FLAGIFIED BOTT MANIFOLDS AND THEIRMAXIMAL TORUS ACTIONS
岡山理科大学理学部応用数学科 黒木 慎太郎
Shintarô Kuroki
Dep. of Applied Mathematics Faculty of Science, Okayama Uniervsityof Science

1. Introduction –Why flagified Bott manifolds?–
This article is a research announcement of the part of the resultsin the progress paper [KLSS] about geometry, combinatorics and algebra (representation theory), so-called Bott’s triangle, of flagified Bottmanifolds.
In this paper, we introduce the following theorem.
Theorem 1.1. The natural torus action on a flagified Bott manifold
is the maximal torus action preserving its complex structure.
We first introduce the motivation of flagified Bott manifolds. A Bottmanifold, defined by Grossberg-Karshon in [GK], is “topologically”
obtained as the iteration of complex projectivization of Whitney sumof two complex line bundles. More precisely, a Bott manifold Bn is asequence
Bn → Bn−1 → · · · → B1 → B0 = {pt}of manifolds Bj = P(Ej ), where Ej is a Whitney sum of two complex
line bundles over Bj−1 and P(Ej ) is its projectivization, i.e., Bn isdiﬀeomorphic to the iterated CP 1 -bundles. We call {Bj | j = 0, . . . , n}a Bott tower of height n (or an n-stage Bott tower). On the other hand,
Grossberg-Karshon also introduce the complex manifold by using thenotion of “Lie theory”, called a Bott-Samelson variety. Here, a BottSamelson variety is defined as follows. Let G be SL(r + 1, C), andB be its Borel subgroup. Let W be the Weyl group of G, which isisomorphic to the symmetric group Sr+1 . Fix the generators (simplereflections) of W (≃ Sr+1 ), and choose some sj , j = 1, . . . , n, fromthe generators. The following submanifold X(s1 , . . . , sn ) ⊂ (G/B)n ,determined by these generators, is called a Bott-Samelson variety:X(s1 , . . . , sn )−1
= {(g1 B, g2 B, . . . , gn B) | gj−1
gj ∈ Bsj B, for all j = 1, . . . , n},1where g0 = e. A Bott manifold is also obtained by deforming thecomplex structure of a Bott-Samelson variety. Using this deformation,
Grossberg-Karshon studies the interesting relations among geometry,combinatorics and algebra of Bott manifolds, so-called Bott’s triangle.
Apart from the original motivation, toric topologists generalize Bottmanifolds from topological point of view in [CMS, KS14, KS15] as aneﬀective testing ground for cohomological rigidity problems. Their definitions are just changed the assumptions of complex vector bundles Ej
in the (topological) definition of Bott manifolds. Namely, if we change
the assumption of each Ej to any Whitney sum of complex line bundles then Bn is called a generalized Bott manifolds in [CMS], and if wechange that to any complex vector bundles then Bn is called a complex
projective towers (or CP -tower) in [KS14, KS15]. Both manifolds are
diﬀeomorphic to the iterated complex projective bundles. However, it
seems to be hard to find the (natural) counterparts corresponding toBott-Samelson variety for these topological generalizations. Therefore,
from the point of view of studying Bott’s triangle, these topologicalgeneralizations are not so eﬀective. So in this paper, we introduce
more natural generalization of Bott manifolds (from the point of viewof studying Bott’s triangle).
In our definition in Section 2, the obtained topological space is theiterated flag bundles instead of the iterated complex projective bundles, so we call it a flagified Bott manifold. We also define a flagified
Bott-Samelson variety as a generalization of a Bott-Samelson variety.
Moreover, we show that the flagified Bott manifold has the structureof a GKM manifold in the sense of Guilleminn-Zara [GZ]; however, aflagified Bott-Samelson variety with the natural torus action does nothave the structure of a GKM manifold though they are diﬀeomorphic.
In Section 3, we give an outline of the proof of Theorem 1.1 by using
the invariant defined in the paper [Ku].

2. Flagified Bott manifolds
We first introduce the definition of flagified Bott manifolds. To dothat, we prepare the general notations. Let F be a (smooth, compact,connected) manifold and E be a complex vector bundle over F . Let
Ep ⊂ E be the fibre over p ∈ F . The associated flag bundle F lag(E) →F is obtained from E by replacing each fiber Ep by the flag manifoldF lag(Ep ), i.e., the set of full flags {0} ⊂ V1 ⊂ V2 · · · ⊂ Vn = Ep in Ep ,where Vi is a complex i-dimensional vector subspace in Ep (≃ Cn ) (Cf.
the definition of projectivization of complex vector bundles).
2Definition 2.1. A flagified Bott tower {Fj | j = 0, . . . , m} of heightm (or an m-stage flagified Bott tower) is a sequence:
Fm → Fm−1 → · · · → F1 → F0 = {pt},of manifolds Fj = F lag(Ej ), where Ej is a complex vector bundle over
Fj−1 which splits into complex line bundles (we often denote Ej =nj +1⊕i=1
ξi,j by a complex line bundle ξi,j over Fj−1 ). We call Fj the jstage flagified Bott manifold of the flagified Bott tower. In particular,
we call Fm a flagified Bott manifold.
2.1. Properties of flagified Bott manifolds. Because each line bundle over Fj−1 is a torus equivariant bundle and the flag manifold has
the natural torus action, there is an (n1 + · · · + nm )-dimensional torus
action on Fm which preserves the complex structure. Moreover, bydefinition, it is easy to check that the flagified Bott manifold Fm isequivariantly diﬀeomorphic to the following twisted product:Fm ∼=m∏j=1

