Cohomological rigidity for Fano Bott manifolds
aa r X i v : . [ m a t h . A T ] A ug COHOMOLOGICAL RIGIDITY FOR FANO BOTT MANIFOLDS
AKIHIRO HIGASHITANI AND KAZUKI KURIMOTO
Abstract.
In the present paper, we characterize Fano Bott manifolds up to diffeomor-phism in terms of three operations on matrix. More precisely, we prove that given twoFano Bott manifolds X and X ′ , the following conditions are equivalent:(1) the upper triangular matrix associated to X can be transformed into that of X ′ bythose three operations;(2) X and X ′ are diffeomorphic;(3) the integral cohomology rings of X and X ′ are isomorphic as graded rings.As a consequence, we affirmatively answer the cohomological rigidity problem for FanoBott manifolds. Introduction
Cohomological rigidity problem.
A toric variety is a normal complex algebraicvariety with an algebraic torus of a C ∗ -torus which has an open dense orbit. In whatfollows, we call a compact smooth toric variety a toric manifold .A fundamental fact on toric varieties claims that each toric variety one-to-one cor-responds to a combinatorial object, called a fan . Moreover, the compactness and thesmoothness of toric varieties can be interpreted in terms of the associated fans. It is wellknown that the set of toric manifolds up to algebraic varieties one-to-one corresponds tothe set of complete nonsingular fans up to unimodular equivalence. Moreover, it is alsoknown by Batyrev [2] that toric varieties are isomorphic as algebraic varieties if and only ifthere is a bijection between primitive collections which preserves their associated primitiverelations (See Subsection 2.2 for the details of them). In particular, the classification oftoric manifolds as algebraic varieties is completely done by using combinatorial objects.However, the classification of toric manifolds as smooth manifolds is not establishedyet. Inspired by a classification of a certain class of Bott manifolds up to diffeomorphismin [13], the following naive problem was proposed: Problem 1.1 (cf. [14], Cohomological rigidity problem for toric manifolds) . Are twotoric manifolds diffeomorphic (or homeomorphic) if their integral cohomology rings areisomorphic as graded rings?
We denote the integral cohomology ring H ∗ ( X ; Z ) of a toric manifold X by H ∗ ( X ). Inwhat follows, we omit “integral” of integral cohomology.We say that a family F of toric manifolds is cohomologically rigid if any two toricmanifolds in F whose cohomology rings are isomorphic as graded rings are diffeomorphic(or homeomorphic). Mathematics Subject Classification.
Primary 57R19; Secondary 14M25, 14J45, 57S15.
Key words and phrases.
Cohomological rigidity, toric Fano manifold, Bott manifold, o counterexample of the cohomological rigidity problem for toric manifolds is known.Towards the solution of Problem 1.1, many results have been obtained, all of which affir-matively answer the problem. Many results are related to Bott manifolds ([4, 5, 6, 7, 8,9, 13, 14]). Those will be explained below.1.2. Bott manifolds and toric Fano manifolds.
The main object of the present paperis Bott manifolds. Note that if B is a toric manifold and E is a Whitney sum of complexline bundles over B , then the projectivization P ( E ) of E is again a toric manifold. Startingwith B as a point and repeating this construction, say d times, we obtain a sequence oftoric manifolds as follows: B d π d −→ B d − π d − −−−→ · · · π −→ B π −→ B = { a point } , where the fiber of π i : B i → B i − for i = 1 , . . . , d is a complex projective space C P n i . Thissequence is called a generalized Bott tower of height d , and we call B d a d -stage generalizedBott manifold . We say that B d is d -stage Bott manifold (omitted “generalized”) when n i = 1 for every i .Bott manifolds are very well-studied objects in the area of toric topology. In fact, thereare many results on cohomological rigidity problem for (generalized) Bott manifolds. Thefollowing theorems are a part of the known results on cohomological rigidity problem fortoric manifolds concerning with (generalized) Bott manifolds. Theorem 1.2 ([9, Theorem 1.1]) . Let X be a d -stage generalized Bott manifold. If H ∗ ( X ) is isomorphic to H ∗ ( Q di =1 C P n i ) as graded rings, then every fibration is topologically triv-ial; in particular, X is diffeomorphic to Q di =1 C P n i . Theorem 1.3 ([4, 9]) . The following results are known: • -stage generalized Bott manifolds are cohomologically rigid ( [9, Theorem 1.3] ); • -stage Bott manifolds are cohomologically rigid ( [9, Theorem 1.3] ); • -stage Bott manifolds are cohomologically rigid ( [4, Theorem 3.3] ). Note that Theorem 1.3 is still open for d -stage Bott manifolds with d ≥ Theorem 1.4 ([6, Theorem 1.1]) . Any graded ring isomorphism between the cohomologyrings of two Bott manifolds preserves their Pontrjagin classes.
Many researchers also investigate toric
Fano manifolds. Here, we say that a nonsingularprojective variety is
Fano if its anticanonical divisor is ample. Note that there are onlyfinitely many toric Fano d -folds for each fixed d . The classification problem of toric Fano d -folds is of particular interest in the study of toric Fano manifolds. The classificationof toric Fano 4-folds was accomplished by Batyrev [2] and Sato [17]. The key tool forthis classification is primitive collections and primitive relations (See Subsection 2.2).After their classification, Øbro [15] succeeded in constructing an algorithm, called SFPalgorithm , which produces the complete list of smooth Fano d -polytopes for a given positiveinteger d . For small d ’s, the database of all smooth Fano d -polytopes is available in thefollowing web page: The classification problem of toric Fano manifolds was solved in some sense.By using those classifications, the authors and Masuda proved the following: heorem 1.5 ([11, Theorem 1.1]) . Toric Fano d -folds with d ≤ are cohomologicallyrigid except for two toric Fano -folds. Note that the exceptional toric Fano 4-folds have ID 50 and ID 57 in the above webpage.Furthermore, the following has been also proved in [5] by Cho, Lee, Masuda and Park:
Theorem 1.6 ([5, Thereom 1.2]) . For two Fano Bott manifolds X and X ′ , if there is anisomorphism between their cohomology rings which preserves the first Chern class, then X and X ′ are isomorphic as algebraic varieties. In [12] and [7], cohomological rigidity problem for real Bott manifolds is investigated.Here, we call a manifold X a real Bott manifold if there is a sequence of R P bundles suchthat for each j = 1 , . . . , d , B j → B j − is the projective bundle of the Whitney sum of areal line bundle and the trivial real line bundle over B j − , where B is a point. It is wellknown that any d -stage real Bott manifold is determined by a certain upper triangularmatrix with its entry in Z / { , } , called a Bott matrix . Moreover, in [7, Section 3],three operations for Bott matrices are introduced. We say that two Bott matrices are
Bottequivalent if those can be transformed by those three matrix operations.The following theorem gives a complete characterization of real Bott manifolds in termsof Bott matrices and answers the cohomological rigidity problem for real Bott manifoldsin some certain sense.
Theorem 1.7 ([7, Theorem 1.1], Classification of real Bott manifolds) . Let A and B beBott matrices and let X ( A ) and X ( B ) be the associated real Bott manifolds to A and B ,respectively. Then the following three conditions are equivalent: (1) A and B are Bott equivalent; (2) X ( A ) and X ( B ) are affinely diffeomorphic; (3) H ∗ ( X ( A )) ⊗ Z Z / and H ∗ ( X ( B )) ⊗ Z Z / are isomorphic as graded rings. Note that the equivalence of (2) and (3) was originally proved in [12, Theorem 1.1].Our motivation to organize the present paper is to obtain an analogue of Theorem 1.7for Fano Bott manifolds.1.3.
