Cohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers
aa r X i v : . [ m a t h . A T ] M a y COHOMOLOGICAL RIGIDITY FOR TORIC FANO MANIFOLDS OFSMALL DIMENSIONS OR LARGE PICARD NUMBERS
AKIHIRO HIGASHITANI, KAZUKI KURIMOTO, AND MIKIYA MASUDA
Abstract.
The cohomological rigidity problem for toric manifolds asks whether toricmanifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings areisomorphic. Many affirmative partial solutions to the problem have been obtained andno counterexample is known. In this paper, we study the diffeomorphism classificationof toric Fano d -folds with d = 3 , ≥ d −
2. In particular, weshow that those manifolds except for two toric Fano 4-folds are diffeomorphic if theirintegral cohomology rings are isomorphic. The exceptional two toric Fano 4-folds (theirID numbers are 50 and 57 on a list of Øbro) have isomorphic cohomology rings and theirtotal Pontryagin classes are preserved under an isomorphism between their cohomologyrings, but we do not know whether they are diffeomorphic or homeomorphic. Introduction
Cohomological rigidity problem.
As is well-known, integral cohomology ring (asa graded ring) is not a complete invariant to distinguish closed smooth manifolds. However,it becomes a complete invariant if we restrict our concern to a small family F of closedsmooth manifolds. For instance, this is the case if F is the family of closed surfaces. Wesay that a family F of manifolds is cohomologically rigid if the integral cohomology ringsdistinguish the manifolds in F up to diffeomorphism (or homeomorphism).A toric variety is a normal complex algebraic variety with an algebraic action of a C ∗ -torus having an open dense orbit. It is well known that there is a one-to-one correspondencebetween toric varieties and a combinatorial object called fans. Therefore, the classificationof toric varieties reduces to the classification of fans. A toric variety is not necessarilycompact or smooth. A compact smooth toric variety, which we call a toric manifold , iswell studied. For instance, its cohomology ring and Chern classes are explicitly describedin terms of the associated fan. As mentioned above, the classification of toric manifolds as varieties reduces to the classification of the associated fans. However, the classificationof toric manifolds as smooth manifolds is unknown. Motivated by the diffeomorphismclassification of a certain family of toric manifolds ([18]), the third author and Dong YoupSuh posed the following naive problem in [19]. Cohomological rigidity problem for toric manifolds.
Are toric manifolds diffeomor-phic (or homeomorphic) if their integral cohomology rings are isomorphic as graded rings?Namely, is the family of toric manifolds cohomologically rigid?
No counterexample to the problem is known and many affirmative partial solutionshave been obtained (see [12, 5, 16, 8] and the reference therein for recent accounts of theproblem). Among those affirmative solutions, Bott manifolds are well studied. A Bottmanifold is a toric manifold associated with a spanning fan of a cross-polytope. It can
Mathematics Subject Classification.
Primary 14M25; Secondary 57R19, 14J45, 57S15.
Key words and phrases.
Cohomological rigidity, toric Fano manifold. e obtained as the total space of an iterated C P -bundle starting with a point. It is notcompletely solved that the family of Bott manifolds is cohomologically rigid but the knownresults are close to the complete solution ([9]). Some partial affirmative solutions are alsoknown for generalized Bott manifolds ([10, 11]) although the known results are far fromthe complete solution. Here, a generalized Bott manifold is a toric manifold associatedwith a spanning fan of the direct sum of simplices (see Definition 3.1 for direct sum ofpolytopes). Similarly to a Bott manifold, a generalized Bott manifold can be obtained asthe total space of an iterated C P n i -bundle starting with a point, where each n i can takeany positive integer.In this paper, we study the diffeomorphism classification of toric Fano d -folds with d = 3 , ≥ d −
2. Our main result (Theorem 1.1) below providesanother affirmative partial solution to the cohomological rigidity problem.1.2.
Smooth Fano d -polytopes and toric Fano d -folds. A lattice polytope is a convexpolytope P ⊂ R d all of whose vertices lie in the integer lattice Z d . We say that P ⊂ R d is a smooth Fano d -polytope if it is a full-dimensional lattice polytope containing the origin inits interior such that the set of vertices of every facet forms a Z -basis of Z d . In particular,smooth Fano d -polytopes are simplicial. To a smooth Fano d -polytope P ⊂ R d , we canassociate a complete nonsingular fan as the spanning fan of P , where each i -dimensionalcone in the fan is spanned by the vertices of an ( i − P .It is known that the set of smooth Fano d -polytopes up to unimodular equivalenceone-to-one corresponds to the set of toric Fano d -folds up to isomorphism as varieties([3]). Moreover, it is known that for a fixed d , there are only finitely many smooth Fano d -polytopes up to unimodular equivalence (see [3]). The number of smooth Fano d -polytopes(up to unimodular equivalence) for small values of d is given as follows:the number ofdimension smooth Fano d -polytopes proved in2 53 18 [2, 25]4 124 [3, 23]5 866 [20]6 7622 [20]7 72256 [20]8 749892 [20]Indeed, Øbro ([20]) provides the algorithm (called SFP algorithm ), which produces acomplete list of smooth Fano d -polytopes (up to unimodular equivalence) for a givenpositive integer d . For d = 2 , , , ,
6, the database of all smooth Fano d -polytopes isopen in the following URL: Each toric Fano d -fold (or smooth Fano d -polytope) has its ID. For example, the IDnumber of Hirzebruch surface of degree 0 (resp. 1) is 4 (resp. 3). Toric Fano 3-folds haveID 6–23 and toric Fano 4-folds have ID 24–147, and so on.One can see that among eighteen toric Fano 3-folds, there are five Bott manifolds (ninegeneralized Bott manifolds including Bott manifolds), and among one hundred twenty-fourtoric Fano 4-folds, there are thirteen Bott manifolds (forty-one generalized Bott manifoldsincluding Bott manifolds). .3. Results.
As explained above, there are only finitely many smooth Fano d -polytopesfor a given d and we know their explicit description. This finiteness and explicitness are agreat advantage to investigate the cohomological rigidity problem for toric Fano d -folds.In this paper we prove the following. Theorem 1.1.
