1}^{j_{1}}{\cdots } t_{r}^{j_{r}}\in \mathbb {C}[t_{1}, \ldots , t_{r}]$, let (k₁,…,k_r) be the maximum tuple among the tuples (j₁,…,j_r) such that c_j≠ 0. Define

$$ v_{\beta}(f)=-(k_{1}, \ldots, k_{r}). $$

This induces the following map

$$ v_{\beta}\!:\! \mathbb{C}(t_{1}, \ldots, t_{r})\setminus \{0\}\!\longrightarrow\! \mathbb{Z}^{r},\! \quad v_{\beta} \!\left( \!\frac{f}{g}\!\right) \!\!:= v_{\beta}(f)- v_{\beta}(g) \text{ for\! } f, g\!\in\! \mathbb{C}[t_{1}, \ldots, t_{r}]\setminus \{0\}. $$

Given an ample line bundle ${\mathscr{L}}$ on Z_i and a non-zero section $\tau \in H^{0}(Z_{\textbf {i}}, {\mathscr{L}})$, we shall regard $H^{0}(Z_{\textbf {i}}, {\mathscr{L}}^{\otimes k})$ as a complex vector subspace of the function field $\mathbb {C}(Z_{\textbf {i}})$ via the map

$$ H^{0}(Z_{\textbf{i}}, \mathcal{L}^{\otimes k}) \longrightarrow \mathbb{C}(Z_{\textbf{i}}), \quad \sigma \longmapsto \frac{\sigma}{\tau^{k}}. $$

As in the theory of Newton-Okounkov bodies, let us now consider the following set

$$ S(Z_{\textbf{i}},\mathcal{L}, v_{\beta}, \tau) := \underset{k>0}{{\bigcup}} \left\{\left( k, v_{\beta}\left( \frac{\sigma}{\tau^{k}}\right)\right): \sigma \in H^{0}(Z_{\textbf{i}}, \mathcal{L}^{\otimes k})\setminus \{0\}\right\} \subset \mathbb{Z}_{\geq 0}\times \mathbb{Z}^{r}. $$

Note that $S(Z_{\textbf {i}},{\mathscr{L}}, v_{\beta }, \tau )$ is a semigroup since v_β is a valuation, which can be easily checked. Let $\mathcal C(Z_{\textbf {i}},{\mathscr{L}}, v_{\beta }, \tau )$ be the real closed convex cone generated by $S(Z_{\textbf {i}},{\mathscr{L}}, v_{\beta }, \tau )$. The Newton-Okounkov body associated to ${\mathscr{L}}$, τ and v_β is defined as

$$ {\Delta}(Z_{\textbf{i}},\mathcal{L}, v_{\beta}, \tau)=\left\{x: (1,x)\in \mathcal C(Z_{\textbf{i}},\mathcal{L}, v_{\beta}, \tau)\right\}. $$

4.2 Properties of the Newton-Okounkov body

In this paper, we consider the Newton-Okounkov body associated to a particular section that we now introduce.

Given a dominant weight λ (w.r.t. B and T), let V (λ) denote the simple G-module with highest weight λ and v_λ ∈ V (λ) be a highest weight vector. For our purpose, we consider the morphism associated to the (very) ample line bundle ${\mathscr{L}}={\mathscr{L}}_{\textbf {i,m}}$

$$ \begin{array}{llll} {\Psi}_{\textbf{i,m}}: & Z_{\textbf{i}} & \longrightarrow & \mathbb{P}(V(m_{1}\varpi_{i_{1}})\otimes\ldots\otimes V(m_{r}\varpi_{i_{r}}))\\ & \left[(p_{1},\ldots,p_{r})\right] & \longmapsto & [p_{1} v_{m_{1}\varpi_{i_{1}}}\otimes\ldots\otimes p_{1}p_{2}{\ldots} p_{r} v_{m_{r}\varpi_{i_{r}}}]. \end{array} $$

Note that the image of the base point z₀ ∈ Z_i through this morphism is [v₀] with

$$ \begin{array}{@{}rcl@{}} v_{0}=s_{i_{1}}v_{m_{1}\varpi_{i_{1}}}\otimes s_{i_{1}}s_{i_{2}}v_{m_{2}\varpi_{i_{2}}}\otimes\ldots\otimes wv_{m_{r}\varpi_{i_{r}}}. \end{array} $$

(4.2)

Moreover, the morphism Ψ_i,m induces an isomorphism of $P_{i_{1}}$-modules

$$ {\Phi}_{\textbf{i,m}}:V_{\textbf{i,m}}^{*}\longrightarrow H^{0}(Z_{\textbf{i}},\mathcal{L}_{\textbf{i,m}}) $$

(4.3)

where V_i,m denotes the so-called generalized Demazure module. As a complex vector space, V_i,m is generated by the following vectors

$$ F_{i_{1}}^{a_{1}}(v_{m_{1}\varpi_{i_{1}}}\otimes F_{i_{2}}^{a_{2}}(v_{m_{2}\varpi_{i_{2}}}\otimes\ldots\otimes F_{i_{r-1}}^{a_{r-1}}(v_{m_{r-1}\varpi_{i_{r-1}}}\otimes F_{i_{r}}^{a_{r}}v_{m_{r}\varpi_{i_{r}}})\ldots)) $$

with $a_{j}\in \mathbb {N}$ and $F_{i_{j}}$ being the root vector associated to the root $-\alpha _{i_{j}}$; see Theorem 6 and Section 4.2 in [28].

We set

$$ f_{0}=s_{i_{1}}v_{m_{1}\varpi_{i_{1}}}^{*}\otimes s_{i_{1}}s_{i_{2}}v_{m_{2}\varpi_{i_{2}}}^{*}\otimes\ldots\otimes wv_{m_{r}\varpi_{i_{r}}}^{*}\quad\text{ and } \quad \varphi_{0} = {\Phi}_{\textbf{i,m}}(f_{0}). $$

Lemma 4.1

The section $\varphi _{0}\in H^{0}(Z_{\mathbf {i}}, {\mathscr{L}}_{\mathbf {i,m}})$ does not vanish on the open subset Bz₀ of Z_i.