Main Result.
The main result of the present paper is the following:
Theorem 1.8 (Main Result) . Let X and X ′ be Fano Bott manifolds and let A ( X ) and A ( X ′ ) be the upper triangular matrices associated to X and X ′ , respectively. Then thefollowing three conditions are equivalent: (1) A ( X ) and A ( X ′ ) are Fano Bott equivalent; (2) X and X ′ are diffeomorphic; (3) H ∗ ( X ) and H ∗ ( X ′ ) are isomorphic as graded rings. See Section 5 for the precise definition of A ( X ) and Fano Bott equivalence. Note thatthose three conditions are equivalent to the fourth condition (4). See Remark 6.4.As an immediate corollary of Theorem 1.8, we conclude the following: Corollary 1.9.
Fano Bott manifolds are cohomologically rigid.
Note that in the case d ≤
4, the cohomological rigidity for d -stage Bott manifolds isalready known by [4, Theorem 3.3] and [4, Theorem 3.3] as well as [11, Theorem 1.1].The key idea of the proof of Theorem 1.8 relies on the identification of Fano Bottmanifolds with rooted signed forests (see Section 4). .4. Structure of the paper.
An overview of the present paper is as follows. In Sec-tion 2, we recall the theory of toric varieties (e.g. primitive collections and primitiverelations) and the description of cohomology rings of toric manifolds. We also introducethe invariants (s.v.e. and maximal basis number) on cohomology rings. We also give asufficient condition (Lemma 2.7) for two Bott manifolds to be diffeomorphic. In Section 3,we introduce the upper triangular matrix A ( X ) arising from a Fano Bott manifold X anddescribe the cohomology ring H ∗ ( X ) of X in terms of A ( X ). We also discuss s.v.e. andmaximal basis number of H ∗ ( X ) (Lemma 3.3). In Section 4, we associate signed rootedforests T X from Fano Bott manifolds X and we prove a key proposition for the proof ofTheorem 1.8 (Proposition 4.5). In Section 5, we introduce three operations on the uppertriangular matrices associated to Fano Bott manifolds and the notion of Fano Bott equiv-alence. We also see that those operations correspond to certain operations on the signsof some edges of the signed rooted forest (Proposition 5.7). Finally, in Section 6, afterpreparing some more lemmas, we give a proof of Theorem 1.8. Acknowledgements.
The authors would like to thank Mikiya Masuda for fruitful dis-cussions about cohomological rigidity problems for toric manifolds. He also gave a lot ofhelpful comments on the first version of the present paper. A. Higashitani was supportedin part by JSPS Grant-in-Aid for Scientific Research (C) 20K03513.2.
Preliminaries
In this section, we recall the well-known description of the cohomology rings of toricmanifolds. For our discussions, we recall the notions, primitive collections and primitiverelations , which represent the toric variety in terms of linear relations of primitive rayvectors of the associated fan of a toric variety. We also introduce the invariants of coho-mology rings to distinguish cohomology rings. At last, we give a sufficient condition fortwo toric manifolds to be diffeomorphic (Lemma 2.7).2.1.
Complete nonsingular fans and toric manifolds.
Please consult e.g. [10] or [16]for the introduction to toric varieties and the associated fans.First, we recall the notion of complete nonsingular fans and their underlying simplicialcomplexes. A fan of dimension d is a collection Σ of rational polyhedral pointed cones in R d such thati) each face of a cone σ in Σ also belongs to Σ, andii) the intersection of two cones in Σ is also a face of each of those cones.It is well known that the set of toric varieties of complex dimension d up to algebraicisomorphism one-to-one corresponds to the set of fans of dimension d up to unimodularequivalence. For a toric variety X , let Σ X be the corresponding fan. Similarly, for a fanΣ, let X Σ be the corresponding toric variety. It is known that a toric variety X is compactif and only if Σ X is complete, i.e., S σ ∈ Σ X σ = R d , and X is smooth if and only if Σ X isnonsingular, i.e., the set of primitive ray generators of every maximal cone in Σ X forms a Z -basis of Z d . Hence, the set of toric manifolds (namely, compact smooth toric varieties)one-to-one corresponds to the set of complete nonsingular fans.For a fan Σ, let Σ be the set of primitive ray generators of 1-dimensional cones in Σ.Given a complete nonsingular fan Σ, since nonsingular fans are simplicial, we see that Σhas a structure of an abstract simplicial complex. Let Σ = { v , . . . , v m } . We define K (Σ) = { I ⊂ [ m ] : cone( v i : i ∈ I ) ∈ Σ } , here [ m ] = { , . . . , m } . We call K (Σ) the underlying simplicial complex of Σ.2.2. Primitive collections and primitive relations.
Next, we recall what primitivecollections and primitive relations are.
Definition 2.1 (Primitive collection, [1, Definition 2.6]) . For a complete nonsingular fanΣ, we call a nonempty subset P = { x , . . . , x k } ⊂ Σ of Σ a primitive collection of Σ iffor each generator x i ∈ P the elements of P \ { x i } generate a ( k − P does not generate any k -dimensional cone in Σ. In other words, P is a minimalnon-face of K (Σ). Let PC(Σ) be the set of all primitive collections of Σ. Definition 2.2 (Primitive relation, [1, Definition 2.8]) . For a complete nonsingular fanΣ, let P = { x , . . . , x k } be a primitive collection of Σ. Let σ be the cone in Σ of thesmallest dimension containing x + · · · + x k and let y , . . . , y m ∈ Σ be the minimalsystem of primitive ray generators of σ . Then there exists a unique linear combination n y + · · · + n m y m with positive integer coefficients n i which is equal to x + · · · + x k . Wecall the linear relation x + · · · + x k = n y + · · · + n m y m (2.1)the primitive relation associated with P .Given a primitive collection P with its associated primitive relation (2.1), let deg( P ) = k − ( n + · · · + n m ), which we call the degree of P .We can verify whether the toric manifold X is Fano or not by the degrees of primitiverelations of Σ X . Proposition 2.3 ([2, Proposition 2.3.6]) . A toric manifold X is Fano if and only if wehave deg( P ) > for every primitive collection P of Σ X . Given two complete nonsingular fans Σ and Σ ′ , we say that PC(Σ) and PC(Σ ′ ) areisomorphic if there is a bijection between Σ and Σ ′ which induces a bijection betweennot only PC(Σ) and PC(Σ ′ ) but also their primitive relations. Proposition 2.4 ([2, Proposition 2.1.8 and Theorem 2.2.4]) . Two toric Fano manifolds X and X ′ are isomorphic as algebraic varieties if and only if PC(Σ X ) and PC(Σ X ′ ) areisomorphic. Cohomology rings of toric manifolds and their invariants.
Next, we recallthe description of the cohomology rings of toric manifolds.
Proposition 2.5 ([3, Theorem 5.3.1]) . Let X be a toric manifold of complex dimension d , let Σ = Σ X and let Σ = { v , . . . , v m } ⊂ R d . Then the cohomology ring of X can bedescribed as follows: H ∗ ( X ) ∼ = Z [ x , . . . , x m ] / ( I X + J X ) , where I X = Y i ∈ F x i : F ⊂ [ m ] , { v i : i ∈ F } ∈ PC(Σ X ) ! and J X = m X j =1 v ij x j : i = 1 , . . . , d and v ij denotes the i -th entry of v j ∈ R d . In order to distinguish cohomology rings up to isomorphism, we prepare the invariantson cohomology rings. efinition 2.6 (s.v.e., maximal basis number) . Let X be a toric manifold.A nonzero linear form in H ∗ ( X ) is said to be s.v.e. (square vanishing element) if it isprimitive and its second power vanishes in H ∗ ( X ).Let V be the set of all s.v.e. of the cohomology ring H ∗ ( X ) of X . Define B = { S ⊂ V : S is a part of a Z -basis of H ∗ ( X ) } . Then there exists S max ∈ B such that | S | ≤ | S max | for any S ∈ B . We call a set S max a maximal basis of s.v.e. of H ∗ ( X ) and | S max | a maximal basis number of H ∗ ( X ).Note that the number of s.v.e. and the maximal basis number are invariants of H ∗ ( X ).We refer the readers to [11, Examples 2.7 and 2.9] for examples of s.v.e and maximal basisnumbers of H ∗ ( X ) and how to compute them.2.4. Diffeomorphism lemma.
