The Gromov width of Bott-Samelson varieties

We prove that the Gromov width of any Bott-Samelson variety birational to a Schubert variety and equipped with a rational K\"ahler 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.


Introduction
The Gromov width of a 2n-dimensional symplectic manifold is the largest capacity a for which a ball in R 2n of radius a/π, 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 ( [Gro85]). There have been many works dedicated to the computation or estimates of the Gromov width of symplectic manifolds (see e.g. [Bir01,KT05,LMZ15,Cas16,FLP18,HLS21] 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 [FLP18]. Gromov widths are closely related to Seshadri constants of line bundles (see [MDP94]). The latter have been busily investigated in algebraic geometry as a measure of local positivity (see e.g. Chap.5 in [Laz04]). 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 [Kav19,Pab18]). This approach has been already followed by several authors; see e.g. [Pab18] for some review. This method applied to concrete simplices included in Newton-Okounkov bodies unimodular to generalized string polytopes (as defined in [Fuj18]) 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 ( [BK21]) 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 α 1 , . . . , α n and α ∨ 1 , . . . , α ∨ n 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 ̟ 1 , . . . , ̟ 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 representativeṡ i of s i .
Let w = s i 1 s i 2 · · · s ir . 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 [BK21] 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 C ω of any minimal curve C of Z i . The latter is denoted by ω([C]) below.
In Section 4, we show the following theorem (and Corollary 1.3) giving a lower bound for the Gromov width.
Theorem 1.2. Let m = (m 1 , . . . , m r ) ∈ Q r >0 and ω m be the associated rational Kähler 2-form of the Bott-Samelson variety Z i . Then 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 [AHLL20,HL20] for the state of the art on this conjecture.
As a straightforward consequence of Theorem 1.1 and Theorem 1.2, we get the following: Corollary 1.5. Let m = (m 1 , . . . , m r ) ∈ Q r >0 and ω m be the associated rational Kähler 2-form of the Bott-Samelson 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 L on X, the Seshadri constant ε(X, L, x) of L at some point x ∈ X is defined as the infimum of L·C/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 [BC01, Proposition 2.6.1], in case of an ample line bundle L, the Seshadri constant ε(X, L, x) is upper bounded by the Gromov width of X equipped with the Fubini-Study form associated to L. Together with Corollary 1.5, this yields 1 Corollary 1.6. Let m ∈ Z r >0 . Then the following inequality holds for the Seshadri constants of the Bott-Samelson variety Z i at any point Remark 1.7. Lower bounds of Seshadri constants can be derived also by embedding methods (see e.g. [Ito13]). 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 [GK94,Pas10,HY18]. The Gromov widths of polarized generalized Bott manifolds are computed in [HLS21] 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.

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 (i 1 , i 2 , . . . , i r ) ∈ I r such that w = s i 1 · · · s ir . 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 [Dem74,LT04].
1 While this paper was being reviewed, Biswas, Hanumanthu and Kannan computed Seshadri constants of equivariant bundles on Bott-Samelson varieties at some points ( [BHK22]).
2 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).
Given a (not necessarily reduced) expression i = (i 1 , . . . , 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 We can realise Z i as an iterated P 1 -bundle. Specifically, let w ′ = s i 1 · · · s i r−1 and i ′ = (i 1 , . . . , i r−1 ). Let f : G/B −→ G/P ir be the map given by gB → gP ir , and let p i ′ : Z i ′ −→ G/P ir be the map given by [(p 1 , . . . , p r−1 )] → p 1 · · · p r−1 P ir . We have the following commutative diagram: For 1 ≤ j ≤ r, let Z i (j) denote the Bott-Samelson variety associated to the sub-expression (i 1 , . . . , i j ) of i = (i 1 , . . . , i r ).
Henceforth, we assume the expression i is reduced. Let X(s i 1 · · · s i j ) be the Schubert variety associated to the Wel group element s i 1 · · · s i j . Then π j (Z i (j)) = X(s i 1 · · · s i j ) and Z i (j) is a desingularization of X(s i 1 · · · s i j ).
The Bott-Samelson 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 B-orbits Bz 0 ≃ Bwx 0 (2.1) (see e.g. [Jan03, §II.13.6] for details). In particular, z 0 is fixed by T .
For 1 ≤ j ≤ r, the line bundle L j on Z i is defined as denotes the line bundle on G/B associated to ̟ i j . For m = (m 1 , . . . , m r ) ∈ Z r , we set L i,m := L m 1 1 ⊗ · · · ⊗ L mr r .  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 [BK21].
First, let us briefly recall the notion of rational curves on any projective variety X. For more details, we refer to [Kol99, Chapter II.2].
Let RatCurves(X) denote the normalization of the space of rational curves on X. Every irreducible component 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 : U → K and a projection µ : U → X. For any x ∈ X, let U x = µ −1 (x) and K x = p(U x ). If K x is non-empty and projective for a general point x ∈ X then K is called a family of minimal rational curves and any member of 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 = (i 1 , . . . , i r ) of w ∈ W , let (2.2) 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 β j being the root subgroup of G corresponding to β j . (1) The T -stable curves in Z i through z 0 are exactly the curves C j .
(2) Every curve C j is isomorphic to the projective line P 1 .
(3) For 1 ≤ j ≤ r and 1 ≤ k ≤ r, we have The following result gives a description of the minimal rational curves in Z i . (1) Every minimal family K on Z i satisfies 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 ir · · · s i j+1 (α i j ) is a simple root.