The family F of toric Fano d -folds satisfying one of the following condi-tion:(1) Picard number ≥ d − ,(2) d = 3 ,(3) d = 4 except for ID numbers 50 and 57,is cohomologically rigid. Namely, two toric Fano d -folds in the family F are diffeomorphicif and only if their integral cohomology rings are isomorphic as graded rings. We actually identify which toric Fano d -folds in the family F are diffeomorphic (seeTables 1 and 6). Those diffeomorphic toric Fano d -folds are in fact weakly equivariantlydiffeomorphic with respect to the restricted actions of the compact subtorus of the C ∗ -torus. They also show that the main theorem in [17] is incorrect, see Remark 2.4 (3) fordetails. Toric Fano 4-folds with ID numbers 50 and 57 have isomorphic integral cohomologyrings and their Pontryagin classes are preserved under an isomorphism between theircohomology rings. We do not know whether they are diffeomorphic or homeomorphic, butthey are not weakly equivariantly homeomorphic with respect to the restricted actions ofthe compact subtorus.The Picard number of a toric Fano d -fold is the number of the vertices of the associatedsmooth Fano d -polytope minus d . It is known that smooth Fano d -polytopes have at most3 d vertices and those with at least 3 d − H ∗ ( X ) of a toric manifold X is the quotient of a polynomial ringin b ( X ) variables by an ideal, where b ( X ) is the rank of H ( X ), i.e. the Picard number of X . In some cases, one can check by an elementary method whether two cohomology ringsare isomorphic or not. However, in general, the elementary method requires a formidablecomputation and does not work well. In order to distinguish our cohomology rings, wepay attention to elements of H ( X ; R ) whose k -th power vanish, where we take R = Z , Z / Z / k = 2 , Conjecture ([7]) . If there is a cohomology ring isomorphism between toric Fano manifoldswhich preserves their first Chern classes, then they are isomorphic as varieties.
Based on the classification of cohomology rings of our toric Fano manifolds, we prove heorem 1.2. The conjecture above is true for toric Fano d -folds with d = 3 , or withPicard number ≥ d − . Structure of the paper.
In Section 2, we briefly recall the theory of toric varietiesand the well-known presentation of the cohomology ring of a toric manifold (Proposi-tion 2.1). We introduce the invariants of cohomology rings used to distinguish our coho-mology rings. We also give a lemma (Lemma 2.3) mentioned above to find diffeomorphicor homeomorphic toric manifolds. We recall the notion of Gr¨obner basis and normalforms. After those preparations, we prove Theorem 1.1. We will verify cases (1), (2), (3)of Theorem 1.1 in Sections 3, 4, 5 respectively. Theorem 1.2 will be proved in Section 6.
Acknowledgements.
A. Higashitani was supported in part by JSPS Grant-in-Aid forScientific Research 18H01134 and M. Masuda was supported in part by JSPS Grant-in-Aid for Scientific Research 16K05152. This work was partly supported by Osaka CityUniversity Advanced Mathematical Institute (MEXT Joint Usage/Research Center onMathematics and Theoretical Physics).2.
Preliminaries
In this section, we recall the well-known presentation of the cohomology ring of a toricmanifold (i.e. a compact smooth toric variety) and give a sufficient condition (Lemma 2.3)for two compact smooth toric varieties to be homeomorphic or diffeomorphic. We alsointroduce naive invariants of cohomology rings used in this paper. We show by an examplehow to compute those invariants using Gr¨obner basis.2.1.
Toric manifolds and their cohomology rings.
We briefly review the theory oftoric varieties and refer the reader to [15] or [22] for details.A toric variety of complex dimension d is a normal algebraic variety X over the complexnumbers C with an algebraic action of ( C ∗ ) d having an open dense orbit. A fan in a lattice N := Hom( C ∗ , ( C ∗ ) d )( ∼ = Z d ) is a set of rational strongly convex polyhedral cones in N ⊗ R such that(1) each face of a cone in ∆ is also a cone in ∆;(2) the intersection of two cones in ∆ is a face of each.The fundamental theorem in the theory of toric varieties says that there is a one-to-onecorrespondence between toric varieties of complex dimension d and fans in N , and thattwo toric varieties are isomorphic if and only if the corresponding fans are unimodularlyequivalent.For a fan ∆, we denote the corresponding toric variety by X (∆). The toric variety X (∆) is compact (or complete) if and only if ∆ is complete, i.e. the union of cones in ∆covers the entire space N ⊗ R . Moreover, X (∆) is smooth (or nonsingular) if and only if∆ is nonsingular, i.e. for any cone σ in ∆, the primitive vectors lying on the 1-dimensionalfaces of σ form a part of a basis of N .In the following, we assume that our fan ∆ is complete and nonsingular. Let ρ , . . . , ρ m be 1-dimensional cones in ∆. We denote the primitive vector lying on ρ i by v i and call ita primitive ray vector . We denote the set of all primitive ray vectors by V (∆). The set ofall cones in ∆ defines an abstract simplicial complex on [ m ] = { , . . . , m } . It is called the underlying simplicial complex of ∆ and denoted by K (∆). Indeed, a subset I of [ m ] is amember of K (∆) if and only if v i ’s for i ∈ I span a cone in ∆. Since ∆ can be recoveredfrom two data K (∆) and V (∆), we may think of ∆ as the pair ( K (∆) , V (∆)). ach 1-dimensional cone ρ i in the fan ∆ corresponds to an invariant divisor X i of X (∆)and we denote the Poincar´e dual to X i by x i . Since X i is of real codimension two, x i liesin H ( X (∆)). With this understanding, we have the following well-known presentation ofthe cohomology ring of a toric manifold. Throughout this paper, we will use it withoutmentioning it. Proposition 2.1 ([4, Theorem 5.3.1]) . The cohomology ring of a toric manifold X (∆) can be described as follows: H ∗ ( X (∆)) = Z [ x , . . . , x m ] / J where deg x i = 2 for i = 1 , . . . , m and J is the ideal generated by all (i) x i · · · x i k for { i , . . . , i k } 6∈ K (∆) ; (ii) m X j =1 u ( v j ) x j for u ∈ Hom( N, Z ) .Remark . (1) The total Chern class and the total Pontryagin class of X (∆) are respec-tively given by c ( X (∆)) = m Y i =1 (1 + x i ) , p ( X (∆)) = m Y i =1 (1 + x i ) . (2) Since the fan ∆ is complete and nonsingular, one can eliminate d variables among x , . . . , x m using the linear relations (ii), so H ∗ ( X (∆)) is actually the quotient of a poly-nomial ring in m − d variables by an ideal, where m − d agrees with the rank of H ( X (∆))because k ≥ Diffeomorphism lemma.