Finally, we recall the key lemma for the proof of Theo-rem 1.8. The following lemma directly follows from [11, Lemma 2.3].
Lemma 2.7 (cf. [11, Lemma 2.3]) . Let
X, X ′ be d -stage Fano Bott manifolds and let Σ , Σ ′ be the associated complete nonsingular fans, respectively. Let Σ = { v , . . . , v d } and let Σ ′ = { v ′ , . . . , v ′ d } . If v i = ± v ′ i for each i = 1 , . . . , d by reordering v i ’s if necessary suchthat K (Σ) is unchanged, then X and X ′ are diffeomorphic. Fano Bott manifolds and their cohomology rings
In this section, we introduce upper triangular matrices ( n ij ) associated to Fano Bottmanifolds. The cohomology rings of Fano Bott manifolds can be described by using suchmatrices (see (3.3)). We discuss the s.v.e. of their cohomology rings (Lemma 3.3). Werefer the reader to [3, Section 7.8] for the introduction to Bott manifolds.3.1. Upper triangular matrices associated to Fano Bott manifolds.
Let X be a d -stage Fano Bott manifold and let Σ = Σ X . Then | Σ | = 2 d . Let Σ = { v ± i : i = 1 , . . . , d } .In general, the underlying simplicial complex of the associated fan of any Bott manifold isthe boundary complex of the cross-polytope of dimension d . Thus, the primitive collectionslook like { v + i , v − i } for each i . Moreover, by Proposition 2.3, the associated primitiverelations of PC(Σ) look as follows: v +1 + v − = v σ (1) ϕ (1) , v +2 + v − = v σ (2) ϕ (2) , ... v + d + v − d = v σ ( d ) ϕ ( d ) , (3.1)where we let v ± d +1 = , ϕ is a map ϕ : [ d ] → [ d + 1] \ { } satisfying that there is k i with ϕ k i ( i ) = d + 1 for each i ∈ [ d ] and σ is a map σ : [ d ] → {±} . Note that the condition“there is k i with ϕ k i ( i ) = d + 1 for each i ∈ [ d ]” is derived from the linear independence of v ǫ , . . . , v ǫ d d for any choice of ǫ i ∈ {±} . Up to the equivalence of PC(Σ), we may assumethat ϕ satisfies that i < ϕ ( i ) ≤ d + 1 for each i . (3.2)In what follows, we will always assume (3.2). n general, d -stage Bott manifolds are determined by a collection of integers ( n ij ) ≤ i Given a d -stage Fano Bott manifold X , let A ( X ) = ( n ij ) ≤ i,j ≤ d ∈ { , ± } d × d be the upper triangular matrix whose upper triangle part is equal to n ij defined as aboveand whose remaining parts are all 0. Moreover, let F B ( d ) = { A ( X ) ∈ { , ± } d × d : X is a d -stage Fano Bott manifold } . Namely, F B ( d ) consists of all upper triangular matrices satisfying the condition in Theo-rem 3.1.3.2. Cohomology rings of Fano Bott manifolds and their s.v.e. For a d -stageFano Bott manifold X , by using the upper triangular matrix ( n ij ) associated to X , thecohomology ring H ∗ ( X ) can be written as follows (see, e.g., [5, (2.5)]): H ∗ ( X ) ∼ = Z [ x , . . . , x d ] / I , where I = ( f i : i = 1 , . . . , d ) and f i = x i − ( n i x + · · · + n i − ,i x i − ) x i for i = 1 , . . . , d. (3.3) or the analysis of s.v.e. of H ∗ ( X ), we perform the following computations. Let a i ∈ Z .Then we have( a x + · · · + a d x d ) = d X i =1 a i x i + X ≤ i Work with the same notation as above. Then S ( H ∗ ( X )) can be dividedinto three disjoint subsets { g i } i ∈ I , { g ′ i } i ∈ I and { h j } j ∈ J with some I, J ⊂ [ d ] satisfying thefollowing: (1) { g i } i ∈ I ′ ∪ { g ′ i } i ∈ I \ I ′ ∪ { h j } j ∈ J is a maximal basis of S ( H ∗ ( X )) for any (possiblyempty) I ′ ⊂ I ; (2) H ∗ ( X ) / ( g i ) ∼ = H ∗ ( X ) / ( g ′ i ) for each i .Proof. (The first step) : Let g = a x + · · · + a d x d ∈ S ( H ∗ ( X )). First, we determinewhat kind of g appears.The case a p = 0 and a i = 0 for any i with i = p : Then g = x p and g = x p = 0.The case a p = 0, a q = 0 and a i = 0 for any i with i = p, q : Let p < q . Then g = 0 im-plies that n pq a q = − a p = 0. Thus, n pq = ± 1, i.e., a q = ± a p . Moreover, n iq a q = − a i = 0 (1 ≤ i = p < q ) ⇒ n iq = 0 (1 ≤ i = p < q ) . Hence, we may assume that n pq = 1, i.e., a q = − a p . Namely, we have ( x p − x q ) = 0.Moreover, in this case, we have n ip a p = − a i = 0 (1 ≤ i < p ) ⇒ n ip = 0 (1 ≤ i < p ) , so we obtain that x p = 0.The case a p = 0, a q = 0 and a r = 0 with p < q < r : Then it follows from (3.4) that n pq a q = − a p = 0 , n pr a r = − a p = 0 and n qr a r = − a q = 0 . Thus, n pq , n pr , n qr ∈ {± } . However, in this case, we have ( a r = ± a q a r = ± a q , implying that a q = a r = 0, a contradiction.Therefore, we see that S ( H ∗ ( X )) ⊂ { x p : 1 ≤ p ≤ d } ∪ { x p − x q : 1 ≤ p < q ≤ d } . The second step) : Let ( x p − x q ) = 0 and ( x p − x q ) = 0. We claim that { p , q } 6 = { p , q } implies { p , q } ∩ { p , q } = ∅ . By the above discussions, we mayassume that n p q = 1 and n p q = 1, and we also have n iq = 0 (1 ≤ i < q , i = p ) , n ip = 0 (1 ≤ i < p ) ,n iq = 0 (1 ≤ i < q , i = p ) , n ip = 0 (1 ≤ i < p ) . Then p = q and p = q never happen. If q = q , then p = p . If p = p , Theorem 3.1(2) implies that q = q since we assume n p q = n p q = 1. Hence, { p , q } ∩ { p , q } = ∅ holds if { p , q } 6 = { p , q } (The third step) : Now, we consider the following subsets of S ( H ∗ ( X )): { g i } i = { x p : ( x p − x q ) = 0 for some q } , { h j } j = { x p : x p = 0 } \ { g i } i , { g ′ i } i = { x p − x q : ( x p − x q ) = 0 } . Since x p ∈ { g i } i if and only if x p − x q ∈ { g ′ i } i (see the first step), we can simultaneouslyindex both of the sets { g i } and { g ′ i } by I ⊂ [ d ]. Moreover, it follows from the second stepthat the condition (1) holds.We claim that the condition (2) also holds. More precisely, we prove the ring isomor-phism H ∗ ( X ) / ( x p ) ∼ = H ∗ ( X ) / ( x p − x q ). We may assume that n pq = 1. By Theorem 3.1(2), we have n pi = 0 ( p < i ), and by n iq = 0 (1 ≤ i < q, i = p ), we see that a generator of I which is divisible by x p is either x p or x q ( x q − x p ). Therefore, the ring endomorphism F : H ∗ ( X ) → H ∗ ( X ) defined by F ( x i ) = x i ( i = p ) , F ( x p ) = − x p + 2 x q is well-defined and induces an isomorphism between H ∗ ( X ) / ( x p ) and H ∗ ( X ) / ( x p − x q ). (cid:3) Remark . For a Fano Bott manifold, a maximal basis of S ( H ∗ ( X )) is not unique asclaimed in the condition (1) of Lemma 3.3. However, the condition (2) of Lemma 3.3 saysthat for any maximal basis S max