Proof

Note that f₀ does not vanish on $U_{\beta _{1}}\times \ldots \times U_{\beta _{r}}[v_{0}]$, by a simple consideration on weights. Thanks to (4.3) and the isomorphism $Bz_{0}\simeq U_{\beta _{1}}\times {\cdots } \times U_{\beta _{r}}$ recalled in (4.1), the result follows. □

Lemma 4.1 enables us to introduce the Newton-Okounkov body associated to φ₀, that is

$$ \begin{array}{@{}rcl@{}} {\Delta}_{\textbf{i,m}}:={\Delta}(Z_{\textbf{i}},\mathcal{L}_{\textbf{i,m}}, v_{\beta}, \varphi_{0}). \end{array} $$

(4.4)

Remark 4.2

• (1) By considering the open embedding $s_{i_{1}}U^{-}_{\alpha _{i_{1}}}\times \ldots \times s_{i_{r}} U^{-}_{\alpha _{i_{r}}}\hookrightarrow Z_{\textbf {i}}$ given by $(s_{i_{1}}u_{1},\ldots , s_{i_{r}}u_{r})\mapsto [(s_{i_{1}}u_{1},\ldots , s_{i_{r}} u_{r})]$, one naturally identifies $\mathbb {C}(Z_{\textbf {i}})$ with $\mathbb {C}(t^{\prime }_{1},\ldots , t^{\prime }_{r})$. In [14, §8], Fujita introduces the valuation $v^{\prime }_{\textbf {i}}$ on $\mathbb {C}(t^{\prime }_{1},\ldots , t^{\prime }_{r})$ defined up to the lexicographic order on $\mathbb {Z}^{r}$ as above and studies the Newton-Okounkov body ${\Delta }(Z_{\textbf {i}},{\mathscr{L}},v^{\prime }_{\textbf {i}},\varphi _{0})$.

• (2) By noticing that $\exp (a_{1} E_{\beta _{1}})\cdots \exp (a_{r} E_{\beta _{r}})wx_{0}= s_{i_{1}}\exp (a_{1} F_{i_{1}})\cdots$ $s_{i_{r}}\exp (a_{r} F_{i_{r}})x_{0}$ for any $(a_{1},\ldots ,a_{r})\in \mathbb {C}^{r}$, one easily sees that the semigroups $S(Z_{\textbf {i}},{\mathscr{L}},v^{\prime }_{\textbf {i}},\varphi _{0})$ and $S(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i},\textbf {m}}, v_{\beta }, \varphi _{0})$ coincide and so do the convex bodies ${\Delta }(Z_{\textbf {i}},{\mathscr{L}},v^{\prime }_{\textbf {i}},\varphi _{0})$ and Δ_i,m.

Thanks to Remark 4.2(2), [14, Cor. 8.3] reads as follows.

Theorem 4.3 (Fujita)

• (1) The semigroup $S(Z_{\mathbf {i}},{\mathscr{L}}_{\mathbf {i},\mathbf {m}}, v_{\beta }, \varphi _{0})$ is finitely generated.

• (2) The Newton-Okounkov body Δ_i,m is a convex polytope.

4.3 Embedding method

We are now ready to state Kaveh’s result which is a consequence of the following theorem.

Theorem 4.4

[29, Section 4.2] Let (X,ω) be a proper connected symplectic toric 2n-dimensional manifold equipped with a momentum map. Suppose there exists a n-dimensional simplex of size κ contained in the momentum image. Then the Gromov width of (X,ω) is at least κ.

Here a simplex in $\mathbb {R}^{m}$ is said to be of size κ if it can be obtained from the simplex $\{(x_{i})_{i}\in \mathbb {R}_{>0}^{m}: x_{1}+\ldots + x_{m}<\kappa \}$ by a linear transformation in $\text {GL}(m, \mathbb {Z})$ and a translation of $\mathbb {R}^{m}$.

Corollary 11.4 in [26] specialized to the case of Bott-Samelson varieties and the Newton-Okounkov bodies Δ_i,m reads as follows.

Corollary 4.5 (Kaveh)

Let $(Z_{\mathbf {i}},{\mathscr{L}}_{\mathbf {i,m}})$ be a Bott-Samelson variety equipped with a very ample line bundle. Then the Gromov width of $(Z_{\mathbf {i}}, {\mathscr{L}}_{\mathbf {i,m}})$ is bounded from below by the supremum of the size of (open) simplices that fit in the interior of Newton-Okounkov body Δ_i,m.

Proof

As a sake of convenience, we outline Kaveh’s proof in our particular setting.

Since the monoid $S(Z_{\mathbf {i},\mathbf {m}},{\mathscr{L}}_{\mathbf {i},\mathbf {m}}, v_{\beta }, \varphi _{0})$ is finitely generated (Theorem 4.3), the variety Z_i admits a flat degeneration to the projective (non necessarily normal) toric variety $X_{0}=\text {Proj} \mathbb {C}[S(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i},\textbf {m}}, v_{\beta }, \varphi _{0})]$ thanks to [2, Theorem 1]. The normalization of X₀ is the projective (normal) toric associated to the polytope Δ_i,m. Moreover, by Theorem A along with Theorem B in [17], there exists a Kähler form ω₀ on the smooth locus U₀ of X₀ such that

• (1) (U₀,ω₀) is symplectomorphic to $(U,\omega _{{\mathscr{L}}_{\textbf {i,m}}})$ for some open subset U of Z_i and

• (2) the momentum image of the symplectic toric manifold (U₀,ω₀) contains the interior of the Newton-Okounkov body Δ_i,m.