In this subsection, we give a sufficient condition for toricmanifolds to be homeomorphic or diffeomorphic. By definition, a toric manifold X (∆) ofcomplex dimension d has an algebraic action of ( C ∗ ) d . Let S be the unit circle group of C . It is known that the orbit space Q of X (∆) by the restricted action of ( S ) d is a d -dimensional manifold with corners such that all faces (even Q itself) are contractible. Thedual face poset of Q defines a simplicial complex which agrees with the simplicial complex K (∆), to be more precise, if Q , . . . , Q m are the facets of Q , then Q I := T i ∈ I Q i = ∅ for I ⊂ [ m ] if and only if I ∈ K (∆). The orbit space Q is often a simple polytope. In fact, thisis the case when X (∆) is projective, more generally when K (∆) is the boundary complexof a simplicial polytope.Remember that V (∆) = { v , . . . , v m } is the set of primitive ray vectors in ∆. Since N = Hom( C ∗ , ( C ∗ ) d ) = Hom( S , ( S ) d ), we may regard v i ∈ N as a homomorphism from S to ( S ) d . We note that Q is the disjoint union of the interior part Int Q I of Q I over I ∈ K (∆), where we allow I = ∅ and Q ∅ = Q . We consider the quotient space X ( Q, V (∆)) := Q × ( S ) d / ∼ (2.1)where ( x, g ) ∼ ( y, h ) if and only if x = y ∈ Int Q I and gh − belongs to the subtorusgenerated by circle subgroups v i ( S ) of ( S ) d for i ∈ I . The space X ( Q, V (∆)) is calledthe canonical model of X (∆) because X ( Q, V (∆)) is homeomorphic to X (∆) ([14], [4,Chapter 7]). Lemma 2.3.
Let ∆ and ∆ ′ be complete nonsingular fans with the same underlying simpli-cial complex, i.e. K (∆) = K (∆ ′ ) . If the sets of primitive ray vectors V (∆) = { v , . . . , v m } and V (∆ ′ ) = { v ′ , . . . , v ′ m } agree up to sign (which means v i = ± v ′ i for each i = 1 , . . . , m ) ,then X (∆) and X (∆ ′ ) are homeomorphic. Moreover, if K (∆) = K (∆ ′ ) is the boundary omplex of a simplicial polytope, then “homeomorphic” above can be improved to “diffeo-morphic”.Proof. As remarked above, X (∆) is homeomorphic to the canonical model X ( Q, V (∆)).We see from the construction that the canonical model does not depend on the signs of v i ’s, so the former statement follows. If K (∆) = K (∆ ′ ) is the boundary complex of asimplicial polytope, then the orbit space Q is a simple polytope; so the latter statementfollows from [13, Proposition 6.4 (iii)]. (cid:3) Remark . (1) The multiplication by ( S ) d on the second factor of X ( Q, V (∆)) descendsto an action of ( S ) d on X ( Q, V (∆)) and Lemma 2.3 holds in this equivariant setting, see[4, Chapter 7].(2) If two complete nonsingular fans are unimodularly equivalent, then the associatedtwo toric manifolds are not only isomorphic but also weakly equivariantly isomorphic withrespect to the actions of ( C ∗ ) d as is known.(3) It is proved in the paper [17] by the third author that if the equivariant cohomologyrings of two toric manifolds X and X ′ are isomorphic as algebras over H ∗ ( BT ), where T is the C ∗ -torus acting on X and X ′ , then v i = ± v ′ i (i.e. the condition in Lemma 2.3 issatisfied). After that, the author claimed that it implies that X and X ′ are isomorphic asvarieties but this is incorrect. The mistake occurs at the very end of the proof of Theorem1 in [17] and the author thanks Hiraku Abe for pointing out the mistake. Indeed, wewill see in this paper that there are toric Fano manifolds which satisfy the condition inLemma 2.3 but they are not isomorphic as varieties. The author also thanks HiroshiSato for providing such an example. In order to correct the theorem, we need to takeequivariant first Chern class into account. Namely, if the equivariant cohomology algebraisomorphism between X and X ′ preserves their equivariant first Chern classes (those are P mi =1 τ i and P mi =1 τ ′ i in [17]), then we can conclude v i = v ′ i for every i so that X and X ′ are isomorphic as varieties.2.3. Invariants of a cohomology ring.
Let ∆ be a complete nonsingular fan of dimen-sion d . It is well-known that the number of i -dimensional cones in ∆ coincides with the2 i -th Betti number of the toric manifold X (∆) for 0 ≤ i ≤ d −
1. Therefore, when ∆is the spanning fan of a simplicial d -polytope P , the face numbers of P are cohomologyinvariants. Namely, if toric manifolds associated with simplicial polytopes P and P ′ haveisomorphic cohomology rings, then the face numbers of P and P ′ coincide. Especially, thenumber of vertices of P and the number of facets of P are cohomology invariants. We willuse this fact without mentioning it throughout this paper.Betti numbers are invariants of a cohomology ring but they depend only on its additivestructure. We now introduce invariants of a cohomology ring, which depend on its ringstructure. Definition 2.5 ( s.v.e. , c.v.e. , k - v.e. ) . Let k ≥ R = Z or Z /p where p is a primenumber. We say that a nonzero element of H ( X ) ⊗ R is k -v.e. over R if it is primitiveand its k -th power vanishes in H ∗ ( X ) ⊗ R . When R = Z , the word “over Z ” will beomitted. Also, 2-v.e. will be called s.v.e. (square vanishing element) and 3-v.e. will becalled c.v.e. (cube vanishing element). Definition 2.6 ( maximal basis number ) . Let V be the set of all s.v.e. of H ∗ ( X ) and weconsider B := { S ⊂ V | S is a part of a Z -basis of H ( X ) } . learly there exists an S max ∈ B such that | S | ≤ | S max | for ∀ S ∈ B . We call | S max | the maximal basis number of H ∗ ( X ).The number of k -v.e. over R and the maximal basis number are invariants of H ∗ ( X ).Note that there may exist infinitely many s.v.e. although the maximal basis number isfinite. Example 2.7.
We compute s.v.e. and the maximal basis numbers of H ∗ ( F ) and H ∗ ( F ),where F a is the Hirzebruch surface of degree a . As is well-known, we have H ∗ ( F ) ∼ = Z [ x, y ] / ( x , y ) and H ∗ ( F ) ∼ = Z [ x, y ] / ( x , y ( y − x )) . Then we easily obtain the following table.s.v.e. (up to sign) maximal basis number F x, y F x, x − y H ∗ ( F ) and H ∗ ( F ) are not isomorphic to each other.2.4. Gr¨obner basis and normal forms.