of S ( H ∗ ( X )), { H ∗ ( X ) / ( f ) : f ∈ S max } is unique up toring isomorphisms. Therefore, when we consider the family of quotient rings by s.v.e., wemay only consider s.v.e. of the form x p , namely, we do not need to treat s.v.e. of the form x p − x q .4. Correspondence between Fano Bott manifolds and signed rootedforests In this section, we introduce the notion of signed rooted forests. We construct the signedrooted forest T ϕ,σ from a map ϕ : [ d ] → [ d + 1] \ { } with (3.2) and a map σ : [ d ] → {±} .Namely, we can construct a signed rooted forest T X from a Fano Bott manifold X . Byusing this, we establish a correspondence between Fano Bott manifolds and signed rootedforests T ϕ,σ . We also observe that an operation on a signed rooted forest, called leaf-cutting by v α , corresponds to taking a quotient by a certain element x α .We call T = ( V, E, v ) a rooted tree on the vertex set V with its root v and the edgeset E if ( V, E ) is a tree, which is a connected graph having no cycle, and a vertex v ∈ V ,called a root , is fixed. A rooted forest is a disjoint union of rooted trees. We recall somenotions on a rooted forest T : A vertex v (resp. v ′ ) in T is called a child (resp. parent ) of a vertex v ′ (resp. v )if { v ′ , v } is an edge of a connected component T of T and v ′ is closer to the rootof T than v . • A descendant of a vertex v in T means any vertex which is either a child of v orrecursively a descendant of v . • We call a vertex v in T a leaf if no child is adjacent to v .We call a rooted forest signed if + or − is assigned to each of its edges. Given an edge e ,we denote the sign of e by sign( e ).Let T and T ′ be two rooted forests. We say that T and T ′ are isomorphic as rootedforests if there is a bijection f : V ( T ) → V ( T ′ ) between the sets of vertices which inducesa bijection between the roots and between the sets of edges, and we call f an isomorphismas signed rooted forests if f also preserves all signs of the edges. Definition 4.1 (Signed rooted forests associated to Fano Bott manifolds) . Let ϕ : [ d ] → [ d + 1] \ { } be a map with (3.2) and let σ : [ d ] → {±} . We define the signed rooted forest T ϕ,σ from ϕ and σ as follows: V ( T ϕ,σ ) = { v , . . . , v d } ,E ( T ϕ,σ ) = {{ v ϕ ( i ) , v i } : 1 ≤ i ≤ d, ϕ ( i ) = d + 1 } with sign( { v ϕ ( i ) , v i } ) = σ ( i ) for each i ,the roots are v i ’s with ϕ ( i ) = d + 1 . As mentioned in Section 3, we can associate the above maps ϕ and σ from Fano Bottmanifolds, so we can construct T ϕ,σ from Fano Bott manifolds. Let T X = T ϕ,σ be suchsigned rooted forest. Remark . Given a Fano Bott manifold X , we see that the signed rooted forest T ϕ,σ associated to X constructed in the above way is well-defined. In fact, for two FanoBott manifolds X and X ′ and their associated signed rooted forests T X and T X ′ , it isstraightforward to check that the equivalence between PC(Σ X ) and PC(Σ X ′ ), which isa bijection between (Σ X ) and (Σ X ′ ) , directly implies the isomorphism between twocorresponding signed rooted forests T X and T X ′ , which is a bijection between V ( T X ) and V ( T X ′ ).On the other hand, given a signed rooted forest T , we can reconstruct the primitiverelations associated to T , so we can associate a Fano Bott manifold from T . Let X T bethe Fano Bott manifold associated to a signed rooted forest T . Example 4.3. (1) Let us consider the Fano Bott manifold X associated to the followingprimitive relations: v +1 + v − = v +2 , v +2 + v − = v − , v +3 + v − = v − , v +4 + v − = v +5 , v +5 + v − = . Then the associated signed rooted forest T X looks as follows. v v + − v v − +(2) Let us consider the Fano Bott manifold X associated to the following primitive rela-tions: v +1 + v − = v +3 , v +2 + v − = v − , v +3 + v − = , v +4 + v − = v +5 , v +5 + v − = . Then the associated signed rooted forest T X looks as follows. v v + v − v v + Remark . Let X be a d -stage Fano Bott manifold and let A ( X ) = ( n ij ) be the uppertriangular matrix associated to X . Then Theorem 3.1 claims that A ( X ) is completelydetermined by the left-most nonzero entry of each row. Moreover, the left-most nonzeroentry of each row, which is 1 or − 1, one-to-one corresponds to the signed edge of theassociated signed rooted forest T X . Namely, we see that n i,i +1 = · · · = n i,j − = 0 , n ij = 1 (resp. n ij = − ⇐⇒ { v j , v i } ∈ E ( T X ) and sign( { v j , v i } ) = + (resp. sign( { v j , v i } ) = − ) (4.1)Furthermore, we can read off all entries of ( n ij ) from T X as follows: for 1 ≤ i < j ≤ d , let n ij = , if { v i , v j } ∈ E ( T X ) and sign( { v i , v j } ) = + , , if there is an upward path ( v i , . . . , v i k ) from v i = v i to v j = v i k such thatsign( { v i ℓ − , v i ℓ } ) = − for ℓ = 1 , . . . , k − { v i k − , v i k } ) = + , − , if there is an upward path from v i to v j all of whose edges have the sign − , , otherwise , where we call a path ( v i , v i , . . . , v i k ) in the signed rooted forest upward if v i j is a childof v i j +1 for all j = 1 , . . . , k − 1. For Example 4.3 (1), we see that n = n = n = 1 , n = n = − , n ij = 0 otherwise . Let T , . . . , T k be the connected components of a signed rooted forest T . It then followsfrom the above construction that the upper triangular matrix associated to T becomes adirect sum of those of T , . . . , T k . iven a Fano Bott manifold X , we observe the following: • By definition of T ϕ,σ , we see that v α is a leaf of T ϕ,σ if and only if α ϕ ([ d ]).Thus, for a leaf v α of T ϕ,σ , we obtain that n iα = 0 for each i = 1 , . . . , α − 1, i.e., x α = 0 in H ∗ ( X ). • On the other hand, if x α = 0 in H ∗ ( X ), then n α = · · · = n α − ,α = 0 by (3.4).Thus, we obtain that α ϕ ([ d ]), i.e., v α is a leaf of T X .Therefore, the set of leaves of T X one-to-one corresponds to { x i : x i = 0 in H ∗ ( X ) } .Now, we consider the quotient ring H ∗ ( X ) / ( x α ) where x α = 0 in H ∗ ( X ). Then we have n iα = 0 for each i = 1 , . . . , α − 1. By H ∗ ( X ) / ( x α ) ∼ = Z [ x , . . . , x d ] / ( I X +( x α )), where I X = I X + J X in Proposition 2.5, we see that H ∗ ( X ) / ( x α ) ∼ = Z [ x , . . . , ˆ x α , . . . , x d ] / I X , where I X is the image of I X by the