be a Bott-Samelson 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 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 [Kol99, 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).

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 ω ∈ H 2 (X, R) by using Gromov-Witten invariants.
We thus start by recalling some basics on Gromov-Witten invariants.
Proof. For any T -stable curve C of X going through z 0 , we have: The proposition follows; see Section 2 in [Lee04].
Let ev k : M X 0,k (A) −→ X k be the evaluation map sending a stable map to the k-tuple of its values at the k marked points. Proof. By Proposition 2.3, A = [C j ] for some j such that s ir · · · s i j+1 (α 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 ir is reduced hence the natural morphism X → Z (i 1 ,....i j−1 ,i j+1 ,...ir) is a P 1 -fibration with fiber over [ṡ i 1 , . . . ,ṡ i j−1 ,ṡ i j+1 , . . . ,ṡ ij−1 ] 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 α i ∈ H * (X, Q) with i = 1, ..., k, the Gromov-Witten 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.
Proof. Note that c 1 (A) = 2 by Proposition 2.2(4). It follows that P D[pt] satisfies the codimension condition that is, its degree equals the expected dimension which is d = dimX + 2 − 2. By Proposition 3.1 along with Corollary 3.2, we have the equality 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 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 [Beh97] and [Sieb98] resp. coincide; see [Sieb98]. As a consequence, Theorem 4.1 in [HLS21] (for peculiar cocycles) reads as follows.
Corollary 3.5. Let [ω] ∈ H 2 (Z i , 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.
for every curve C in the minimal family containing C j . This along with Theorem 2.3 yields the inequality for any minimal curve.

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 [KK12] and [LM08]. 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.

Let us thus start by defining the Newton-Okounkov 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 β i denotes the root subgroup of G corresponding to β i .
We shall regard U β 1 × · · · × U βr as an affine (open) neighbourhood of the base point z 0 of Z i via the natural isomorphisms (4.1) The first isomorphism is well-known ; it follows mainly from the characterisation of the roots β i recalled in (2.3) (see e.g. [Jan03, §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 C(Z i ) = C(U β 1 × · · · × U βr ) with the rational function field C(t 1 , . . . , t r ) via the isomorphism of algebraic varieties C r ≃ U β 1 × · · · × U βr given by where E β i denotes the root vector associated to the (positive) root β i .
This induces the following map v β : Given an ample line bundle L on Z i and a non-zero section τ ∈ H 0 (Z i , L), we shall regard H 0 (Z i , L ⊗k ) as a complex vector subspace of the function field C(Z i ) via the map As in the theory of Newton-Okounkov bodies, let us now consider the following set Note that S(Z i , L, v β , τ ) is a semigroup since v β is a valuation, which can be easily checked. Let C(Z i , L, v β , τ ) be the real closed convex cone generated by S(Z i , L, v β , τ ). The Newton-Okounkov body associated to L, τ and v β is defined as 4.2. 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 Note that the image of the base point z 0 ∈ Z i through this morphism is (4.2) Moreover, the morphism Ψ i,m induces an isomorphism of P i 1 -modules 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 ) with a j ∈ N and F i j being the root vector associated to the root −α i j ; see Theorem 6 and Section 4.2 in [LLM02].
Lemma 4.1 enables us to introduce the Newton-Okounkov body associated to ϕ 0 , that is (4.4) Remark 4.2.
(1) By considering the open embedding . . , t ′ r ) defined up to the lexicographic order on Z r as above and studies the Newton-Okounkov body ∆(Z i , L, v ′ i , ϕ 0 ). (2) By noticing that exp(a 1 E β 1 ) · · · exp(a r E βr )wx 0 = s i 1 exp(a 1 F i 1 ) · · · s ir exp(a r F ir )x 0 for any (a 1 , . . . , a r ) ∈ C r , one easily sees that the semigroups S(Z i , L, v ′ i , ϕ 0 ) and S(Z i , L i,m , v β , ϕ 0 ) coincide and so do the convex bodies ∆(Z i , L, v ′ i , ϕ 0 ) and ∆ i,m .