The corollary thus follows from Theorem 4.4. □

Remark 4.6

Kaveh’s result does not require that the Newton-Okounkov body be a convex polytope, but the proof becomes longer.

4.4 Proofs of Theorem 1.2 and Corollary 1.3

We now proceed to the proof of Theorem 1.2: we shall exhibit a simplex of the advertized size in the Newton-Okounkov body Δ_i,m and apply Corollary 4.5.

Given $\textbf {m}\in \mathbb {Z}_{>0}^{r}$, for each 1 ≤ j ≤ r, we set

$$ \lambda_{j}=m_{j}\omega_{i_{j}}\quad\text{ and }\quad \ell_{j}=m_{j}+\sum\limits_{j<\ell\leq r} \langle \lambda_{\ell}, s_{i_{\ell}}{\cdots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle. $$

Lemma 4.7

The roots $s_{i_{k+1}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ and $s_{i_{\ell }}{\ldots } s_{i_{j+1}}(\alpha _{i_{j}})$ are positive for every 1 ≤ k < j ≤ r and every j < ℓ ≤ r respectively.

Proof

We show the assertion for the roots $s_{i_{k+1}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$, the proof being similar for the other roots. Let us fix j and proceed by induction on k. We thus start by showing that $s_{i_{2}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ is a positive root for all j ≥ 2, the case k = 2. Because $s_{i_{1}}s_{i_{2}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})=\beta _{j}$ is a positive root, if $s_{i_{2}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ were a negative root, it would be equal to $-\alpha _{i_{1}}$. This would imply that β_j = β₁ – a contradiction since the roots β_j are pairwise distinct and j≠ 1. Next, we consider $s_{i_{\ell }}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ with 2 ≤ ℓ ≤ j − 1 < r. By induction hypothesis, $s_{i_{\ell -1}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ is a positive root hence by arguing similarly as for the case k = 2, we get that $s_{i_{\ell }}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})$ is also a positive root; otherwise it would be equal to $-\alpha _{i_{\ell -1}}$ and in turn we would have: $\beta _{j}=s_{i_{1}}{\ldots } s_{i_{\ell }}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})=s_{i_{1}}{\ldots } s_{i_{\ell -2}}(\alpha _{i_{\ell -1}})=\beta _{\ell -1}$ – a contradiction. □

Recall the definition of the linear form f₀ ∈ V (λ₁)^∗⊗… ⊗ V (λ_r)^∗ as well as the morphism Φ_i,m introduced in Section 4.2. Set

$$ f_{j}=F_{\beta_{j}}^{\ell_{j}}f_{0} \quad\text{ and }\quad \varphi_{j}={\Phi}_{\textbf{i}}(f_{j}) $$

with $F_{\beta _{j}}=E_{-\beta _{j}}$ being the root operator associated to the negative root − β_j. Note that ℓ_j is positive by Lemma 4.7.

Lemma 4.8

For each j, we have $E_{\beta _{j}}(s_{i_{1}}{\ldots } s_{i_{k}}v_{\lambda _{k}})=0$ for every 1 ≤ k < j ≤ r. In particular, we have the equality

$$ f_{j}=s_{i_{1}}v_{\lambda_{1}}^{*}\otimes\ldots\otimes F_{\beta_{j}}^{\ell_{j}}\left( s_{i_{1}}{\ldots} s_{i_{j}}v_{\lambda_{j}}^{*}\otimes\ldots\otimes wv_{\lambda_{r}}^{*}\right). $$

(4.5)

Proof

Lemma 4.7 implies that $\langle s_{i_{1}}{\ldots } s_{i_{k}}\lambda _{k},\beta _{j}^{\vee }\rangle =\langle \lambda _{k},s_{i_{k+1}}{\ldots } s_{i_{j-1}}(\alpha _{i_{j}})^{\vee }\rangle \geq 0$ and, in turn $E_{\beta _{j}}(s_{i_{1}}{\cdots } s_{i_{k}}v_{\lambda _{k}}) =0$ for all j≠ 1 because $s_{i_{1}}{\cdots } s_{i_{k}}v_{\lambda _{k}}$ is an extremal weight vector. By duality, this proves the lemma. □

Recall the definition of the weight vector v₀ given in (4.2).

Lemma 4.9

The equality $E^{\ell _{j}+1}_{\beta _{j}}(v_{0})=0$ holds for every 1 ≤ j ≤ r. Moreover, $E^{\ell _{j}}_{\beta _{j}}(v_{0})\neq 0$.

Proof

Note that we have

$$ a_{j\ell}:=\langle s_{i_{1}}{\ldots} s_{i_{\ell}}\lambda_{\ell},\beta_{j}^{\vee}\rangle= \begin{cases}-m_{j} & \text{if}~j=\ell \\ -\langle \lambda_{\ell}, s_{i_{\ell}}{\cdots} s_{i_{j+1}}(\alpha_{i_{j}}^{\vee})\rangle & \text{if}~ j<\ell\leq r \end{cases}. $$

By Lemma 4.7, the integers a_jℓ are negative and since $s_{i_{1}}{\cdots } s_{i_{\ell }}v_{\lambda _{\ell }}$ is an extremal weight vector, $E_{\beta _{j}}^{1-a_{j\ell }}(s_{i_{1}}{\cdots } s_{i_{\ell }}v_{\lambda _{\ell }}) =0$ for all j ≤ ℓ ≤ r. Moreover, by definition of ℓ_j, the integers − a_jℓ, for ℓ = j,…,r, sum up to ℓ_j. All this implies the equality $E^{\ell _{j}+1}_{\beta _{j}}(s_{i_{1}}{\ldots } s_{i_{j}}v_{\lambda _{j}}\otimes \ldots \otimes wv_{\lambda _{r}})=0$. We conclude the proof by applying Lemma 4.8. □

Lemma 4.10

The sections $\varphi _{j}\in H^{0}(Z_{\mathbf {i}},{\mathscr{L}}_{\mathbf {i,m}})$ are not trivial.