natural projection Z [ x , . . . , x d ] → Z [ x , . . . , ˆ x α , . . . , x d ]. Then Z [ x , . . . , ˆ x α , . . . , x d ] / I X is isomorphic to the cohomology ring of a Fano Bott manifoldwhose primitive relations are v +1 + v − = v σ (1) ϕ (1) , ... v + α − + v − α − = v σ ( α − ϕ ( α − , v + α +1 + v − α +1 = v σ ( α +1) ϕ ( α +1) , ... v + d + v − d = v σ ( d ) ϕ ( d ) , where ϕ : [ d ] \ { α } → [ d + 1] \ { , α } (resp. σ : [ d ] \ { α } → {±} ) is defined by ϕ ( i ) = ϕ ( i )(resp. σ ( i ) = σ ( i )). This implies that T X is isomorphic to T X \ v α as signed rooted forests.Therefore, when x α = 0 or ( x α − x β ) = 0 in H ∗ ( X ), taking the quotient H ∗ ( X ) / ( x α )(which is isomorphic to H ∗ ( X ) / ( x α − x β )) is equivalent to a leaf-cutting by v α (seeRemark 3.4). Moreover, Lemma 3.3 says that for two Fano Bott manifolds X and X ′ , H ∗ ( X ) ∼ = H ∗ ( X ′ ) implies { H ∗ ( X ) / ( x i ) : x i = 0 } ∼ = { H ∗ ( X ′ ) / ( x ′ i ) : x ′ i = 0 } . These meanthat the number of leaves of T X is equal to the maximal basis number of H ∗ ( X ).The following proposition will play a crucial role in the proof of Theorem 1.8. Proposition 4.5. Let X and X ′ be two Fano Bott manifolds. Assume that H ∗ ( X ) ∼ = H ∗ ( X ′ ) and let F : H ∗ ( X ) → H ∗ ( X ′ ) be a ring isomorphism. (1) F induces an isomorphism between T X and T X ′ as rooted forests. (2) Let f : T X → T X ′ be an isomorphism induced by F . Take a rooted subtree T ⊂ T X whose root is the same as that of T X . Then H ∗ ( X T ) ∼ = H ∗ ( X f ( T ) ) .Proof. (1) Since { x α ∈ H ∗ ( X ) : x α = 0 } one-to-one corresponds to the leaves of T X , andsince we may assume that F sends { x α ∈ H ∗ ( X ) : x α = 0 } to { x ′ α ∈ H ∗ ( X ′ ) : x ′ α = 0 } (see the above discussion), we obtain a bijection f between the leaves of T X and those of T X ′ induced by F , as discussed above.Let v l , . . . , v l k be the leaves of T X and let v ′ l ′ i = f ( v l i ) ( i = 1 , . . . , k ). Remark thattaking a quotient by x l i is equivalent to taking a quotient by x ′ l ′ i . We prove the assertionby induction on k .Let k = 1. Then F induces the correspondence v l and v ′ l ′ . Consider F : H ∗ ( X ) / ( x l ) → H ∗ ( X ′ ) / ( x ′ l ′ ). Since F is also an isomorphism and H ∗ ( X ) / ( x l ) (resp. H ∗ ( X ′ ) / ( x ′ l ′ )) orresponds to a rooted tree T X \ v l (resp. T X ′ \ v ′ l ′ ) which contains only one leaf, wealso obtain the correspondence between their leaves, which are the parents of v l and v ′ l ′ ,respectively. By repeating this procedure, we obtain a bijection of all vertices of T X and T X ′ which induces an isomorphism between T X and T X ′ .Let k > 1. Then F gives a bijection between the leaves of T X and T X ′ . Take one leaf v l and consider F : H ∗ ( X ) / ( x l ) → H ∗ ( X ′ ) / ( x ′ l ′ ). Note that F gives a bijection betweenthe leaves of T X \ v l and T X ′ \ v ′ l ′ . Since F is induced from F , we see that F preservesthe bijectivity between { v l , . . . , v l k } and { v ′ l ′ , . . . , v ′ l ′ k } . Note that the number of leaves of T X \ v l is either the same as that of T X or minus one. • When the number of leaves of T X \ v l decreases from that of T X , by the hypothesisof induction, we obtain an isomorphism g : T X \ v l → T X ′ \ v ′ l ′ . Let v (resp. v ′ )be the parent of v l (resp. v ′ l ′ ). Once we can see that g ( v ) = v ′ , (4.2)by combining the correspondence between v l and v ′ l ′ , we obtain an isomorphism T X and T X ′ induced by F . • When the number of leaves of T X \ v l stays the same, we see that new leaves v l ′′ and v l ′′′ appear both in T X \ v l and T X ′ \ v ′ l ′ , respectively. Since F still givesa bijection between { v l , . . . , v l k } and { v ′ l ′ , . . . , v ′ l ′ k } , v l ′′ should correspond to v l ′′′ .Consider H ∗ ( X ) / ( x l , x l ′′ ) and H ∗ ( X ′ ) / ( x ′ l ′ , x ′ l ′′′ ). By the same discussion, we canrepeat this procedure until the number of leaves decreases. Hence, the assertionfollows.Our remaining task is to show (4.2). Note that there is another leaf, say v l , which isa descendant of v . When v l is a child of v , by the hypothesis of induction, we ob-tain an isomorphism h : T X \ v l → T X ′ \ v ′ l ′ . Remark that h gives a bijection be-tween { v l , v l , . . . , v l k } and { v ′ l ′ , v ′ l ′ , . . . , v ′ l ′ k } since h comes from the isomorphism between H ∗ ( X ) / ( x l ) and H ∗ ( X ′ ) / ( x ′ l ′ ). Thus, in particular, h sends v l to v ′ l ′ . Since v (resp. v ′ )is the unique vertex of T X (resp. T X ′ ) adjacent to v l (resp. v ′ l ′ ), we obtain that h ( v ) = v ′ .Even if v l is not a child of v , by removing the leaves until the number of leaves decreases,we obtain the same conclusion. Hence, we see that h ( v ) = g ( v ) = v ′ .(2) Let f : T X → T X ′ be an isomorphism constructed in the above way. For any rootedsubtree T ⊂ T X whose root is the same as that of T X , since H ∗ ( X T ) ∼ = H ∗ ( X ) / ( x α : v α ∈ V ( T X \ T ))and f sends V ( T X \ T ) to V ( T X ′ \ f ( T )) by construction of f , we obtain that H ∗ ( X T ) ∼ = H ∗ ( X f ( T ) ), as required. (cid:3) Three operations on matrices and Fano Bott equivalence In this section, we introduce the equivalence relation in F B ( d ) (see Definition 3.2),which we call Fano Bott equivalence. The notion of Fano Bott equivalence is derived fromBott equivalence defined in [7]. efinition 5.1 (Three operations on F B ( d )) . For A ∈ Z d × d and i ∈ [ d ], let A i (resp. A i )denote the i -th row (resp. column) vector of A . Let e i (resp. e i ) denotes the i -th unitcolumn (resp. row) vector. Given a permutation π on [ d ], we define a permutation matrix P by setting P j = e π ( j ) for each j . (Op ) : For a permutation matrix P corresponding to a permutation π on [ d ], we definea map Φ P : Z d × d → Z d × d by Φ P ( A ) := P AP − for A ∈ Z d × d . Namely, we see that A ij = (Φ P ( A )) π ( i ) π ( j ) . (Op ± ) : For k ∈ [ d ], we define a map Φ ± k : Z d × d → Z d × d as follows: for A ∈ Z d × d , wemultiply − k -th column and add the k -th column times ( k, j )-entry to the j -thcolumn for each j ∈ [ d ] \ { k } . Namely, we see that(Φ ± k ( A )) j = ( − A k , if j = k,A j + A kj A k , otherwise . Notice