P U (nj + 1)/m∏

T nj

j=1:= P U (n1 + 1) ×T n1 (P U (n2 + 1) ×T n2 (· · · × P U (nm + 1)/T nm ) · · · ),where P U (n+1) ≃ SU (n+1)/Zn+1 , i.e., the quotient group of SU (n+1) by its center, T nj ⊂ P U (nj + 1) is the maximal torus and T nj action on the latter factors, i.e., the action on P U (nj+1 + 1) × · · · ×
P U (nm + 1) factor, is determined by the structure of fibres. Note thata flag manifold is a GKM manifold (see [GZ] for details), i.e., the oneskeleton of orbit space of torus action has the structure of a graph;more precisely, zero-dimensional orbits are regarded as vertices andinvariant 2-spheres consisting of one-dimensional orbits are regarded asedges in the orbit space. By using this fact and the above equivariant
diﬀeomorphism, we have the following proposition.
Proposition 2.2. The flagified Bott manifold is a GKM manifold.
2.2. Flagified Bott-Samelson varieties. There is the Bott-Samelson
variety counterpart for the flagified Bott manifolds, called a flagified
Bott-Samelson variety. Let us briefly introduce it (see the upcoming
paper [KLSS] for details). We first choose an element wj ∈ W from theWeyl group of G for j = 1, . . . , n. Moreover, we assume that each wj isthe longest element of some Weyl subgroup Wj in W , i.e., Wj is generated by some (fixed) generators in W . Then, the following subvariety
3X(w1 , . . . , wn ) ⊂ (G/B)n is called a flagified Bott-Samelson variety:X(w1 , . . . , wn )−1= {(g1 B, g2 B, . . . , gn B) | gj−1
gj ∈ Bwj B, for all j = 1, . . . , n},where g0 = e. If we choose each wj as the simple reflection sj , thenthis is nothing but the Bott-Samelson variety. In the above case, it
is known that X(wj ) is the flag manifold whose dimension depends
on the length of wj ; for example, X(sj ) is CP 1 . Therefore, by iterating the projections from the 1st factor of X(w1 , . . . , wn ), we see thatX(w1 , . . . , wn ) is the iterated flag bundle. In particular, we have thefollowing fact:
Proposition 2.3. The flagified Bott-Samelson variety X(w1 , . . . , wn ) isa smooth manifold.
In [KLSS], we show that X(w1 , . . . , wn ) is diﬀeomorphic to a flagified
Bott manifold. However, the natural complex structure induced from(G/B)n (i.e., regarding as a complex submanifold of (G/B)n ) is diﬀerent from that of the flagified Bott manifold induced from the flagified
bundles (i.e., regarding as a CP -tower, see Section 2.3); in [KLSS],
we also construct a deformation of complex structures from a flagified
Bott-Samelson variety to a flagified Bott manifold.
2.3. Relations among the generalizations of Bott manifolds. Asa final part of this section, we remark the relations among the generalizations of Bott manifolds. Due to the examples in [KS15, Section 2],
the flag manifold may be regarded as a CP -tower. From this observation, we also know that the flagified Bott manifold is also a CP -tower.
Moreover, the lowest dimension of flag manifolds is the one-dimensional
complex projective space CP 1 . This shows that the intersection of theset of flagified Bott manifolds and the set of generalized Bott manifolds is nothing but the set of Bott manifolds. Therefore, there is therelations in Figure 1 among the manifolds which are generalizations ofBott manifolds.
3. Outline of the proof of main theorem
We next give an outline of the proof of Theorem 1.1, see [KLSS]
for details. Recall that we can define a free abelian group with finite
rank A(Γ, α, ∇) for the GKM graph (Γ, α, ∇) in the paper [Ku], called agroup of axial functions. We have the following fact (see [Ku, Corollary3.1]):
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CP-tower