Proof

Let $u_{j}=\exp (E_{\beta _{j}})\in U_{\beta _{j}}$. By Lemma 4.9 together with Lemma 4.8, we have

$$ \begin{array}{@{}rcl@{}} u_{j} (v_{0}) &=& E_{\beta_{j}}^{0} (v_{0}) + {\ldots} + \frac{E_{\beta_{j}}^{k}}{k!}(v_{0})+\ldots+ \frac{E_{\beta_{j}}^{\ell_{j}}}{\ell_{j}!}(v_{0})\\ & = & v_{0} + {\ldots} + s_{i_{1}}v_{\lambda_{1}}\otimes\ldots\otimes s_{i_{1}}{\ldots} s_{i_{j-1}}v_{\lambda_{j-1}}\\&\otimes& \frac{E_{\beta_{j}}^{k}}{k!}\left( s_{i_{1}}{\ldots} s_{i_{j}}v_{\lambda_{j}}\otimes\ldots\otimes wv_{\lambda_{r}}\right)+\ldots. \end{array} $$

Besides, Equality (4.5) implies $f_{j}(E_{\beta _{j}}^{k}(v_{0}))=0$ for all k < ℓ_j. Moreover, $f_{j}(E_{\beta _{j}}^{\ell _{j}}(v_{0}))\neq 0$ since $E_{\beta _{j}}^{\ell _{j}}(v_{0})\neq 0$ (Lemma 4.9). Therefore, we have f_j(u_j(v₀))≠ 0 and the lemma follows. □

Proposition 4.11

For each 1 ≤ j ≤ r, we have $\varphi _{j}/\varphi _{0}=a_{j} t_{j}^{\ell _{j}}\in \mathbb {C}[t_{1},\ldots , t_{r}]$ for some $a_{j}\in \mathbb {C}$.

Proof

Take a section $\varphi \in H^{0}(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i,m}})$. Thanks to Lemma 4.1, the quotient φ/φ₀ is a regular function on Bz₀ and in turn can be regarded as an element of $\mathbb {C}[t_{1},\ldots ,t_{r}]$.

By arguing as in the proof of Lemma 4.10 and using the fact that the roots β_j are pairwise distinct, we show that $f_{j}(U_{\beta _{k}}v_{0})=0$ for all k≠j. It follows that $\varphi _{j}/\varphi _{0}\in \mathbb {C}[t_{j}]$. To conclude the proof, we observe that in the course of the proof of Lemma 4.10, we got that $f_{j}(u_{\beta _{j}}v_{0})=f_{j}((aE_{\beta _{j}})^{\ell _{j}}v_{0}/\ell _{j} !)$ with $u_{\beta _{j}}=\exp (a E_{\beta _{j}})$ and $a\in \mathbb {C}$. □

Corollary 4.12

The polytope Δ_{i, m} contains a simplex of size

$$ \kappa=\min\{ \ell_{j}: 1\leq j\leq r\}. $$

Proof

First, note that the polytope Δ_{i, m} contains the origin since v_β(φ₀) = 0.

From the definition of the valuation v_β and Proposition 4.11, we get

$$ v_{{\beta}}(\varphi_{j})=-(0,\ldots,0,\ell_{j},0,\ldots,0)=-\ell_{j} e_{j}. $$

This together with the convexity of Δ_i,m (Theorem 4.3) implies the corollary. □

Proof Proof of Theorem 1.2

We first consider the case of an integral Kähler form ω of Z_i, that is ω is the pullback of the Fubini-Study form on the projectivization of $H^{0}(Z_{\textbf {i}},{\mathscr{L}})$ for some very ample line bundle ${\mathscr{L}}$ of Z_i. By Theorem 2.1, the line bundle ${\mathscr{L}}$ equals ${\mathscr{L}}_{\textbf {i, m}}$ for some $\textbf {m}\in \mathbb {Z}_{>0}^{r}$. We thus write ω = ω_m.

Corollaries 4.5 and 4.12 give the inequality:

$$ w_{G}(Z_{\textbf{i}}, \omega_{\textbf{m}})\geq \min\big\{ \ell_{j}: 1\leq j\leq r\big\}. $$

(4.6)

We next consider any 2-form $\omega _{\textbf {m}^{\boldsymbol {\prime }}}$ of Z_i with $\textbf {m}^{\boldsymbol {\prime }}\in \mathbb {Q}_{>0}^{r}$, namely $\omega _{\textbf {m}^{\boldsymbol {\prime }}}$ is the 2-form associated to ${\mathscr{L}}_{\textbf {m}^{\boldsymbol {\prime }}}\in \text {Pic}(Z_{\textbf {i}})\otimes _{\mathbb {Z}} \mathbb {Q}$. Let $a\in \mathbb {Z}_{>0}$ be such that $a\omega _{\textbf {m}^{\boldsymbol {\prime }}}$ is an integral Kähler form, i.e., $a\omega _{\textbf {m}^{\boldsymbol {\prime }}}=\omega _{\textbf {m}}$ with $\textbf {m}=a\textbf {m}^{\boldsymbol {\prime }}\in \mathbb {Z}_{>0}^{r}$. Since $w_{G}(Z_{\textbf {i}}, a\omega _{\textbf {m}^{\boldsymbol {\prime }}})=aw_{G}(Z_{\textbf {i}}, \omega _{\textbf {m}^{\boldsymbol {\prime }}})$, we shall derive Inequality (1.1) from Inequality (4.6).

