Abstract
We prove that the Gromov width of any BottSamelson 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 BottSamelson varieties.
1 Introduction
The Gromov width of a 2ndimensional 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 nonsqueezing 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 BottSamelson 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 BottSamelson varieties.
Our computation of the Gromov width is performed in two steps. First, by using Gromov’s Jholomorphic curves techniques applied to minimal curves, we prove that the Gromov width of the BottSamelson 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 NewtonOkounkov 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 NewtonOkounkov bodies unimodular to generalized string polytopes (as defined in [14]) enables us to get a lower bound for the Gromov width in case the BottSamelson varieties are equipped with a positive integral 2form (Theorem 1.2). Finally, by making use of Brion–Kannan’s characterization [8] of the minimal curves of BottSamelson 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 α_{1},…,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 ϖ_{1},…,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 BottSamelson varietyZ_{i} is the quotient
where B^{r} acts on \(P_{i_{1}}\times {\cdots } \times P_{i_{r}}\) by
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 BottSamelson variety is a desingularization of the Schubert variety in G/B associated to w.
In Section 2, the reader can find some material on BottSamelson 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 2form 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
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 2form of the BottSamelson variety Z_{i}. Then
Theorem 1.2 extends the results obtained by FangLittelmannPabiniak in [13] concerning the lower bound of the Gromov width of rational coadjoint orbits equipped with the KirillovKostantSouriau symplectic form. Namely, we recover
Corollary 1.3 (FangLittelmannPabiniak)
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 KirillovKostantSouriau form. Then
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 2form of the BottSamelson variety Z_{i}. Then
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_{x}C 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 FubiniStudy form associated to \({\mathscr{L}}\). Together with Corollary 1.5, this yields^{Footnote 1}
Corollary 1.6
Let \(\mathbf {m}\in \mathbb {Z}^{\mathbf {r}}_{>\mathbf {0}}\). Then the following inequality holds for the Seshadri constants of the BottSamelson variety Z_{i} at any point x ∈ Z_{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 BottSamelson 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]^{Footnote 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 BottSamelson varieties under consideration equals the symplectic area of a line. This method can not be applied further to any BottSamelson variety to recover fully Corollary 1.5 as Example 5.13 shows.
2 BottSamelson 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 (i_{1},i_{2},…,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 BottSamelson varieties; for more details, see [12, 30].
Given a (not necessarily reduced) expression i = (i_{1},…,i_{r}) of w ∈ W, recall the definition of the BottSamelson 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
We can realize Z_{i} as an iterated \(\mathbb {P}^{1}\)bundle. Specifically, let \(w^{\prime }=s_{i_{1}}{\cdots } s_{i_{r1}}\) and \(\textbf {i}^{\boldsymbol {\prime }}=(i_{1}, \ldots , i_{r1})\). 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 \([(p_{1}, \ldots , p_{r1})]\mapsto p_{1}{\cdots } p_{r1}P_{i_{r}}\). We have the following commutative diagram:
For 1 ≤ j ≤ r, let Z_{i}(j) denote the BottSamelson variety associated to the subexpression (i_{1},…,i_{j}) of i = (i_{1},…,i_{r}).
Consider the natural morphisms
Explicitly, f_{j} maps [(p_{1},…,p_{r})] to [(p_{1},…,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
and Z_{i}(j) is a desingularization of \(X(s_{i_{1}}{\cdots } s_{i_{j}})\).
The BottSamelson variety Z_{i} is equipped with a base point that is
Moreover, we have π_{r}(z_{0}) = wx_{0} with x_{0} being the base point of G/B and π_{r} yields an isomorphism between the Borbits
(see, e.g., [23, §II.13.6] for details). In particular, z_{0} is fixed by T.
For 1 ≤ j ≤ r, the line bundle \({\mathscr{L}}_{j}\) on Z_{i} is defined as
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
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 Tstable curves and the minimal rational curves on BottSamelson 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) quasiprojective variety equipped with a quasifinite 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 nonempty 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 BottSamelson varieties associated to a reduced expression.
Given a reduced word i = (i_{1},…,i_{r}) of w ∈ W, let
Note that
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}
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 BottSamelson variety associated to a reduced expression i.