that Φ ± k is nothing but a kind of unimodular transformation. (Op ± ) : Given A ∈ Z d × d , assume that A ℓ = and A k = ± e ℓ for some k and ℓ . Thenwe define Φ ± k,ℓ ( A ) as follows: we multiply − A kℓ , and multiply A ik to A iℓ if A ik = 0 for i ∈ [ d ] \ { ℓ } , and the other entries stay the same. Namely, we see that(Φ ± k,ℓ ( A )) iℓ = − A kℓ , if i = k,A ik · A iℓ , if A ik = 0 ,A iℓ , otherwise , and (Φ ± k,ℓ ( A )) j = A j for j = ℓ. Definition 5.2 (Fano Bott equivalence) . Given two matrices A and A ′ in F B ( d ), we saythat A and A ′ are Fano Bott equivalent if A can be transformed into A ′ through a sequenceof the three operations (Op1), (Op2 ± ) and (Op3 ± ). Proposition 5.3. Let A ∈ F B ( d ) . Then Φ ± k ( A ) ∈ F B ( d ) for any k ∈ [ d ] , and Φ ± k,ℓ ( A ) ∈F B ( d ) for any k, ℓ ∈ [ d ] satisfying A ℓ = and A k = e ℓ .Proof. Given A ∈ F B ( d ), we prove that Φ ± k ( A ) and Φ ± k,ℓ ( A ) satisfy the conditions inTheorem 3.1.For each p ∈ [ d ], it follows from Theorem 3.1 that we have A p = , or A p = e q for some q with p < q ≤ d , or A p = − e q + A q for some q with p < q ≤ d .Let A p = . Then we easily see that (Φ ± k ( A )) p = (Φ ± k,ℓ ( A )) p = .Let A p = e q for some q with p < q ≤ d . • We see that (Φ ± k ( A )) p = e q if k = q and (Φ ± k ( A )) p = − e q + A q if k = q . • Note that (Φ ± k,ℓ ( A )) ℓ = . We see that (Φ ± k,ℓ ( A )) p = e q if q = ℓ . When q = ℓ , if p = k then A pk = 0. If p = k , then (Φ ± k,ℓ ( A )) p = − e q , so the assertion holds.Let A p = − e q + A q for some q with p < q ≤ d . Then A pj = A qj holds for any j = q . • We see that (Φ ± k ( A )) p = − e q + (Φ ± k ( A )) q if k = q and (Φ ± k ( A )) p = e q if k = q . • If q = ℓ , then A q = A ℓ = . Thus, (Φ ± k,ℓ ( A )) p = − e q (resp. = e q ) when p = k (resp. p = k ). If q = ℓ , then we see that (Φ ± k,ℓ ( A )) p = − e q + (Φ ± k,ℓ ( A )) q . (cid:3) emark that (Op1) does not necessarily preserve F B ( d ), while this corresponds to anisomorphism of the associated signed rooted forests. Example 5.4. Let us consdier A = − − 10 0 0 0 0 − 10 0 0 0 − . Then it is straightforwardto check that A ∈ F B (6). Moreover, we see the following:Φ ± ( A ) = − − 10 0 1 0 0 00 0 0 0 0 − 10 0 0 0 − , Φ ± ( A ) = − − 10 0 0 0 0 − 10 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0 , Φ ± , ( A ) = − − , Φ ± , ( A ) = − − 10 0 0 0 0 − 10 0 0 0 − − 10 0 0 0 0 − 10 0 0 0 0 0 . Let X be d -stage Fano Bott manifolds. We discuss the relationship between A ( X ) and T X . As mentioned above, (Op1) on A ( X ) corresponds to an isomorphism of T X . Moreprecisely, given a permutation matrix P associated to a permutation π on [ d ], Φ P corre-sponds to a bijection on V ( T X ) associated to π . For clarifying the relationship between(Op2 ± ) (and (Op3 ± )) and the operations on T X , we introduce a new notion and operationon the signed rooted forest T X . Definition 5.5 (Equivalent signs) . Let T and T ′ be signed rooted forests. We say that T and T ′ have equivalent signs if T and T ′ are isomorphic as rooted forests and the signs ofthe edges of T can be transformed, up to isomorphism as rooted forests, into the signs ofthe edges of T ′ by some replacements of all the signs of the edges { v i , v j } for all j ∈ ϕ − ( i )at the same time. When this is the case, we write T ∼ T ′ .It directly follows from the construction of the signed rooted forests from primitiverelations that if T ∼ T ′ then the corresponding primitive relations are equivalent. Example 5.6. Let us consider the following three signed rooted forests T , T , T (butactually, trees). We see that T ∼ T . In fact, by exchanging the signs of the edges { v , v } , { v , v } , { v , v } and exchanging the vertices v and v by a graph isomorphism,we see that T can be transformed into T .On the other hand, T and T are not equivalent. In fact, the signs of the edges { v , v } and { v , v } in T should be different, while those in T should be the same.The following proposition will be a crucial part of the proof of Theorem 1.8. Proposition 5.7. Let X be a d -stage Bott manifold. Let A = ( n ij ) ∈ F B ( d ) be the uppertriangular matrix associated to X . Let T = T X with V ( T ) = { v , . . . , v d } and let ϕ be themap with (3.2) . T T v v + v − v v + − v v + v − v v ++ v v + v + v v + − (1) For k ∈ [ d ] , the operation Φ ± k of (Op ± ) corresponds to the simultaneous changeof the signs of the edges { v k , v j } of T for all j ∈ ϕ − ( k ) . (2) For k, ℓ ∈ [ d ] , if A ℓ = and A k = e ℓ , then v ℓ is a root of T and v k is its child.Moreover, the operation Φ ± k,ℓ of (Op ± ) corresponds to the change of the sign ofthe edge { v ℓ , v k } of T .Proof. (1) By (4.1), it suffices to show that for each i = 1 , . . . , d − 1, the left-most nonzeroentry of (Φ ± k ( A )) i stays the same as that of A i when it is not in the k -th column, whilethe left-most nonzero entry of (Φ ± k ( A )) i changes from that of A i when it is in the k -thcolumn. This directly follows from the definition of the operation Φ ± k .(2) By our assumption, we know that the left-most nonzero entry of A k is in the ℓ -thcolumn. Similarly to (1), it suffices to show that for each i , the the left-most nonzeroentry of (Φ ± k,ℓ ( A )) i stays the same as that of A i when i = k , while the left-most nonzeroentry of (Φ ± k,ℓ ( A )) k changes from that of A k . This also directly follows from the definitionof the operation Φ ± k,ℓ . (cid:3) Remark . From Proposition 5.7 (1) together with Proposition 2.4 and Remark 4.2, wesee that given two Fano Bott manifolds, the following three conditions are equivalent: • X and X ′ are isomorphic as varieties; • PC(Σ X ) and PC(Σ X ′ ) are equivalent; • A ( X ) can be transformed into A ( X ′ ) through a sequence of the two operations(Op1) and (Op2 ± ). 