Recall the definition of the T-stable curve C_j from Section 2. Thanks to Corollary 2.4, Inequality (4.6) reads as

$$ w_{G}(Z_{\textbf{i}}, \omega_{\textbf{m}}) \geq \min\big\{ \omega_{\textbf{m}}([C_{j}]): 1\leq j\leq r \} = \min\{ \omega_{\textbf{m}}([C]): C \text{ minimal}\}. $$

This ends the proof of Theorem 1.2. □

Proof Proof of Corollary 1.3

Arguing as in the proof of Theorem 1.2, we can assume that λ is integral. Under this assumption, the coadjoint orbit O_λ equipped with its K-invariant complex structure is thus a flag variety and, in turn, a Schubert variety X(w) associated to the longest element w of the Weyl group of a parabolic subgroup P_λ of G containing the Borel subgroup B.

Let i be any reduced expression of w and let ${\mathscr{L}}_{\lambda }$ denote the ample line bundle on O_λ associated to λ. The pullback via the morphism $\pi _{r}: Z_{\textbf {i}}\rightarrow O_{\lambda }$ of ${\mathscr{L}}_{\lambda }$ being generated by its global sections, it is equal to ${\mathscr{L}}_{\textbf {i,m}}$ for some $m\in \mathbb {Z}_{\geq 0}^{r}$ by Theorem 2.1. Note that the construction of the Newton-Okounkov body ${\Delta }_{\textbf {i,m}}={\Delta }(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i,m}},v_{\beta },\tau )$ can also be performed for ${\mathscr{L}}_{\textbf {i,m}}$ generated by its global sections and non necessarily ample. Thanks to the birationality of the morphism π_r, we can regard the Newton-Okounkov body ${\Delta }(O_{\lambda },{\mathscr{L}}_{\lambda }, v_{\beta },\tau _{\lambda })$ as the Newton-Okounkov body Δ_i,m with τ_λ being the lowest weight vector of the dual of V (λ) and such that τ_λ(v_λ) = 1.

Let y₀ be the base point of G/P_λ. Recall that the T-stable curves through wy₀ are the U_α-orbit closures of wy₀ within O_λ = G/P_λ with α being a positive root non-orthogonal to w(λ). Note that the set of these roots α coincides with the set of roots β_j (defined in (2.3)). Moreover, for any such curve, say C_α, the equalities $(w(\lambda ),-\alpha ^{\vee })={\mathscr{L}}_{\lambda }\cdot C_{\alpha }=\omega _{\textbf {m}}(C_{j})$ hold (see, e.g., [8]). This enables to conclude the proof. □

5 Gromov Widths of Bott Manifolds and of Bott-Samelson Varieties

The main goal of this section is to derive the Gromov widths of Bott-Samelson varieties which can be degenerated into Bott manifolds from the Gromov widths of the latter manifolds (computed in [21]). We would like also to draw the reader’s attention on Proposition 5.6 which gives a reformulation of the combinatorial expression of the Gromov width obtained in loc. cit. in terms of the symplectic areas of minimal curves. This result is independent from the rest of the paper.

5.1 Generalized Bott manifolds

We start by reviewing a few basic facts on generalized Bott manifolds.

An m-stage generalized Bott tower is a sequence of complex projective space bundles

$$ B_{m}\overset{\pi_{m}}{\longrightarrow} B_{m-1} \overset{\pi_{m-1}}{\longrightarrow} {\cdots} \overset{\pi_{2}}{\longrightarrow} B_{1}\overset{\pi_{1}}{\longrightarrow}B_{0}=\{pt\}, $$

where $B_{j}=\mathbb {P}({\mathscr{L}}_{j}^{(1)} \oplus {\cdots } \oplus {\mathscr{L}}_{j}^{(n_{j})} \oplus \mathcal {O}_{B_{j-1}})$ for some line bundles ${\mathscr{L}}_{j}^{(1)}, \ldots , {\mathscr{L}}_{j}^{(n_{j})}$ over B_j− 1.

Since the Picard group of B_j− 1 is isomorphic to $\mathbb {Z}^{j-1}$ for any j = 1,…,m, each line bundle ${\mathscr{L}}_{j}^{(k)}$ corresponds to a (j − 1)-tuple of integers $(a_{j, 1}^{(k)}, \ldots , a_{j, j-1}^{(k)}) \in \mathbb {Z}^{j-1}$. The variety B_m is thus determined by a collection of integers

$$ a_{j, l}^{(k)} \quad\text{with}~ 2 \leq j \leq m, 1 \leq k \leq n_{j} ~\text{and}~ 1 \leq l \leq j-1. $$

B_m is called the m-stage generalized Bott manifold associated to the collection $(a_{j, l}^{(k)})$.

Recall that any projective bundle of sum of line bundles over a smooth toric variety is again a smooth toric variety (see [11, Proposition 7.3.3]). Therefore B_m is a smooth projective toric variety.

We now describe the fan of the m-stage generalized Bott manifold B_m associated to the collection $(a_{j, l}^{(k)})$. Let n = n₁ + ⋯ + n_m and let $\{{e_{1}^{1}}, \ldots , e_{1}^{n_{1}}, \ldots , {e_{m}^{1}}, \ldots , e_{m}^{n_{m}}\}$ be the standard basis for $\mathbb {Z}^{n}$. For l = 1,…,m, we define

$$ {u_{l}^{k}}={e_{l}^{k}} \quad \text{ for } \quad k=1, \ldots, n_{l} \quad \text{and} $$

$$ {u_{l}^{0}}=-\sum\limits_{k=1}^{n_{l}}{e_{l}^{k}} + \sum\limits_{j=l+1}^{m} \sum\limits_{k=1}^{n_{j}} a_{j, l}^{(k)}{e_{j}^{k}}. $$

Note that ${u_{1}^{0}}, \ldots , u_{1}^{n_{1}}, \ldots , {u_{m}^{0}}, \ldots , u_{m}^{n_{m}} (\in \mathbb {Z}^{n})$ are ray generators.