For the computation of k -v.e. of H ∗ ( M ), werecall what Gr¨obner basis is. We refer the reader to [24] for the introduction to Gr¨obnerbasis. The terminologies not defined in this section for Gr¨obner basis can be found there.Let S = k [ x , . . . , x m ] be a polynomial ring with m variables over a field k and let M be the set of all monomials in S . We say that a total order < in M a monomial order on S if it satisfies that • < u for any u ∈ M with u = 1, and • uw < vw holds for any u, v, w ∈ M with u < v .Fix a monomial order < on S . Given a polynomial f ∈ S , we call the leading monomialwith respect to < appearing in f the initial monomial , denoted by in < ( f ). Given an ideal I ⊂ S , the ideal generated by the initial monomials f in I is called the initial ideal of I ,denoted by in < ( I ). Namely, in < ( I ) = (in < ( f ) | f ∈ I ) . For a system of generator { g , . . . , g s } of an ideal I , it is not necessarily the case thatin < ( I ) = (in < ( g ) , . . . , in < ( g s )) holds. We say that { g , . . . , g s } is a Gr¨obner basis for I with respect to a monomial order < if this equality holds. Even if a system of generator G of an ideal is not a Gr¨obner basis for an ideal, there is an algorithm, so called Buchbergeralgorithm , to compute its Gr¨obner basis from G by appending some additional generatorsto G .Let I ⊂ S be an ideal and let { g , . . . , g s } ⊂ I be a system of generator of I . For f ∈ S which is not equal to 0, we can get an equation f = f g + · · · + f s g s + f ′ satisfying the following: • u / ∈ (in < ( g ) , . . . , in < ( g s )) for all monomials u appearing in f ′ if f ′ = 0, and • in < ( f i g i ) ≤ in < ( f ) if f i = 0. e call f ′ a normal form of f and write NF( f ). It is known that NF( f ) is well-defined if { g , . . . , g s } is a Gr¨obner basis for I .Once we get a Gr¨obner basis of an ideal I , we have many advantages. One of suchadvantages is the following proposition: Proposition 2.8.
Let I ⊂ S be an ideal and let { g , . . . , g s } be a Gr¨obner basis for I withrespect to a monomial oder < . Given a polynomial f ∈ S , we have the following: f ∈ I ⇐⇒ NF( f ) = 0 . Example 2.9.
Let X = X be the toric Fano manifold corresponding to ID number 12,which will appear in Section 4, and let P be the associated smooth Fano 3-polytope. Then P has six vertices.Let us consider the cohomology ring H ∗ ( X ), and demonstrate how to compute s.v.e. ofthis ring. According to the database, the vertices of P are as follows: v = (1 , , , v = (0 , , , v = (0 , , , v = ( − , , , v = (0 , , − , v = (0 , − , .
1) First, we compute the defining ideal I of the cohomology ring. By Proposition 2.1,we can compute I as follows: H ∗ ( X ) ∼ = Z [ x , . . . , x ] / ( I X + J X ) , where I X = ( x x , x x , x x ) and J X = ( x − x , x + x − x , x + x − x ) . Note that I X (resp. J X ) corresponds to (i) (resp. (ii)). So, we obtain that H ∗ ( X ) ∼ = Z [ x , . . . , x ] / (( x x , x x , x x ) + ( x − x , x + x − x , x + x − x )) ∼ = Z [ x, y, z ] / ( x , ( z − y ) z, ( y − x ) y ) . Note that we apply the change of variables x = x , x = y , x = z , and x = x , x = z − y and x = y − x .2) Next, we compute a Gr¨obner basis for the ideal I = ( x , z ( z − y ) , y ( y − x )). As amonomial oder, let < be the graded lexicographic order induced by x < y < z . Then wecan see that { x , z ( z − y ) , y ( y − x ) } is a Gr¨obner basis for I with respect to < .3) Finally, we compute s.v.e. of H ∗ ( X ). Let f = x , f = z − yz and f = y − xy .Let f = ax + by + cz . Then the normal form of f can be computed as follows: f = a x + b y + c z + 2 abxy + 2 acxz + 2 bcyz = a f + c f + b f + (2 a + b ) bxy + 2 acxz + (2 b + c ) cyz. Hence, NF( f ) = (2 a + b ) bxy + 2 acxz + (2 b + c ) cyz . Thus, we can see that NF( f ) = 0 ifand only if (2 a + b ) b = 0 , ac = 0 , (2 b + c ) c = 0 . When a = 0, we have b = c = 0. When a = 0, we have c = 0 and (2 a + b ) b = 0, i.e., b = 0or b = − a . Hence, we conclude that NF( f ) = 0 if and only if f = ax or f = ax − ay .Therefore, s.v.e. of H ∗ ( X ) are x and x − y , and its maximal basis number is 1 since x and x − y cannot form a Z -basis. Example 2.10.
Let us consider the cohomology ring H ∗ ( X ), which will appear inSection 5, and demonstrate how to compute s.v.e. of this ring. Let X = X and let P e the associated smooth Fano 4-polytope. Then P has seven vertices. According to thedatabase, the vertices of P are as follows: v = (1 , , , , v = (0 , , , , v = (0 , , , , v = (0 , , , ,v = ( − , − , − , , v = (0 , , , − , v = (0 , , , − .
1) First, we compute the defining ideal I of the cohomology ring. By Proposition 2.1,we can compute I as follows: H ∗ ( X ) ∼ = Z [ x , . . . , x ] / ( I X + J X ) , where I X = ( x x x x , x x , x x , x x , x x x x ) and J X = ( x − x , x − x , x − x + x , x + 3 x − x − x ) , ∼ = Z [ x, y, z ] / ( x ( x − y ) , ( x − y ) z, ( − x + y + z ) z, ( − x + y + z ) y, x y )= Z [ x, y, z ] / ( x , ( x − y ) z, ( − y + z ) z, ( − x + y ) y, x y ) . Note that we apply the change of variables x = x , x = y , x = z , and x = x = x , x = x − y and x = − x + y + z .2) Next, we compute a Gr¨obner basis for the ideal I = ( x , ( x − y ) z, ( − y + z ) z, ( − x + y + z ) y, x y ) . As a monomial order, let < be the graded lexicographic order induced by x < y < z . AGr¨obner basis for I with respect to a monomial order < is { x , ( x − y ) z, ( − y + z ) z, ( − x + y ) y, x y, x z } . (2.2)Note that the generator itself does not become a Gr¨obner basis since I ∋ ( x − y ) · ( x − y ) z − z · ( − x + y ) y = x z in < ( I ) , but it follows from Buchberger algorithm that (2.2) is a Gr¨obner basis for I with respectto < .3) Finally, we compute s.v.e. of H ∗ ( X ). Let f = ax + by + cz . Then NF( f ) = a x + 2 b ( a + b ) xy + 2 c ( a + b + c ) xz . Therefore, we conclude that N F ( f ) = 0 if and onlyif a = b = c = 0. This means that H ∗ ( X ) has no s.v.e.3. The case with large Picard number