4.3.
We are now ready to state Kaveh's result which is a consequence of the following theorem.
Theorem 4.4 ([LMS13, Section 4.2]). Let (X, ω) be a proper connected symplectic toric 2ndimensional 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 R m is said to be of size κ if it can be obtained from the simplex {(x i ) i ∈ R m >0 : x 1 + . . . + x m < κ} by a linear transformation in GL(m, Z) and a translation of R m . Corollary 11.4 in [Kav19] specialised to the case of Bott-Samelson varieties and the Newton-Okounkov bodies ∆ i,m reads as follows. 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. We now proceed to the proof of Theorem 1.2: we shall exhibit a simplex of the advertized size in the Newton-Okounkov body ∆ i,m and apply Corollary 4.5.
Lemma 4.7. The roots s i k+1 . . . s i j−1 (α i j ) and s i ℓ . . . s i j+1 (α 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 . . . s i j−1 (α 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 . . . s i j−1 (α i j ) is a positive root for all j ≥ 2, the case k = 2. Because s i 1 s i 2 . . . s i j−1 (α i j ) = β j is a positive root, if s i 2 . . . s i j−1 (α i j ) were a negative root, it would be equal to −α 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 ℓ . . . s i j−1 (α i j ) with 2 ≤ ℓ ≤ j − 1 < r. By induction hypothesis, s i ℓ−1 . . . s i j−1 (α i j ) is a positive root hence by arguing similarly as for the case k = 2, we get that s i ℓ . . . s i j−1 (α i j ) is also a positive root; otherwise it would be equal to −α i ℓ−1 and in turn we would have: 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 Subsection 4.2. Set with F β j = E −β 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 β j (s i 1 . . . s i k v λ k ) = 0 for every 1 ≤ k < j ≤ r. In particular, we have the equality Proof. Lemma 4.7 implies that s i 1 . . . s i k λ k , β ∨ j = λ k , s i k+1 . . . s i j−1 (α i j ) ∨ ≥ 0 and, in turn E β j (s i 1 · · · s i k v λ k ) = 0 for all j = 1 because s i 1 · · · s i k v λ 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 Proof. Note that we have By Lemma 4.7, the integers a jℓ are negative and since s i 1 · · · s i ℓ v λ ℓ is an extremal weight vector, E 1−a jℓ β j (s i 1 · · · s i ℓ v λ ℓ ) = 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 ℓ j +1 β j (s i 1 . . . s i j v λ j ⊗ . . . ⊗ wv λr ) = 0. We conclude the proof by applying Lemma 4.8.
Lemma 4.10. The sections ϕ j ∈ H 0 (Z i , L i,m ) are not trivial.
Proof. Let u j = exp(E β j ) ∈ U β j . By Lemma 4.9 together with Lemma 4.8, we have Lemma 4.9). Therefore, we have f j (u j (v 0 )) = 0 and the lemma follows.
Proof. Take a section ϕ ∈ H 0 (Z i , L 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 C[t 1 , . . . , 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 β k v 0 ) = 0 for all k = j. It follows that ϕ j /ϕ 0 ∈ 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 β j v 0 ) = f j ((aE β j ) ℓ j v 0 /ℓ j !) with u β j = exp(aE β j ) and a ∈ C.
Proof. First, note that the polytope ∆ i,m contains the origin since v β (ϕ 0 ) = 0. 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 i , L) for some very ample line bundle L of Z i . By Theorem 2.1, the line bundle L equals L i,m for some m ∈ Z r >0 . We thus write ω = ω m .
Recall the definition of the T -stable 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 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 L λ denote the ample line bundle on O λ associated to λ. The pull-back via the morphism π r : Z i → O λ of L λ being generated by its global sections, it is equal to L i,m for some m ∈ Z r ≥0 by Theorem 2.1. Note that the construction of the Newton-Okounkov body ∆ i,m = ∆(Z i , L i,m , v β , τ ) can also be performed for L 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 ∆(O λ , L λ , v β , τ λ ) as the Newton-Okounkov body ∆ 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 T -stable 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 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(λ), −α ∨ ) = L λ · C α = ω m (C j ) hold (see e.g. [BK21]). This enables to conclude the proof.