(1)
The Tstable curves in Z_{i} through z_{0} 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 BottSamelson 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 z_{0} 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 BottSamelson variety equipped with an ample line bundle. Assume i is a reduced expression. Then
Proof
Note that the curves containing the base point z_{0} are dominant and recall that the minimal curves of Z_{i} containing z_{0} are all Tstable (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 BottSamelson 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 GromovWitten invariants.
We thus start by recalling some basics on GromovWitten 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
where c_{1} denotes the first Chern class of the tangent bundle T_{X} of X.
Proposition 3.1
Let A be the class of a Tstable curve of X through the generic point z_{0} of X. Then the moduli space \(\overline {{\mathscr{M}}^{X}_{0,k}}(A)\) is smooth and has the expected dimension. Moreover,
Proof
For any Tstable curve C of X going through z_{0}, 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
be the evaluation map sending a stable map to the ktuple 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 z_{0} 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 \(s_{i_{1}}...s_{i_{j1}}s_{i_{j+1}}...s_{i_{r}}\) is reduced hence the natural morphism \(X\rightarrow Z_{(i_{1},....i_{j1},i_{j+1},...i_{r})}\) is a \(\mathbb {P}^{1}\)fibration with fiber over \([\dot {s}_{i_{1}},\ldots ,\dot {s}_{i_{j1}},\dot {s}_{i_{j+1}},\ldots ,\dot {s}_{i_{}j1}]\) the curve C_{j} itself. It follows that the evaluation map ev^{1} 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 GromovWitten invariant is defined to be the rational number
whenever the degrees of α_{1},…,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 z_{0}. Then
Proof
Note that c_{1}(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 \(d={\dim }X+22\). By Proposition 3.1 along with Corollary 3.2, we have the equality
The right hand side being obviously nonequal 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 Jholomorphic maps of genus 0 into X of class A and with kmarked 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
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 NewtonOkounkov bodies.
NewtonOkounkov 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 nonequivalent definitions of NewtonOkkounkov bodies for BottSamelson varieties. Here, we are concerned with one construction which parallels one of Fujita’s definitions.
4.1 Definition of the NewtonOkounkov body
Let us thus start by defining the NewtonOkounkov body of interest to us.
Fix a reduced expression i = (i_{1},…,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 z_{0} of Z_{i} via the natural isomorphisms
The first isomorphism is wellknown; 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 x_{0} 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
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 t_{1},…,t_{r} (also denoted by ≤) by setting
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 (k_{1},…,k_{r}) be the maximum tuple among the tuples (j_{1},…,j_{r}) such that c_{j}≠ 0. Define
This induces the following map
Given an ample line bundle \({\mathscr{L}}\) on Z_{i} and a nonzero 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
As in the theory of NewtonOkounkov bodies, let us now consider the following set
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 NewtonOkounkov body associated to \({\mathscr{L}}\), τ and v_{β} is defined as
4.2 Properties of the NewtonOkounkov body
In this paper, we consider the NewtonOkounkov 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 Gmodule 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 \({\mathscr{L}}={\mathscr{L}}_{\textbf {i,m}}\)
Note that the image of the base point z_{0} ∈ Z_{i} through this morphism is [v_{0}] with
Moreover, the morphism Ψ_{i,m} induces an isomorphism of \(P_{i_{1}}\)modules
where V_{i,m} denotes the socalled generalized Demazure module. As a complex vector space, V_{i,m} is generated by the following vectors
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
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 Bz_{0} of Z_{i}.
Proof
Note that f_{0} 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 NewtonOkounkov body associated to φ_{0}, that is
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 NewtonOkounkov 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 NewtonOkounkov 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 2ndimensional manifold equipped with a momentum map. Suppose there exists a ndimensional 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 BottSamelson varieties and the NewtonOkounkov bodies Δ_{i,m} reads as follows.
Corollary 4.5 (Kaveh)
Let \((Z_{\mathbf {i}},{\mathscr{L}}_{\mathbf {i,m}})\) be a BottSamelson 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 NewtonOkounkov 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 X_{0} 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 ω_{0} on the smooth locus U_{0} of X_{0} such that

(1)
(U_{0},ω_{0}) 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 (U_{0},ω_{0}) contains the interior of the NewtonOkounkov body Δ_{i,m}.
The corollary thus follows from Theorem 4.4. □
Remark 4.6
Kaveh’s result does not require that the NewtonOkounkov 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 NewtonOkounkov body Δ_{i,m} and apply Corollary 4.5.