This section is devoted to verifying the case (1) in Theorem 1.1. The Picard numberof a smooth Fano d -fold associated with a smooth Fano d -polytope P is the number ofvertices of P minus d . On the other hand, smooth Fano d -polytopes are known to haveat most 3 d vertices and those with 3 d , 3 d − d − Definition 3.1 ( direct sum ) . Let P ⊂ R d and Q ⊂ R e be polytopes. Then P ⊕ Q = conv( P × { e } ∪ { d } × Q ) ⊂ R d + e , where d (resp. e ) denotes the origin of R d (resp. R e ), is called the direct sum of P and Q . The direct sum is also called the free sum of P and Q . Definition 3.2 ( skew bipyramid ) . Let P ⊂ R d be a polytope. Then we call a polytope B ⊂ R d +1 a skew bipyramid (or simply bipyramid ) over P if P is contained in affinehyperplane H such that there are two vertices v and w of B which lie on either side of H such that B = conv( { v, w } ∪ P ) and the line segment [ v, w ] meets P in its (relative)interior. hroughout this section, e , . . . , e d will denote the standard basis of Z d and P (resp. P ) will denote the hexagon (resp. pentagon) with vertices ± e , ± e , ± ( e − e ) (resp. e , ± e , ± ( e − e )) . The classification results on smooth Fano d -polytopes with 3 d , 3 d − d − Theorem 3.3 (Casagrande [6]) . A smooth Fano d -polytope P has at most d vertices. Ifit does have exactly d vertices, then d is even and P is unimodularly equivalent to P ⊕ d . Theorem 3.4 (Øbro [21]) . Let P be a smooth Fano d -polytope with exactly d − vertices.If d is even, then P is unimodularly equivalent to P ⊕ P ⊕ d − . If d is odd, then P isunimodularly equivalent to a bipyramid over P ⊕ d − . Theorem 3.5 (Assarf-Joswig-Paffenholz [1]) . Let P be a smooth Fano d -polytope withexactly d − vertices. If d is even, then P is unimodularly equivalent to (1) a double bipyramid over P ⊕ d − or (2) P ⊕ P ⊕ P ⊕ d − or (3) DP (4) ⊕ P ⊕ d − , where DP (4) is the convex hull of vertices ± e , ± e , ± e , ± e , ± ( e + e + e + e ) in R . If d is odd, then P is unimodularly equivalent to a bipyramid over P ⊕ P ⊕ d − .Remark . The polytopes in (1), (2), (3) in Theorem 3.5 have different face numbers.Indeed, their facet numbers are respectively 24 · d − , 25 · d − , 30 · d − .3.1. Cohomology of toric Fano -folds associated to P and P . Let X P be thetoric Fano 2-fold associated to P . We number the vertices of P e , e , − e + e , − e , − e , e − e from 1 to 6 and denote the corresponding elements in H ( X P ) by µ , . . . , µ . Then wehave H ∗ ( X P ) ∼ = Z [ µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ , µ + µ − µ − µ )) ∼ = Z [ x, y, z, w ] / ( x ( x + y ) , y ( x + y ) , z ( y − w ) , y ( x − z ) , z ( z + w ) , w ( z + w ) , xz, xw, yw ) , (3.1)where x = µ , y = µ , z = µ and w = µ . Lemma 3.7.
The maximal basis number of H ∗ ( X P ) is .Proof. It follows from (3.1) that any element of H ( X P ) is of the form ax + by + cz + dw with integers a, b, c, d and an elementary computation shows that( ax + by + cz + dw ) = ( a − ab + b − bc + c − cd + d ) w . Therefore, the s.v.e of H ∗ ( X P ) are primitive elements in the set { ax + by + cz + dw | ( a − b ) + ( c − d ) = 2 bc } . (3.2) n particular, x + y , y + z and z + w are s.v.e. and since they form a part of a Z -basis of H ( X P ), the maximal basis number of H ∗ ( X P ) is at least 3.On the other hand, it follows from (3.2) that s.v.e. of H ∗ ( X P ) over Z / { ax + by + cz + dw | a + b + c + d = 0 } , where a, b, c, d are regarded as elements of Z /
2. Therefore, the dimension of H ( X P ) ⊗ Z / H ∗ ( X P ) is at most 3, proving thelemma. (cid:3) Lemma 3.8.
For any s.v.e. f of H ∗ ( X P ) , there exist infinitely many s.v.e. of H ∗ ( X P ) such that f together with the s.v.e. does not form a part of a Z -basis of H ( X P ) .Proof. As observed in (3.2), any s.v.e. ax + by + cz + dw of H ∗ ( X P ) must satisfy thecondition ( a − b ) + ( c − d ) = 2 bc. Therefore, the parity of a − b and c − d must be the same. Moreover, at least one of a, b, c, d must be odd because the s.v.e. ax + by + cz + dw is primitive. In fact, one cansee from the identity above that the parity of ( a, b, c, d ) must be one of the following upto symmetry:(i) (even, even, odd, odd); (ii) (even, odd, odd, even).Since any two elements with the same parity do not form a part of a Z -basis of H ( X P ),it suffices to show that there are infinitely many s.v.e. in each of (i) and (ii) above andhere are examples of infinitely many s.v.e. in each case:(i) (2 k ( k + 1) , k , , , (ii) (2( k + 1) , k ( k + 1) + 1 , , , where k is any integer. (cid:3) Let X P be the toric Fano 2-fold associated to P . We number the vertices of P e , e , − e + e , − e , e − e from 1 to 5 and denote the corresponding elements in H ( X P ) by µ , . . . , µ . Then wehave H ∗ ( X P ) ∼ = Z [ µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ + µ , µ + µ − µ − µ )) ∼ = Z [ x, y, z ] / ( x , y ( x − z ) , y , z ( y + z ) , xz ) , (3.3)where x = µ , y = µ and z = µ . Lemma 3.9.
The maximal basis number of H ∗ ( X P ) is .Proof. The proof is essentially the same as in Lemma 3.7. It follows from (3.3) that anyelement of H ( X P ) is of the form ax + by + cz with integers a, b, c and an elementarycomputation shows that ( ax + by + cz ) = ( − ab − bc + c ) z . Therefore, s.v.e of H ∗ ( X P ) are primitive elements in the set { ax + by + cz | c = 2 b ( a + c ) } . (3.4)In particular, x and y are s.v.e. and since they form a part of Z -basis, the maximal basisnumber of s.v.e. of H ∗ ( X P ) is at least 2. n the other hand, (3.4) shows that c must be even. This implies that the basis maximalnumber must be at most 2, proving the lemma. (cid:3) Lemma 3.10.