6. Proof of Theorem 1.8 The goal of this section is to complete our proof of Theorem 1.8. Proposition 6.3 is anessential part of the proof, which is the statement on the distinguishments of cohomologyrings by the precise observations of the signed rooted forests associated to Fano Bottmanifolds.Here, we introduce a special kind of rooted trees: Definition 6.1. Let V = { v , v , . . . , v p , w , w , . . . , w q } ( p ≥ , q ≥ ,E = {{ v i − , v i } : i = 1 , . . . , p } ∪ {{ v p , w j } : j = 1 , . . . , q } . (6.1)We call the rooted tree ( V, E, v ) a broom with q leaves . emma 6.2. Let B and B ′ be brooms and assign the certain signs of the edges of B and B ′ . Let X and X ′ be the Fano Bott manifolds corresponding to B and B ′ , respectively.Assume that H ∗ ( X ) ∼ = H ∗ ( X ′ ) . Then B ∼ B ′ .Proof. Let B and B ′ be the same broom with q leaves ( V, E, v ) as in (6.1).First, we prove the assertion in the case q = 2. Then there are only two possibilities B and B ′ of assignments of signs up to equivalence as follows: B : sign( { v i − , v i } ) = + for i = 1 , . . . , p, sign( { v p , w } ) = sign( { v p , w } ) = + ,B ′ : sign( { v i − , v i } ) = + for i = 1 , . . . , p, sign( { v p , w } ) = + , sign( { v p , w } ) = − . Namely, B B ′ . Let R = H ∗ ( X ) ⊗ Z ( Z / 2) and R ′ = H ∗ ( X ′ ) ⊗ Z ( Z / R and R ′ , we rename the vertices as follows: w , w , v p , v p − , . . . , v −→ v , v , v , . . . , v p +3 . Once we can prove that R = R ′ , we conclude H ∗ ( X ) = H ∗ ( X ′ ) as required.Let ( n ij ) (resp. ( n ′ ij )) be the upper triangular matrix corresponding to B (resp. B ′ ).By Remark 4.4, we can compute them as follows: n = 1 , n i,i +1 = 1 for i = 2 , . . . , p + 2 , n ij = 0 otherwise ,n ′ = − , n ′ = n ′ = 1 , n ′ i,i +1 = 1 for i = 3 , . . . , p + 2 n ′ ij = 0 otherwise . Here, we consider the entries of ( n ij ) and ( n ′ ij ) as the elements of Z / 2. More concretely,we identify − D (resp. D ′ ) denote the acyclic digraph corresponding to ( n ij ) ∈ ( Z / ( p +3) × ( p +3) (resp. ( n ′ ij ) ∈ ( Z / ( p +3) × ( p +3) ). In this case, we see that both L ( D ) and L ( D ′ ) correspond to { , } and ρ D ( L ( D )) = rank (cid:18)(cid:18) · · · 01 0 · · · (cid:19)(cid:19) = 1 , and ρ D ′ ( L ( D ′ )) = rank (cid:18)(cid:18) · · · 01 1 · · · (cid:19)(cid:19) = 2 . Therefore, we obtain that R = R ′ by [7, Theorem 1.1 and Proposition 8.6 (i)].Now, let us consider the general case. We assign the signs of the edges { v p , w j } for j = 1 , . . . , q as follows: B : |{ j ∈ [ q ] : sign( { v p , w j } ) = + }| = a, |{ j ∈ [ q ] : sign( { v p , w j } ) = −}| = b,B ′ : |{ j ∈ [ q ] : sign( { v p , w j } ) = + }| = a ′ , |{ j ∈ [ q ] : sign( { v p , w j } ) = −}| = b ′ . Here, we have a + b = a ′ + b ′ = q . When B B ′ , we have { a, b } 6 = { a ′ , b ′ } . Then, forany isomorphism f : V → V between signed rooted trees B and B ′ , we can find a pair ofleaves w i , w j such that(sign( { v p , w i } ) , sign( { v p , w j } )) = (sign( { v p , f ( w i ) } ) , sign( { v p , f ( w j ) } )) . Let us fix such an isomorphism f and such leaves w i , w j . Let B be the subtree of B induced by v , v , . . . , v p and w i , w j . In particular, B is a broom with 2 leaves. Then itfollows from the above discussion that H ∗ ( X B ) = H ∗ ( X f ( B ) ). Therefore, Proposition 4.5(2) implies that H ∗ ( X ) = H ∗ ( X ′ ), as desired. (cid:3) The following proposition plays the crucial role for the proof of Theorem 1.8. roposition 6.3. Let X and X ′ be Fano Bott manifolds. Assume that H ∗ ( X ) ∼ = H ∗ ( X ′ ) .Then T X and T X ′ have equivalent signs by exchanging the signs assigned to the edgesadjacent to the roots if necessary.Proof. By Proposition 4.5 (1), there is an isomorphism f : T X → T X ′ as rooted forests.Suppose, on the contrary, that T X T X ′ . Then Definition 5.5 says that there is a vertex v of T X such that the signs of the edges adjacent to v are not preserved by f even if weexchange all those signs at the same time. Since the equivalence of signs is considered upto isomorphism as rooted forests, we have to treat a certain part of children of v which canbe transferred by an isomorphism as rooted forests. Let w ( i ) j ( i = 1 , . . . , p , j = 1 , . . . , q i )be all the children of v such that w ( i ) j can be transferred into w ( i ′ ) j ′ by f if and only if i = i ′ .Let v ′ = f ( v ), let w ′ ( j ) i = f ( w ( i ) j ) for each i and j , let a i = |{ j ∈ [ q i ] : sign( { v, w ( i ) j } ) = + }| (resp. a ′ i = |{ j ∈ [ q i ] : sign( { v ′ , w ′ ( i ) j } ) = + }| ) and let b i = q i − a i (resp. b ′ i = q ′ i − a ′ i ) foreach i . The assumption T T ′ implies that there is I ⊂ [ p ] with (X i ∈ I a i , X i ∈ I b i ) = (X i ∈ I a ′ i , X i ∈ I b ′ i ) , otherwise it is a contradiction to the choice of v . Let us fix such I .Note that the length from the root v of T X to v is at least 1. Let v , v , . . . , v k = v bethe vertices between v and v . Consider the broom B defined by v , v , . . . , v k and w ( i ) j for i ∈ I and j ∈ [ q i ]. Let B ′ = f ( B ). Note that B B ′ . By Lemma 6.2, we see that H ∗ ( X B ) = H ∗ ( X B ′ ).On the other hand, since B and B ′ are the rooted subtrees of T X and T X ′ whose rootsare the same as that of T X and T X ′ , respectively, we see that H ∗ ( X B ) ∼ = H ∗ ( X B ′ ) byProposition 4.5 (2), a contradiction.Therefore, we conclude that T X ∼ T X ′ , as required. (cid:3) Now, we are ready to give a proof of Theorem 1.8. Proof of Theorem 1.8. Note that (2) ⇒ (3) is trivial and (3) ⇒ (1) directly follows fromPropositions 5.7 and 6.3.In what follows, we prove (1) ⇒ (2). Let X and X ′ be d -stage Fano Bott manifoldsand let T = T X and T ′ = T X ′ be the associated signed rooted forests, respectively. ByProposition 5.7, the assumption of (1) says that T ∼ = T ′ as (non-signed) rooted forests and T and T ′ have equivalent signs by exchanging the signs assigned to the edges adjacentto the roots. Namely, there is an isomorphism f : T → T ′ as rooted forests whichpreserves the signs of all edges of T and T ′ except for the edges adjacent to roots. Weassume that f is an “identity”, more precisely, f ( v i ) = v ′ i , where V ( T ) = { v , . . . , v d } and V ( T ′ ) = { v ′ , . . . , v ′ d } .Let Σ = (Σ X ) = { v ± i : i = 1 , . . . , d } and let Σ ′ = (Σ X ′ ) = { w ± i : i = 1 , . . . , d } .Let A := A ( X ) = ( n ij ) ≤ i,j ≤ d (resp. A ′ := A ( X ′ ) = ( n ′ ij ) ≤ i,j ≤ d ). As mentioned inRemark 4.4, we can read off ( n ij ) (resp. ( n ′ ij )) from T (resp. T ′ ). Since v +1 , . . . , v + d (resp. w +1 , . . . , w + d ) are linearly independent, we can take v + i = w + i = e i for each i . Thus, we seethat