Given \(\textbf {m}\in \mathbb {Z}_{>0}^{r}\), for each 1 ≤ j ≤ r, we set
Lemma 4.7
The roots \(s_{i_{k+1}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) 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 \(s_{i_{k+1}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\), the proof being similar for the other roots. Let us fix j and proceed by induction on k. We thus start by showing that \(s_{i_{2}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) is a positive root for all j ≥ 2, the case k = 2. Because \(s_{i_{1}}s_{i_{2}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})=\beta _{j}\) is a positive root, if \(s_{i_{2}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) were a negative root, it would be equal to \(\alpha _{i_{1}}\). This would imply that β_{j} = β_{1} – a contradiction since the roots β_{j} are pairwise distinct and j≠ 1. Next, we consider \(s_{i_{\ell }}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) with 2 ≤ ℓ ≤ j − 1 < r. By induction hypothesis, \(s_{i_{\ell 1}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) is a positive root hence by arguing similarly as for the case k = 2, we get that \(s_{i_{\ell }}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})\) is also a positive root; otherwise it would be equal to \(\alpha _{i_{\ell 1}}\) and in turn we would have: \(\beta _{j}=s_{i_{1}}{\ldots } s_{i_{\ell }}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})=s_{i_{1}}{\ldots } s_{i_{\ell 2}}(\alpha _{i_{\ell 1}})=\beta _{\ell 1}\) – a contradiction. □
Recall the definition of the linear form f_{0} ∈ V (λ_{1})^{∗}⊗… ⊗ V (λ_{r})^{∗} as well as the morphism Φ_{i,m} introduced in Section 4.2. Set
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
Proof
Lemma 4.7 implies that \( \langle s_{i_{1}}{\ldots } s_{i_{k}}\lambda _{k},\beta _{j}^{\vee }\rangle =\langle \lambda _{k},s_{i_{k+1}}{\ldots } s_{i_{j1}}(\alpha _{i_{j}})^{\vee }\rangle \geq 0 \) 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 v_{0} 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
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, \(E_{\beta _{j}}^{1a_{j\ell }}(s_{i_{1}}{\cdots } s_{i_{\ell }}v_{\lambda _{\ell }}) =0\) 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
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}(v_{0}))≠ 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/φ_{0} is a regular function on Bz_{0} 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
Proof
First, note that the polytope Δ_{i, m} contains the origin since v_{β}(φ_{0}) = 0.
From the definition of the valuation v_{β} and Proposition 4.11, we get
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 FubiniStudy 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:
We next consider any 2form \(\omega _{\textbf {m}^{\boldsymbol {\prime }}}\) of Z_{i} with \(\textbf {m}^{\boldsymbol {\prime }}\in \mathbb {Q}_{>0}^{r}\), namely \(\omega _{\textbf {m}^{\boldsymbol {\prime }}}\) is the 2form 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 Tstable curve C_{j} from Section 2. Thanks to Corollary 2.4, Inequality (4.6) reads as
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 Kinvariant 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 NewtonOkounkov 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 NewtonOkounkov body \({\Delta }(O_{\lambda },{\mathscr{L}}_{\lambda }, v_{\beta },\tau _{\lambda })\) as the NewtonOkounkov 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 y_{0} be the base point of G/P_{λ}. Recall that the Tstable curves through wy_{0} are the U_{α}orbit closures of wy_{0} within O_{λ} = G/P_{λ} with α being a positive root nonorthogonal 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 BottSamelson Varieties
The main goal of this section is to derive the Gromov widths of BottSamelson 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 mstage generalized Bott tower is a sequence of complex projective space bundles
where \(B_{j}=\mathbb {P}({\mathscr{L}}_{j}^{(1)} \oplus {\cdots } \oplus {\mathscr{L}}_{j}^{(n_{j})} \oplus \mathcal {O}_{B_{j1}})\) 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 \(\mathbb {Z}^{j1}\) for any j = 1,…,m, each line bundle \({\mathscr{L}}_{j}^{(k)}\) corresponds to a (j − 1)tuple of integers \((a_{j, 1}^{(k)}, \ldots , a_{j, j1}^{(k)}) \in \mathbb {Z}^{j1}\). The variety B_{m} is thus determined by a collection of integers
B_{m} is called the mstage 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 mstage generalized Bott manifold B_{m} associated to the collection \((a_{j, l}^{(k)})\). Let n = n_{1} + ⋯ + 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
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 = (k_{1},…,k_{m}) with 0 ≤ k_{l} ≤ n_{l} for 1 ≤ l ≤ m, let
be the ndimensional 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,n_{1}] ×… × [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
For 1 ≤ l ≤ m, we set
Theorem 5.2
[21, Theorem 1.1] Let (B_{m},ω) be the mstage generalized Bott manifold associated to the collection \((a_{j, l}^{(k)})\) and equipped with a symplectic 2form ω given as in (5.1). Then
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 onedimensional cones in the fan Σ.