Let A = Z [ x , . . . , x m ] / I A and B = Z [ y , . . . , y n ] / I B . If f is an s.v.e. of A ⊗ B , then f ∈ A or f ∈ B . In particular, the maximal basis number behaves additivelywith respect to tensor products.Proof. Let f = P mi =1 a i x i + P nj =1 b j y j . Then, f = m X i =1 a i x i ! + n X j =1 b j y j + 2 m X i =1 a i x i ! n X j =1 b j y j = m X i =1 a i x i ! + n X j =1 b j y j + 2 m X i =1 n X j =1 a i b j x i y j Thus, we have a i b j = 0 for any i and j , i.e. we have either a i = 0 for any i or b j = 0 forany j . (cid:3) Under these preparations, we start to prove Theorem 1.1 for the case (1). There are fol-lowing five cases according to the number of vertices V ( P ) and the parity of the dimension d for smooth Fano d -polytopes P :(1) V ( P ) = 3 d (in this case d must be even),(2) V ( P ) = 3 d − d is odd,(3) V ( P ) = 3 d − d is even,(4) V ( P ) = 3 d − d is odd,(5) V ( P ) = 3 d − d is even.In (1) and (3) above, there is only one smooth Fano d -polytope by Theorems 3.3 and 3.4,so it suffices to treat the remaining three cases. In the following, we may assume d ≥ d -folds are distinguished by theircohomology rings when d ≤ The case where V ( P ) = 3 d − with d odd ≥ . By Theorem 3.4, there are two P ’s up to unimodular equivalence and their vertices are as follows: e , − e , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) , ( Y d ) e , − e + e , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) . ( Y d )where the tags denote the corresponding toric Fano d -folds. In each case, the first twovertices correspond to the apices of the bipyramid and {± e k , ± e k +1 , ± ( e k − e k +1 ) } (1 ≤ k ≤ d − ) forms the hexagon P . One can see from the above that H ∗ ( Y d ) = Z [ x ] / ( x ) ⊗ H ∗ ( X P ) ⊗ d − , H ∗ ( Y d ) = H ∗ ( Y ) ⊗ H ∗ ( X P ) ⊗ d − We number the eight vertices e , e , − e + e , − e , − e , e − e , e , − e + e n ( Y ) above from 1 to 8 and denote the corresponding elements in H ( Y ) by µ , . . . , µ .Then we have H ∗ ( Y ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ + µ , µ + µ − µ − µ , µ − µ )) ∼ = Z [ x, y, z, w, v ] / ( x ( x + y − v ) , y ( x + y − v ) , z ( y − w − v ) ,y ( x − z ) , z ( z + w ) , w ( z + w ) , xz, xw, yw, v ) , where x = µ , y = µ , z = µ , w = µ and v = µ . One can check thats.v.e. of H ∗ ( Y ) = { y + 2 z − v, x + 2 y − v, z + w, v } (up to sign), (3.5)and hence the maximal basis number of H ∗ ( Y ) is 2. Therefore, it follows from Lemma 3.7and Lemma 3.10 that the maximal basis numbers of H ∗ ( Y d ) and H ∗ ( Y d ) are as in thefollowing table, so H ∗ ( Y d ) and H ∗ ( Y d ) are not isomorphic to each other. Ring Maximal basis number H ∗ ( Y d ) 1 + 3 · d − H ∗ ( Y d ) − · d − The case where V ( P ) = 3 d − with d odd ≥ . We first treat the case where d = 3. By Theorem 3.5, the vertices of P are one of the following: e , − e , e , ± e , ± ( e − e ) , ( Z ) e , − e + e , e , ± e , ± ( e − e ) , ( Z ) e , − e + e , e , ± e , ± ( e − e ) , ( Z ) e , − e − e , e , ± e , ± ( e − e ) . ( Z )In each case, { e , ± e , ± ( e − e ) } forms P and the first two vertices correspond to theapices of the bipyramid over P .We claim that Z and Z are diffeomorphic. Indeed, the vertices in Z are unimodularlyequivalent to − e , e + e , e , ± e , ± ( e − e )through an automorphism ( x , x , x ) → ( − x , x , x ) of Z , and these vectors agree withthe vectors in ( Z ) up to sign. Therefore, Z and Z are diffeomorphic by Lemma 2.3.We shall observe that the cohomology rings of Z , Z , Z are not isomorphic to eachother. Clearly Z = C P × X P and hence H ∗ ( Z ) ∼ = Z [ x ] / ( x ) ⊗ H ∗ ( X P ) . To describe the cohomology rings of the remaining ones, we number the seven vertices e , e , − e + e , − e , e − e , e , − e + ∗ from 1 to 7, where ∗ = e or ± e . We denote the corresponding elements in H ( Z i ) for i = 2 , µ , . . . , µ and set x = µ , y = µ , z = µ , w = µ . hen we have H ∗ ( Z ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ + µ + µ , µ + µ − µ − µ , µ − µ )) ∼ = Z [ x, y, z, w ] / ( x ( x − w ) , y ( x − z − w ) , y ( y − w ) , z ( y + z ) , xz, w ); H ∗ ( Z ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ + µ , µ + µ − µ − µ + µ , µ − µ )) ∼ = Z [ x, y, z, w ] / ( x , y ( x − z ) , y ( y − w ) , z ( y + z − w ) , xz, w ) . By an elementary computation using these presentations together with Lemmas 3.9 and 3.10,we obtain the following table; so H ∗ ( Z i ) for i = 1 , , Ring s.v.e. Maximal basis number H ∗ ( Z ) infinitely many 3 H ∗ ( Z ) 2 x − w, y − w, w H ∗ ( Z ) x, y − w, w Now we treat the case where d ≥
5. By Theorem 3.5, one can see that the vertices of P are one of the following:the vertices in ( Z i ) , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) ( Z di )for i = 1 , . . . , e , − e + e , e , ± e , ± ( e − e ) , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) . ( Z d )Note that Z d appears in the case d ≥ e and the vertices e , ± e , ± ( e − e ) cannotbe replaced each other by unimodular transformation. In each case, {± e k , ± e k +1 , ± ( e k − e k +1 ) } (2 ≤ k ≤ d − ) forms P and the first two vertices correspond to the apices of thebipyramid. We also note that { e , ± e , ± ( e − e ) } in ( Z d ) forms P . Therefore, one cansee from the above that Z di = Z i × ( X P ) d − for i = 1 , , , Z d = Y × X P × ( X P ) d − , (3.6)where Y denotes the Fano 3-fold Y in the previous subsection. Therefore, H ∗ ( Z di ) for i = 1 , . . . , Z di .Since Z and Z are diffeomorphic as observed before, so are Z d and Z d . It follows fromLemmas 3.7, 3.9 and 3.10 that we obtain the following table. Ring Maximal basis number H ∗ ( Z d ) 3 + 3 · d − H ∗ ( Z d ) 1 + 3 · d − H ∗ ( Z d ) 2 + 3 · d − H ∗ ( Z d ) 1 + 3 · d − Thus, we have to check H ∗ ( Z d ) ∼ = H ∗ ( Z d ) or not. Suppose that there is an isomorphism F : H ∗ ( Z d ) ∼ = H ∗ ( Y ) ⊗ H ∗ ( X P ) ⊗ H ∗ ( X P ) ⊗ d − → H ∗ ( Z d ) ∼ = H ∗ ( Z ) ⊗ H ∗ ( X P ) ⊗ d − . We note that any s.v.e. of H ∗ ( Z d ) belongs to one of the factors of the tensor productsby Lemma 3.10. Let f be an s.v.e. of H ∗ ( Z d ) which belongs to the factor H ∗ ( Y ). Weconsider the following set S ( f ) := { g ∈ H ( Z d ) | g is an s.v.e. and { f, g } is not a part of a Z -basis of H ( Z d ) } . f g ∈ S ( f ), then g must belong to H ∗ ( Y ) because otherwise { f, g } is a part of a Z -basisof H ( Z d ). Therefore S ( f ) is a finite set by (3.5) and hence so is S ( F ( f )) because F isan isomorphism. This together with Lemma 3.8 shows that F ( f ) must be an s.v.e. of H ∗ ( Z ). This means that F sends the set of s.v.e. of H ∗ ( Y ) to the set of s.v.e. of H ∗ ( Z ).However, the cardinality