the matrix M (resp. M ′ ) whose row vectors consist of v +1 , . . . , v + d , v − , . . . , v − d (resp. +1 , . . . , w + d , w − , . . . , w − d ) looks as follows: M = E − n n · · · n d − n · · · n d − · · · n d . . . ... − and M ′ = E − n ′ n ′ · · · n ′ d − n ′ · · · n ′ d − · · · n ′ d . . . ... − , where E ∈ { , } d × d denotes the identity matrix. Namely, the top halves of M and M ′ are E and the bottom half of M (resp. M ′ ) is − E + A (resp. − E + A ′ ).Here, we see that the operation (Op1) is a unimodular transformation on M . In fact,(Op1) permutes the corresponding columns and the corresponding rows of the top halfand the bottom half. Moreover, since (Op2 ± ) is a unimodular transformation expressedby T k := e ... − e k + A k ... e d ∈ Z d × d , we see that E · T k = e ... − e k + A k ... e d and( − E + A ) · T k = − T k + Φ ± k ( A ) = − e + Φ ± k ( A ) ... e k ... − e d + Φ ± k ( A ) d . Let M ′′ be the resulting matrix after applying a certain sequence of unimodular transfor-mations corresponding to (Op1) and (Op2 ± ) to M and let T ′′ be the signed rooted forestassociated to (the bottom half of) M ′′ . Since A and A ′ are Fano Bott equivalent, wemay assume that M ′′ can be transform into M ′ by a certain sequence of transformationscorresponding to (Op3 ± ).First, we assume that T and T ′ are trees, i.e., connected. Let v d be the unique rootof T . By our assumption, we see that M ′′ and M ′ agree except for the d -th columns.(See Proposition 5.7.) Let v i , . . . , v i k be the children of the root v d such that the signs of { v d , v i j } and { v ′ d , v ′ i j } are different. Then, let us consider the subtrees T i j of T induced byall descendants of v i j with its root v i j . By the construction of ( n ij ) from T , we see that n pq = 0 only if both v p and v q are contained in the same T i j for some j . Now, apply theunimodular transformations I j = ( e ( j ) pq ) ≤ p,q ≤ d ∈ Z d × d for j = 1 , . . . , k to M ′′ , where e ( j ) pp = ( − v p ∈ V ( T i j ) , , and e ( j ) pq = 0 if p = q . Then we see that the row vectors of M ′′ · ( I · · · I k ) and M ′ coincideup to sign. Hence, those satisfy the condition in Lemma 2.7, so we conclude that X and X ′ are diffeomorphic. ven if T and T ′ have at least two connected components, we may apply the aboveprocedure to each connected component. (cid:3) Remark . By the above proof, we see that the three conditions of Theorem 1.8 isequivalent to the fourth condition:(4) the signed rooted forests T X and T X ′ have equivalent signs by exchanging the signsassigned to the edges adjacent to the roots if necessary. Example 6.5. Let us consider the following signed rooted forests T and T ′ . T T ′ T ′′ v v v + v − + v v + v ++ v v v − v ++ v v − v −− v v v − v ++ v v − v − +We see that T T ′ by the difference of signs of edges { v , v } and { v , v } , but thecorresponding Fano Bott manifolds are diffeomorphic since the remaining signs can betransformed. The following matrices are M, M ′′ and M ′ appearing in the above proofand the unimodular transformations I which corresponds to the subtree induced by thedescendants of v used in the proof: M = E − − − − − − − − , M ′′ = E − − − − − − − − − − ,I = − − − , M ′ = E − − − − − − − − − − − − − . We can transform M into M ′′ by applying the permutation matrix corresponding to thetransposition (1 , 2) and T appearing in the above proof. Note that T ′′ above is the signedrooted tree corresponding to M ′′ .We can see that M ′′ I coincides with M ′ up to signs of rows. In fact, the first, second,third and seventh rows of the top and bottom halves of M ′′ I are exactly equal to thoseof M ′ , while the fourth, fifth and sixth rows are equal up to signs. eferences [1] V. V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math J. (1991),569–585.[2] V. V. Batyrev, On the classification of toric Fano 4-folds, J. Math. Sci. (New York) (1999), 1021–1050.[3] V. Buchstaber and T. Panov, Toric Topology, Mathematical Surveys and Monographs, . AmericanMathematical Society, Providence, RI, 2015.[4] S. Choi, Classification of Bott manifolds up to dimension 8, Proc. Edinb. Math. Soc. (2) (2015),no. 3, 653–659.[5] Y. Cho, E. Lee, M. Masuda, and S. Park, Unique toric structure on a Fano Bott manifold,arXiv:2005.02740.[6] S. Choi, M. Masuda and S. Murai, Invariance of Pontrjagin classes for Bott manifolds, Algebr. Geom.Topol. (2015), no. 2, 965–986.[7] S. Choi, M. Masuda, S. Oum, Classification of real Bott manifolds and acyclic digraphs, Trans. Amer.Math. Soc. 369 (2017), no. 4, 2987–3011.[8] S. Choi, M. Masuda and D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. (2010), no. 1, 109–129.[9] S. Choi, M. Masuda and D. Y. Suh, Topological classification of generalized Bott towers, Trans. Amer.Math. Soc. (2010), no. 2, 1097–1112.[10] W. Fulton, “Introduction to Toric Varieties”, Ann. of Math. Studies 131, Princeton Univ. Press, 1993.[11] A. Higashitani, K. Kurimoto and M. Masuda, Cohomological rigidity for toric Fano manifolds of smalldimension or large Picard number, arXiv:2005.13795.[12] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol. (2009), no. 4, 2479–2502.[13] M. Masuda and T. Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. (2008), no. 8, 95–122.[14] M. Masuda and D. Y. Suh, Classification problems of toric manifolds via topology, Contemp. Math. , , Amer. Math. Soc., Providence, RI, 2008.[15] M. Øbro, An algorithm for the classification of smooth Fano polytopes, arXiv:0704.0049v1.[16] T. Oda, “Convex Bodies and Algebraic Geometry -An introduction to the theory of toric varieties”,Ergeb. Math. Grenzgeb. (3), Vol. 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris,Tokyo, 1988.[17] H. Sato, Towards the classification of higher-dimensional toric Fano varieties, Tohoku Math J. (2000), 383–413.[18] Y.Suyama, Fano generalized Bott manifolds, Manuscripta Math. (2019),(A. Higashitani) Department of Pure and Applied Mathematics, Graduate School of Infor-mation Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan E-mail address : [email protected] (K. Kurimoto) Department of Mathematics, Graduate School of Science, Kyoto SangyoUniversity, Kyoto 603-8555, Japan E-mail address : [email protected]@cc.kyoto-su.ac.jp