Given k = (k₁,…,k_m) with 0 ≤ k_l ≤ n_l for 1 ≤ l ≤ m, let

$$ \mathcal{C}_{\textbf{k}} = \text{Cone} \big({u_{l}^{k}}: 1\leq l\leq m, 0\leq k\leq n_{l}, k\neq k_{l}\big) $$

be the n-dimensional cone in $\mathbb {R}^{n}$ generated by all ${u_{l}^{k}}$ but the $u_{l}^{k_{l}}$’s.

Proposition 5.1

Let Σ be the fan in $\mathbb {R}^{n}$ whose maximal cones consist of the cones $\mathcal C_{\textbf {k}}$, k ∈ [0,n₁] ×… × [0,n_m]. Then Σ is a smooth complete fan in $\mathbb {R}^{n}$.

Furthermore, the generalized Bott manifold defined by the collection $(a_{j, l}^{(k)})$ is the toric variety associated to the fan Σ.

Since B_m is a toric manifold, $H_{2}(B_{m},\mathbb {R})$ is isomorphic to the $\mathbb {R}$-span of the torus stable prime divisors ${D_{l}^{k}}$ of B_m, the latter corresponding to the ray generators ${u_{l}^{k}}$ of the fan Σ. Given $[\omega ]\in H_{2}(B_{m},\mathbb {R})$, by abuse of notation, we thus write

$$ {[\omega]} = \sum\limits_{l,k} {\lambda_{l}^{k}} [{D_{l}^{k}}]\quad \text{ with } {\lambda_{l}^{k}}\in\mathbb{R}, 1\leq l\leq m \text{ and } 0\leq k\leq n_{l}. $$

(5.1)

For 1 ≤ l ≤ m, we set

$$ u(l)= \sum\limits_{k=0}^{n_{l}} {u_{l}^{k}} \quad \text{and} \quad \lambda(l)= \sum\limits_{k=0}^{n_{l}} {\lambda_{l}^{k}}. $$

Theorem 5.2

[21, Theorem 1.1] Let (B_m,ω) be the m-stage generalized Bott manifold associated to the collection $(a_{j, l}^{(k)})$ and equipped with a symplectic 2-form ω given as in (5.1). Then

$$ w_{G}(B_{m}, \omega)=\min\{ \lambda(l): u(l)=0\}. $$

5.2 Minimal rational curves on toric manifolds

Let us recall the combinatorial description of the minimal rational curves on complete toric varieties obtained in [10].

Let X be any smooth complete toric variety and Σ be its fan. By Σ(1) we denote the set of all primitive generators of one-dimensional cones in the fan Σ.

Definition 5.3

[3] A non-empty subset $\mathfrak {P}=\{x_{1}, \ldots , x_{k}\}$ of Σ(1) is called a primitive collection if, for any 1 ≤ i ≤ k, the set $\mathfrak {P}\setminus \{x_{i}\}$ generates a (k − 1) −dimensional cone in Σ, while $\mathfrak {P}$ does not generate a k-dimensional cone in Σ.

For a primitive collection $\mathfrak {P}=\{x_{1}, \ldots , x_{k}\}$ of Σ(1), let $\sigma (\mathfrak {P})$ be the unique cone in Σ that contains x₁ + ⋯ + x_k in its interior. Let y₁,…,y_m be generators of $\sigma (\mathfrak {P})$. There thus exists a unique equation such that

$$ x_{1}+\cdots+x_{k}=a_{1}y_{1}+\ldots+a_{m}y_{m} \quad\text{ with }\quad a_{i}\in \mathbb{Z}_{>0}. $$

The equation x₁ + ⋯ + x_k − a₁y₁ −… − a_my_m = 0 is called the primitive relation of $\mathfrak {P}$. The degree of $\mathfrak {P}$ is

$$ \deg(\mathfrak{P})=k-\sum\limits_{i=1}^{m}a_{i}. $$

Theorem 5.4

[10, Proposition 3.2 and Corollary 3.3] Let X be a smooth complete toric variety.

• (1) There is a bijection between minimal rational components of degree k on X and primitive collections $\mathfrak {P}=\{x_{1}, \ldots , x_{k}\}$ of Σ(1) such that x₁ + ⋯ + x_k = 0.

• (2) There exists a minimal rational component in RatCurves(X).

For later use, we recall briefly the idea of the proof of this theorem.

Proof The idea of the proof of Theorem 5.4

For a given family $\mathcal {K}$ of minimal rational curves on X of degree k, there exists a torus invariant open subset U ⊂ X such that $U\simeq (\mathbb {C}^{*})^{n+1-k}\times \mathbb {P}^{k-1}$ (see [10, Corollary 2.6]) such that the lines in the factor $\mathbb {P}^{k-1}$ give general members of $\mathcal {K}$. The fan defining U is the fan of $\mathbb {P}^{k-1}$ viewed as a fan in $\mathbb {R}^{n}$. This gives a collection {x₁,…,x_k} of Σ(1) which is primitive and such that x₁ + ⋯ + x_k = 0.

For the converse, assume that $\mathfrak {P}=\{x_{1}, \ldots , x_{k}\}$ is a primitive collection of X such that x₁ + ⋯ + x_k = 0. Now consider the subfan ${\Sigma }^{\prime }$ of Σ given by the collection of cones in Σ generated by the subsets of $\mathfrak {P}$. Then the toric variety $U_{{\Sigma }^{\prime }}$ associated with ${\Sigma }^{\prime }$ is an open subset of X and $U_{{\Sigma }^{\prime }}\simeq (\mathbb {C}^{*})^{n+1-k}\times \mathbb {P}^{k-1}$. Let C_k be a line in the factor $\mathbb {P}^{k-1}$. Then the deformations of C_k give a minimal family of rational curves in X of degree k. □

5.3 Gromov width in terms of minimal curves

For 1 ≤ l ≤ m, define $\mathfrak {P}_{l}:=\{{u_{l}^{0}}, {u_{l}^{1}}, \ldots , u_{l}^{n_{l}}\}$.