of the former set up to sign is 4 by (3.5) while that of the latterset up to sign is 3 (see the previous subsection). This contradicts the injectivity of F .Hence, H ∗ ( Z d ) and H ∗ ( Z d ) are not isomorphic.3.4. The case where V ( P ) = 3 d − with d even ≥ . Theorem 3.5 says that thereare three types of P ’s, i.e. (1), (2) and (3) in the theorem, but they have different facenumbers (Remark 3.6). Therefore, the toric Fano d -folds in these different types can bedistinguished by their cohomology rings. Since there are only one smooth Fano d -polytopein (2) and (3), its suffices to investigate the case (1).We first treat the case where d = 4. One can see that the vertices of P in Theorem.3.5(1)are one of the following: e , − e , e , − e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e + e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e + e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e + e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e − e , ± e , ± e , ± ( e − e ) , ( W ) e , − e + e , e , − e − e , ± e , ± e , ± ( e − e ) . ( W )In each case, {± e , ± e , ± ( e − e ) } forms the hexagon P and the first two vertices andsecond two vertices correspond to the apices of the double bipyramid over P . Note thatthose W , . . . , W can be obtained by considering the pair of the first two vertices and thesecond two vertices, i.e. where each of two segments coming from apices intersect. We cancheck that those are exactly the possible cases.We claim that W is diffeomorphic to W . Indeed, the vertices in W are unimodularlyequivalent to e , − e + e , − e , e + e , ± e , ± e , ± ( e − e )through an automorphism ( x , x , x , x ) → ( x , − x , x , x ) of Z , and these vectors agreewith the vectors in W up to sign. Therefore, W and W are diffeomorphic by Lemma 2.3.The same argument shows that W is diffeomorphic to W .We shall observe that H ∗ ( W i ) for i = 1 , . . . , H ∗ ( W ) ∼ = Z [ x, y ] / ( x , y ) ⊗ H ∗ ( X P ) , and H ∗ ( W ) ∼ = Z [ x, y ] / ( x , y ( y − x )) ⊗ H ∗ ( X P ) . To describe the cohomology rings of the remaining ones, we number the ten vertices e , e , − e + e , − e , − e , e − e , e , − e + ∗ , e , − e + ⋆ from 1 to 10, where ∗ = e or e and ⋆ = 0 , ± e or ± e . We denote the correspondingelements in H ( W i ) for i = 3 , , . . . , µ , . . . , µ and set x = µ , y = µ , z = µ , w = µ , v = µ , u = µ . hen we have H ∗ ( W ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ + µ , µ + µ − µ − µ , µ − µ , µ + µ − µ )) ∼ = Z [ x, y, z, w, v, u ] / ( x ( x + y − u ) , y ( x + y − u ) , z ( y − w − u ) ,y ( x − z ) , z ( z + w ) , w ( z + w ) , xz, xw, yw, v , u ( u + v )) .H ∗ ( W ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ + µ , µ + µ − µ − µ , µ − µ , µ − µ )) ∼ = Z [ x, y, z, w, v, u ] / ( x ( x + y − v ) , y ( x + y − v ) , z ( y − w − v ) ,y ( x − z ) , z ( z + w ) , w ( z + w ) , xz, xw, yw, v , u ) .H ∗ ( W ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ + µ + µ , µ + µ − µ − µ , µ − µ , µ − µ )) ∼ = Z [ x, y, z, w, v, u ] / ( x ( x + y − v − u ) , y ( x + y − v − u ) , z ( y − w − v − u ) ,y ( x − z ) , z ( z + w ) , w ( z + w ) , xz, xw, yw, v , u ) .H ∗ ( W ) ∼ = Z [ µ , µ , µ , µ , µ , µ , µ , µ , µ , µ ] / (( µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ , µ µ )+ ( µ − µ − µ + µ + µ , µ + µ − µ − µ + µ , µ − µ , µ − µ )) ∼ = Z [ x, y, z, w, v, u ] / ( x ( x + y − v ) , y ( x + y − v ) , z ( y − w − v ) ,y ( x − z + u ) , z ( z + w − u ) , w ( z + w − u ) , xz, xw, yw, v , u ) . By an elementary computation using the above presentations together with Lemma 3.7,we obtain the following table; so H ∗ ( W i ) for i = 1 , , Ring s.v.e. Maximal basis number H ∗ ( W ) infinitely many 5 H ∗ ( W ) infinitely many 4 H ∗ ( W ) z + w, v + 2 u, v H ∗ ( W ) 2 y + 2 z − v, x + 2 y − v, v, z + w, u H ∗ ( W ) z + w, v, u H ∗ ( W ) 2 x + 2 y − v, v, z + 2 w − u, u Now we treat the case where d ≥
6. The vertices of P in Theorem 3.5 (1) are eitherthe vertices in ( W i ) , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) ( W di )for i = 1 , . . . , e , − e + e , e , − e + e , ± e , ± e , ± ( e − e ) , . . . , ± e d − , ± e d , ± ( e d − − e d ) . ( W d )Note that W d appears in the case d ≥ e and the vertices ± e , ± e , ± ( e − e ) cannotbe replaced each other by unimodular transformation. The first two vertices and second wo vertices correspond to apices of the double bipyramid and {± e k − , ± e k , ± ( e k − − e k ) } (3 ≤ k ≤ d ) forms P . One can see from the above that W di = W i × ( X P ) d − for i = 1 , . . . , W d = Y × Y × ( X P ) d − , (3.7)where Y is the 3-fold Y in subsection 1.2. Since W (resp. W ) is diffeomorphic to W (resp. W ), W d (resp. W d ) is diffeomorphic to W d (resp. W d ). The maximal basisnumber of H ∗ ( X P ) is 3 by Lemma 3.7 and that of H ∗ ( Y ) is 2 by (3.5). Therefore, itfollows from (3.7), Table 3.4, and Lemma 3.10 that we obtain the following table. Ring Maximal basis number H ∗ ( W d ) 5 + 3 · d − H ∗ ( W d ) 4 + 3 · d − H ∗ ( W d ) 2 + 3 · d − H ∗ ( W d ) 3 + 3 · d − H ∗ ( W d ) 3 + 3 · d − H ∗ ( W d ) 2 + 3 · d − H ∗ ( W d ) − · d − Thus, we have to check whether H ∗ ( W d ) ∼ = H ∗ ( W d ) or not and H ∗ ( W d ) ∼ = H ∗ ( W d )or not, but the same argument as in the last part of the previous subsection shows thatthey are not isomorphic. Indeed, if there is an isomorphism F : H ∗ ( W d ) → H ∗ ( W d ), then F must send the set of s.v.e. of H ∗ ( W ) to that of H ∗ ( W ). However, the cardinalityof the former set up to sign is 4 while that of the latter set is 3 (see Table 3.4). Thiscontradicts the injectivity of F . Therefore, H ∗ ( W d ) is not isomorphic to H ∗ ( W d ). Thesame argument shows that H ∗ ( W d ) is not isomorphic to H ∗ ( W d ).4. The case of dimension q by X q .There are 18 variety-isomorphism classes of toric Fano 3-folds, in other words, 18 uni-modular equivalence classes of smooth Fano 3-polytopes. In this section, we will classifythem up to diffeomorphism. It turns out that the cohomological rigidity holds for them.More precisely, there are 16 diffeomorphism classes as is shown in Table 1 below, whereID numbers whose toric Fano 3-folds are diffeomorphic are enclosed by curly braces, andfive ID numbers before k are Bott manifolds. In Table 1, V ( P ) is the number of verticesof P , P shows the unimodular equivalence classes of smooth Fano 3-polytopes, H ∗ showsthe isomorphism classes of integer cohomology rings, and Diff shows the diffeomorphismclasses. V ( P ) P H ∗ Diff ID4 1 1 1 235 4 4 4 7, 19, 20, 226 7 6 6 { , } , 12, 17, 21 k
6, 167 4 3 3 8, { , } , 148 2 2 2 9, 15total 18 16 16 Table 1.