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 [HLS21]). 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. 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 Since the Picard group of B j−1 is isomorphic to Z j−1 for any j = 1, . . . , m, each line bundle L Note that u 0 1 , . . . , u n 1 1 , . . . , u 0 m , . . . , u nm m (∈ Z n ) are ray generators. Given k = (k 1 , . . . , k m ) with 0 ≤ k l ≤ n l for 1 ≤ l ≤ m, let C k = Cone u k l : 1 ≤ l ≤ m, 0 ≤ k ≤ n l , k = k l be the n-dimensional cone in R n generated by all u k l but the u k l l 's. Proposition 5.1. Let Σ be the fan in R n whose maximal cones consist of the cones C k , k ∈ Since B m is a toric manifold, H 2 (B m , R) is isomorphic to the R-span of the torus stable prime divisors D k l of B m , the latter corresponding to the ray generators u k l of the fan Σ. Given [ω] ∈ H 2 (B m , R), by abuse of notation, we thus write 5.2. Let us recall the combinatorial description of the minimal rational curves on complete toric varieties obtained in [CFH14].
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.
The equation x 1 + · · · + x k − a 1 y 1 − . . . − a m y m = 0 is called the primitive relation of P. The degree of P is (1) There is a bijection between minimal rational components of degree k on X and primitive collections P = {x 1 , . . . , 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.
The idea of the proof of Theorem 5.4. For a given family K of minimal rational curves on X of degree k, there exists a torus invariant open subset U ⊂ X such that U ≃ (C * ) n+1−k × P k−1 (see [CFH14,Corollary 2.6]) such that the lines in the factor P k−1 give general members of K. The fan defining U is the fan of P k−1 viewed as a fan in 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 P = {x 1 , . . . , x k } is a primitive collection of X such that x 1 + · · · + x k = 0. Now consider the subfan Σ ′ of Σ given by the collection of cones in Σ generated by the subsets of P. Then the toric variety U Σ ′ associated with Σ ′ is an open subset of X and U Σ ′ ≃ (C * ) n+1−k × P k−1 . Let C k be a line in the factor P k−1 . Then the deformations of C k give a minimal family of rational curves in X of degree k. 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 K l of minimal rational curves corresponds to a primitive collection P l with u(l) = 0. Given such a primitive collection P l , consider the subfan Σ ′ of Σ given by the collection of cones in Σ generated by the subsets of P l . Then the fan Σ ′ is isomorphic to the fan of P n l (see the proof of Theorem 5.4). To any primitive relation u(l) = 0 viewed in the fan of P n l , we can associate two maximal cones in Σ ′ , say σ and σ ′ , such that the intersection τ = σ ∩ σ ′ is a cone of codimension 1 in Σ ′ . Let C l be the torus invariant curve in P n l associated to τ . Note that C l is isomorphic to the projective line P 1 (see [CLS11, Section 6.3, p.289]). Then the family K l associated to P l is obtained by deformation of the curve C l (see the proof of Theorem 5.4).
By [CLS11, Proposition 6.4.1], we see that the intersection number L · R(l) equals λ(l). Finally, since L · C = L · C ′ for any C, C ′ ∈ K l , the proof follows.
Corollary 5.7. w G (X, L) = min{L · C : C ⊂ X is a minimal rational curve }.
Proof. This follows readily from Theorem 5.2 and Proposition 5.6. 5.4. For some appropriate choice of (i, m), Bott-Samelson varieties Z i equipped with an ample line bundle L i,m can be degenerated into Bott manifolds using the theory of Newton-Okounkov bodies. These toric degenerations were derived in [HY18] from some previous constructions of Grossberg and Karshon (see [GK94] and [Pas10] 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 The polytope P i,m consists of the points (x 1 , . . . , x r ) ∈ 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 (P-k) when the following holds: If (x k+1 , . . . , x r ) satisfies the inequalities . . .
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 [HY15] for details and original references.
Theorem 5.9. Let m ∈ Z r >0 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 and equipped with the ample line bundle Theorem 5.10 ([HY18, Theorem 3.4]). Let m ∈ Z r >0 and suppose that (i, m) satisfies condition (P). Then the polytope P i,m is a Newton-Okounkov 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 [HY18], under the condition (P ) the polytope P i,m coincides with the generalized string polytope introduced in [Fuj18]; the latter turns out to be unimodular to the polytope ∆ i,m we are considering in Section 4 (by [Fuj18,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 i , L i,m ) whenever the pair (i, m) satisfies the condition (P).
Corollary 5.12. Let m ∈ Z r >0 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 ω G (Z i , L i,m ) = ω G (X(P i,m )) = min{m j : α i k , α ∨ i j = 0 ∀ k > j} = min{L i,m · C j : C j line of Z i }.
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 i , L 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 P j = {e 1 j , u 0 j } with e 1 j + u 0 j = 0. By (5.3), it is clear that e 1 j + u 0 j = 0 if and only if α i k , α ∨ i j = 0 ∀ k > j.
(5.5) Let C j ∈ K j be a curve of X(P i,m ) defined by the primitive relation e 1 j + u 0 j = 0. Then by [CLS11, Proposition 6.4.1], M i,m · C j = a j and in turn, M i,m · C j = m j thanks to (5.4) and (5.5).
The curve C j is a line if and only if L k · C j = 0 for all j < k and this is equivalent to the assertion that s i j commutes with s i j+1 , . . . , s ir ; see Remark 4.2 in [BK21]. Besides, by Proposition 2.2(3), L k · C j = 0 for all j > k. It follows L i,m · 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.