Lemma 5.5

The set of all primitive collections of the generalized Bott manifold B_m is $\{\mathfrak {P}_{l}: 1\leq l\leq m\}$.

Proof

This follows readily from the description of the fan of B_m (Proposition 5.1) together with the definition of primitive collections (Definition 5.3). □

Here is the advertised reformulation of Theorem 5.2 in terms of minimal rational curves.

Proposition 5.6

Keep the notation as in Theorem 5.2.

$$ \min\{\mathcal{L}\cdot C: C\subset X \text{ is a minimal rational curve }\}= \min\{ \lambda(l): u(l)=0\}. $$

Proof

Recall the definition of the fan Σ of X; see Proposition 5.1. By Theorem 5.4, any family $\mathcal {K}_{l}$ of minimal rational curves corresponds to a primitive collection $\mathfrak {P}_{l}$ with u(l) = 0. Given such a primitive collection $\mathfrak {P}_{l}$, consider the subfan ${\Sigma }^{\prime }$ of Σ given by the collection of cones in Σ generated by the subsets of $\mathfrak {P}_{l}$. Then the fan ${\Sigma }^{\prime }$ is isomorphic to the fan of $\mathbb {P}^{n_{l}}$ (see the proof of Theorem 5.4). To any primitive relation u(l) = 0 viewed in the fan of $\mathbb {P}^{n_{l}}$, we can associate two maximal cones in ${\Sigma }^{\prime }$, say σ and $\sigma ^{\prime }$, such that the intersection $\tau = \sigma \cap \sigma ^{\prime }$ is a cone of codimension 1 in ${\Sigma }^{\prime }$. Let C_l be the torus invariant curve in $\mathbb {P}^{n_{l}}$ associated to τ. Note that C_l is isomorphic to the projective line $\mathbb {P}^{1}$ (see [11, Section 6.3, p. 289]). Then the family $\mathcal {K}_{l}$ associated to $\mathfrak {P}_{l}$ is obtained by deformation of the curve C_l (see the proof of Theorem 5.4).

Besides, the relation u(l) = 0 corresponds to the element $R(l)=(b_{\rho })_{\rho }\in N_{1}(X)\subset \mathbb {R}^{|{\Sigma }(1)|}$, the group of numerical classes of 1-cycles on X, with

$$ b_{\rho}=\begin{cases} 1 & \text{if } \rho \in \mathfrak{P}_{l} \\ 0 & \text{otherwise}. \end{cases} $$

By [11, Proposition 6.4.1], we see that the intersection number ${\mathscr{L}}\cdot R(l)$ equals λ(l). Finally, since ${\mathscr{L}}\cdot C ={\mathscr{L}}\cdot C^{\prime }$ for any $C,C^{\prime }\in \mathcal {K}_{l}$, the proof follows. □

Corollary 5.7

$w_{G}(X, {\mathscr{L}})=\min \limits \{{\mathscr{L}}\cdot C: C\subset X~\text {is a minimal rational curve}~\}$.

Proof

This follows readily from Theorem 5.2 and Proposition 5.6. □

5.4 A weaker version of Corollary 1.5

For some appropriate choice of (i, m), Bott-Samelson varieties Z_i equipped with an ample line bundle ${\mathscr{L}}_{\textbf {i,m}}$ can be degenerated into Bott manifolds using the theory of Newton-Okounkov bodies. These toric degenerations were derived in [19] from some previous constructions of Grossberg and Karshon (see [16] and [37] also). Next, we recall the main properties of these toric degenerations.

To define properly the relevant Newton-Okounkov bodies, denoted below by P_i,m, we need the following functions. Given (i, m), we set

$$ \begin{array}{@{}rcl@{}} A_{r} &=& m_{r}, \\ A_{r-1}(x_{r}) &=& \langle m_{r-1}\varpi_{i_{r-1}}+m_{r}\varpi_{i_{r}}-x_{r}\alpha_{i_{r}},\alpha_{i_{r-1}}^{\vee}\rangle,\\ A_{r-2}(x_{r-1},x_{r}) &=& \langle m_{r-2}\varpi_{i_{r-2}}+m_{r-1}\varpi_{i_{r-1}}+ m_{r}\varpi_{i_{r}}-x_{r-1}\alpha_{i_{r-1}}-x_{r}\alpha_{i_{r}},\alpha_{i_{r-2}}^{\vee}\rangle,\\ \quad {\vdots} \\ A_{1}(x_{2},\dots,x_{r}) &=& \langle m_{1}\varpi_{i_{1}}+m_{2}\varpi_{i_{2}}+{\ldots} + m_{r}\varpi_{i_{r}}-x_{2}\alpha_{i_{2}}-\cdots-x_{r}\alpha_{i_{r}},\alpha_{i_{1}}^{\vee}\rangle.\\ \end{array} $$

(5.2)

The polytope P_i,m consists of the points $(x_{1},\ldots , x_{r})\in \mathbb {R}^{r}$ satisfying the inequalities

$$ 0\leq x_{j}\leq A(x_{j+1},...,x_{n}),\quad 1\leq j\leq r. $$

We now recall the definition of the technical assumption on the pair (i, m) needed for the construction of the toric degenerations.

Definition 5.8

We say that the pair (i, m) satisfies the condition (P-k) when the following holds: If (x_k+ 1,…,x_r) satisfies the inequalities

$$ \begin{array}{ccccl} 0 & \leq & x_{r} & \leq & A_{r} \\ 0 & \leq & x_{r-1} & \leq & A_{r-1}(x_{r}) \\ & & {\vdots} && \\ 0 & \leq & x_{k+1} & \leq & A_{k+1}(x_{k+2}, \ldots, x_{r}) \end{array} $$

then A_k(x_k+ 1,…,x_r) ≥ 0.