ﬂagiﬁedgeneralized

Bott

Bott

Bottmanifolds

Figure 1. The relations among the generalizations ofBott manifolds.
Theorem 3.1. Let (ΓM , αM , ∇M ) be the GKM graph induced from the2m-dimensional GKM manifold with n-dimensional torus T n -action.
If rank A(ΓM , αM , ∇M ) = n, then this T n -action is the maximal torus
action which preserving the almost complex structure.
Therefore, because
the flagified Bott manifold M = Fm defined in∑
Section 2.1 is a mn
j=1 j (nj + 1)-dimensional GKM manifold with thenatural (n1 + · · · + nm )-dimensional torus action, it is enough to checkthat its induced GKM graph (Γm , αm , ∇m ) satisfies thatrank A(Γm , αm , ∇m ) = n1 + · · · + nm .
For the flagified Bott manifold Fm , because its sequence of flag bundles
is equivariant, it is easy to show that there is the following sequence ofGKM fibrations (see [GSZ] for details) on its induced GKM graph:(Γm , αm , ∇m ) → (Γm−1 , αm−1 , ∇m−1 ) → · · · → (Γ1 , α1 , ∇1 ),where (Γ1 , α1 , ∇1 ) is the GKM graph of the n1 (n1 + 1)-dimensional flag
manifold with T n1 -action with the fixed connection, i.e., the connection
induced from the invariant complex structure on SL(n1 + 1, C)/B. In
this paper, we only give the outline of the proof for the case of m = 2,i.e., 2-stage flagified Bott manifold, because we can easily generalize
the proof for this case to the general m-stage flagified Bott manifold.
We first claim the following fact:
Proposition 3.2. The GKM graph (Γ(n) , α(n) , ∇(n) ) induced from theflag manifold M = SL(n + 1, C)/B satisfies that A(Γ(n) , α(n) , ∇(n) ) ≃Zn .
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Therefore, the T n -action on the flag manifold SL(n + 1)/B is themaximal torus action. Moreover, as a corollary of this proposition, wehave thatA(Γ1 , α1 , ∇1 ) = A(Γ(n1 ) , α(n1 ) , ∇(n1 ) ) ≃ Zn1 .
We next consider (Γ2 , α2 , ∇2 ), the induced GKM graph from F2 ,where the 2-stage flagified Bott manifold F2 satisfies the fibration SL(n2 +1)/B → F2 → F1 = SL(n1 + 1)/B. We can check the graph Γ2 is combinatorially equivalent to Γ(n1 ) × Γ(n2 ) . The following fact is the key
fact to show the main theorem:
Proposition 3.3. Up to weak equivariant diﬀeomorphism on F2 , we may
change the axial function on (Γ2 , α2 , ∇2 ) which satisfies the following
properties: there exists two GKM subgraphs G1 and G2 in (Γ2 , α2 , ∇2 )such that• Gj = (Γ(nj ) , α(nj ) , ∇(nj ) );• the intersection of G1 and G2 is the one vertex.
Note that the maximality of the dimension of torus acting on a manifold is not changed by the weak equivariant diﬀeomorphism, i.e., equivariant diﬀeomorphism up to automorphism on the torus.
By the computation of the group of axial functions, this propositionshows the following lemma:
Lemma 3.4. rank A(Γ2 , α2 , ∇2 ) ≤ rank A(G1 )+rank A(G2 ) = n1 +n2 .
On the other hand, because there is an (n1 +n2 )-dimensional torus action on F2 , it follows from [Ku, Theorem 3.1] that n1 +n2 ≤ rankA(Γ2 , α2 , ∇2 ).
Together with the above lemma, we establish the following equality:
rank A(Γ2 , α2 , ∇2 ) = n1 + n2 .
Consequently, by Theorem 3.1, we have that
Theorem 3.5. The natural (n1 + n2 )-dimensional torus action on F2
is the maximal torus action.
The detailed proofs of the above facts and Theorem 1.1 will be appeared in [KLSS].
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 15K17531.
The author would like to thank Eunjeong Lee for her valuable comments.
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References[CMS] S. Choi, M. Masuda, D. Y. Suh, Topological classification of generalized Bottmanifolds, Trans. Amer. Math. Soc. 362, 2, (2010) 1097–1112.
[GK] M. Grossberg, Y. Karshon, Bott towers, complete integrability, and the extended char- acter of representations, Duke Math. J. 76, 1 (1994), 23–58.
[GSZ] V. Guillemin, S. Sabatini, C. Zara Cohomology of GKM fiber bundles J.
Algebraic Combin. 35 (2012), no. 1, 19–59.
[GZ] V. Guillemin, C. Zara One-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J. 107, 2 (2001), 283–349.
[Ku] S. Kuroki, Upper bounds for the dimension of tori acting on GKM manifolds,
arXiv:1510.07216 (preprint 2016).
[KLSS] S. Kuroki, E. Lee, J. Song, D. Y. Suh, in preparation.
[KS14] S. Kuroki, D. Y. Suh, Complex projective towers and their cohomologicalrigidity up to dimension six, Proc. of Steklov Inst. Math., 2014, 286, 1, 285–307.
[KS15] S. Kuroki, D. Y. Suh, Cohomological non-rigidity of eight-dimensional complex projective towers, Alge. Geom. Top. 15, 2 (2015), 769–782.
Department of Applied Mathematics Faculty of Science, Okayama
Uniervsity of Science, 1-1 Ridai-cho Kita-ku Okayama-shi Okayama 7000005, Okayama, Japan
E-mail address: kuroki@ms.u-tokyo.ac.jp

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