Definition 5.3
[3] A nonempty 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 kdimensional 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 x_{1} + ⋯ + x_{k} in its interior. Let y_{1},…,y_{m} be generators of \(\sigma (\mathfrak {P})\). There thus exists a unique equation such that
The equation x_{1} + ⋯ + x_{k} − a_{1}y_{1} −… − a_{m}y_{m} = 0 is called the primitive relation of \(\mathfrak {P}\). The degree of \(\mathfrak {P}\) is
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 x_{1} + ⋯ + 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 \(U\simeq (\mathbb {C}^{*})^{n+1k}\times \mathbb {P}^{k1}\) (see [10, Corollary 2.6]) such that the lines in the factor \(\mathbb {P}^{k1}\) give general members of \(\mathcal {K}\). The fan defining U is the fan of \(\mathbb {P}^{k1}\) viewed as a fan in \(\mathbb {R}^{n}\). This gives a collection {x_{1},…,x_{k}} of Σ(1) which is primitive and such that x_{1} + ⋯ + x_{k} = 0.
For the converse, assume that \(\mathfrak {P}=\{x_{1}, \ldots , x_{k}\}\) is a primitive collection of X such that x_{1} + ⋯ + 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 \(U_{{\Sigma }^{\prime }}\simeq (\mathbb {C}^{*})^{n+1k}\times \mathbb {P}^{k1}\). Let C_{k} be a line in the factor \(\mathbb {P}^{k1}\). 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.
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 1cycles on X, with
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), BottSamelson varieties Z_{i} equipped with an ample line bundle \({\mathscr{L}}_{\textbf {i,m}}\) can be degenerated into Bott manifolds using the theory of NewtonOkounkov 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 NewtonOkounkov bodies, denoted below by P_{i,m}, we need the following functions. Given (i, m), we set
The polytope P_{i,m} consists of the points \((x_{1},\ldots , x_{r})\in \mathbb {R}^{r}\) satisfying the inequalities
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 (Pk) when the following holds: If (x_{k+ 1},…,x_{r}) satisfies the inequalities
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 (Pk) 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 NewtonOkounkov body. In particular, there exists a oneparameter 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 BottSamelson 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
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
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 = SL_{3}. Take i = (1,2,1) and \(\textbf {m}=(m_{1}, m_{2}, m_{3})\in \mathbb {Z}^{3}_{>0}\) with m_{1} + m_{2} < m_{3}. Then (i, m) does not satisfy the condition (P) since (P3) does not hold for (x_{2},x_{3}) = (0,m_{3}). Besides, ℓ_{1} = m_{1} + m_{2}, ℓ_{2} = m_{2} + m_{3} and ℓ_{3} = m_{3} 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.
Notes
While this paper was being reviewed, Biswas, Hanumanthu and Kannan computed Seshadri constants of equivariant bundles on BottSamelson varieties at some points [7].
As a side result, we prove that the Gromov widths of generalized Bott manifolds can be expressed as the symplectic areas of minimal curves of these manifolds (Proposition 5.6).
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The authors thank the referees for several useful suggestions and comments, which improved the paper.
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Open Access funding enabled and organized by Projekt DEAL. This research was supported by the CRC/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” of the Deutsche Forschungsgemeinschaft.
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Bonala, N.C., CupitFoutou, S. The Gromov Width of BottSamelson Varieties. Transformation Groups (2022). https://doi.org/10.1007/s00031022097651
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DOI: https://doi.org/10.1007/s00031022097651
Keywords
 Gromov width
 BottSamelson varieties