We say that the pair (i, m) satisfies the condition (P) if m_r ≥ 0 and (i, m) satisfies the conditions (P-k) for all k = 1,…,r − 1.

The following theorem gathers several results on the aforementioned degenerations; see [18] for details and original references.

Theorem 5.9

Let $\mathbf {m} \in \mathbb {Z}_{>0}^{r}$ and suppose that (i,m) satisfies condition (P). Then

• (1) The polytope P_i,m is a smooth lattice polytope.

• (2) The symplectic toric manifold X(P_i,m) associated to the polytope P_i,m is the Bott tower with

  $$ {u_{j}^{0}} = -{e^{1}_{j}} - \sum\limits_{k>j}^{r}\langle \alpha_{i_{k}} \alpha_{i_{j}}^{\vee} \rangle {e^{1}_{k}} \quad \text{for all } 1\leq j\leq r $$

  (5.3)

  and equipped with the ample line bundle

  $$ \mathcal{M}_{\mathbf{i,m}} = \sum\limits_{{j=1}}^{{r}} a_{j}[{D_{j}^{0}}] \quad \text{ with } a_{j} = \langle m_{j} \varpi_{i_{j}} + {\cdots} + m_{r} \varpi_{i_{r}}, \alpha_{i_{j}}^{\vee} \rangle . $$

  (5.4)

Theorem 5.10

[19, Theorem 3.4] Let $\mathbf {m} \in \mathbb {Z}_{>0}^{r}$ and suppose that (i,m) satisfies condition (P). Then the polytope P_i,m is a Newton-Okounkov body. In particular, there exists a one-parameter flat family with generic fiber being isomorphic to Z_i and special fiber isomorphic to the toric variety X(P_i,m).

Remark 5.11

The polytope P_i,m enjoys further nice properties. For instance, as proved in [19], under the condition (P) the polytope P_i,m coincides with the generalized string polytope introduced in [14]; the latter turns out to be unimodular to the polytope Δ_i,m we are considering in Section 4 (by [14, Theorem 8.2] along with Remark 4.2).

Finally, we apply Theorem 5.2 to compute the Gromov widths of the polarized Bott-Samelson varieties $(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i,m}})$ when the pair (i, m) satisfies the condition (P).

Corollary 5.12

Let $\mathbf {m} \in \mathbb {Z}_{>0}^{r}$ and (i,m) satisfy condition (P). Let X(P_i,m) be the symplectic projective toric manifold associated to the polytope P_i,m. Then

$$ \begin{array}{@{}rcl@{}} \omega_{G}(Z_{\mathbf{i}},\mathcal{L}_{\mathbf{i,m}}) &=& \omega_{G} (X(P_{\mathbf{i,m}})) \\ &=& \min\{m_{j}: \langle\alpha_{i_{k}}, \alpha_{i_{j}}^{\vee} \rangle=0 ~ \forall~ k>j\}\\ &=& \min\{\mathcal{L}_{\mathbf{i,m}}\cdot C_{j}: C_{j} \text{ line of } Z_{\mathbf{i}}\}. \end{array} $$

Proof

Note that the toric degeneration is smooth since it is a Bott manifold by Theorem 5.9. By Theorem 5.10 along with the recalls made in the proof of Corollary 4.5, $(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i,m}})$ and X(P_i,m) are symplectomorphic. The first equality thus follows.

To prove the second equality, we apply Theorem 5.2. We thus consider the primitive collections $\mathfrak {P}_{j}=\{{e^{1}_{j}}, {u_{j}^{0}}\}$ with ${e^{1}_{j}}+{u_{j}^{0}}=0$. By (5.3), it is clear that

$$ {e^{1}_{j}}+{u_{j}^{0}}=0 \quad \text{if and only if }\quad \langle \alpha_{i_{k}}, \alpha_{i_{j}}^{\vee} \rangle=0 ~ \forall~ k>j. $$

(5.5)

Let $\widetilde {C}_{j}\in \mathcal {K}_{j}$ be a curve of X(P_i,m) defined by the primitive relation ${e^{1}_{j}}+{u_{j}^{0}}=0$. Then by [11, Proposition 6.4.1], ${\mathscr{M}}_{\textbf {i,m}}\cdot \widetilde {C}_{j}=a_{j}$ and in turn, ${\mathscr{M}}_{\textbf {i,m}}\cdot \widetilde {C}_{j}=m_{j}$ thanks to (5.4) and (5.5).

The curve C_j is a line if and only if ${\mathscr{L}}_{k}\cdot C_{j}=0$ for all j < k and this is equivalent to the assertion that $s_{i_{j}}$ commutes with $s_{i_{j+1}},\ldots ,s_{i_{r}}$; see Remark 4.2 in [8]. Besides, by Proposition 2.2(3), ${\mathscr{L}}_{k}\cdot C_{j}=0$ for all j > k. It follows ${\mathscr{L}}_{\textbf {i}, \textbf {m}} \cdot C_{j}=m_{j}$ if C_j is a line of Z_i. This yields the last equality and concludes the proof. □

The following example shows that the equalities in Corollary 5.12 may not hold when the condition (P) is not satisfied.

Example 5.13

Let G = SL₃. Take i = (1,2,1) and $\textbf {m}=(m_{1}, m_{2}, m_{3})\in \mathbb {Z}^{3}_{>0}$ with m₁ + m₂ < m₃. Then (i, m) does not satisfy the condition (P) since (P-3) does not hold for (x₂,x₃) = (0,m₃). Besides, ℓ₁ = m₁ + m₂, ℓ₂ = m₂ + m₃ and ℓ₃ = m₃ hence $\omega _{G}(Z_{\textbf {i}},{\mathscr{L}}_{\textbf {i,m}}) = {\min \limits } \{\ell _{j}: j=1,2,3\}=m_{1}+m_{2}$ by Corollary 1.5.