Diffeomorphism classification of toric Fano 3-folds e shall explain how we obtain Table 1. There is only one smooth Fano 3-polytope P with V ( P ) = 4, so there is nothing to prove in this case. The case where V ( P ) = 7or 8 is treated in Section 3. Indeed, toric Fano 3-folds X q with q = 8 , , , , ,
15 arerespectively Z , Z , Z , Z , Y , Y in Section 3. Therefore, it suffices to investigate thecase where V ( P ) = 5 or 6. Convention. (1) The vertices of a smooth Fano 3-polytope P are shown in the database of Øbroand we number them as 1 , , . . . in the order shown in the database.(2) The first three vertices of P are the standard basis of Z , so we omit them andwrite the vertices from 4th in Tables 2 and 4 below.(3) Minimal nonfaces of P are described using the numbering of the vertices of P .(4) I denotes the ideal of the cohomology ring H ∗ ( X q ) and its minimal generators aredescribed in the tables.(5) s.v.e. and c.v.e. in the tables are up to sign unless the coefficient is Z / The case where V ( P ) = 5 . In this case, there are four smooth Fano 3-polytopes asshown in Table 2. They are all combinatorially equivalent to a direct sum of a 2-simplexand a 1-simplex, so the corresponding toric Fano 3-folds are generalized Bott manifolds.
ID vertices of P from 4th minimal nonfaces7 ( − , − , , (0 , , −
1) 35 , − , , , (0 , − , −
1) 14 , − , − , , (0 , , −
1) 35 , − , , , (0 , − , −
1) 14 , Table 2.
Vertices and minimal nonfaces of P with V ( P ) = 5We denote the degree two cohomology element corresponding to 4th and 5th vertices by x and y , respectively. Then the cohomology ring of each toric Fano 3-fold with ID numberin Table 2 is the quotient of a polynomial ring Z [ x, y ] by an ideal I . By an elementarycomputation, we obtain Table 3 which shows that those four cohomology rings are notisomorphic to each other. ID I s.v.e. s.v.e. Z / Z / x , y (2 x − y ) ∅ ( y )19 x , y ( x − y ) x ( x ) ( x )20 x , y ( x − y ) ∅ ∅ x , y x ( x ) ( x, y ) Table 3.
Ideals and invariants when V ( P ) = 54.2. The case where V ( P ) = 6 . In this case, there are seven smooth Fano 3-polytopes P as shown in Table 4. The polytopes with ID numbers 11, 12, 18, 18, 21 are combinatoriallyequivalent to a cross-polytope, so the corresponding toric Fano 3-folds are Bott manifolds.
18D vertices of P from 4th minimal nonfaces11 ( − , , , (0 , − , , (0 , , −
1) 14 , , − , , , (0 , − , , (0 , , −
1) 14 , , − , , , (0 , − , , (0 , , −
1) 14 , , − , , , (0 , − , − , (0 , , −
1) 14 , , − , , , (0 , − , , (0 , , −
1) 14 , ,
366 ( − , − , , (0 , , − , (0 , , −
1) 26 , , , , − , , , (1 , , − , ( − , − ,
0) 14 , , , , Table 4.
Vertices and minimal nonfaces of P with V ( P ) = 6 Remark . We interchanged 5th and 6th vertices in [20] for ID numbers 12 and 18 sothat the minimal nonfaces have the same numbering as others.Through an automorphism ( x , x , x ) → ( x , − x , x ) of Z , the vertices of ID number11 are unimodularly equivalent to (1 , , , (0 , − , , (0 , , , ( − , , , (0 , , , (0 , , − X is diffeomorphic to X by Lemma 2.3.We denote the degree two cohomology element corresponding to 4th, 5th and 6th ver-tices by x , y and z respectively. Then the cohomology ring of each toric Fano 3-fold withID number in Table 4 is the quotient of a polynomial ring Z [ x, y, z ] by an ideal I . By anelementary computation, we obtain Table 5, which shows that the cohomology rings inthe tables are not isomorphic to each other: ID I s.v.e. maximal basis number11(18) x , y , z ( x + y − z ) x, y x , y ( y − z ) , z ( x − z ) x, x − z x , y , z ( x − z ) x, y, x − z x , y , z x, y, z z ( x − y ) , y (2 x − y − z ) , z ( x − z ) , x , x y ∅ x ( x + z ) , y , xy, z , z ( x − y ) y Table 5.
Ideals and invariants when V ( P ) = 65. The case of dimension P of dimension 4. In this sectionwe will classify them up to diffeomorphism. It turns out that the cohomological rigidityholds for them except for X and X . The X and X have isomorphic cohomologyrings and their total Pontryagin classes are preserved under an isomorphism between theircohomology rings but we do not know whether they are diffeomorphic or not.In Table 6 below, ID numbers whose toric Fano 4-folds are diffeomorphic are enclosedby curly braces as before. Thirteen toric Fano 4-folds with the ID numbers in the uppertwo lines in the row of ( V ( P ) , F ( P )) = (8 ,
16) are Bott manifolds and toric Fano 4-folds for( V ( P ) , F ( P )) = (6 , , (6 , , (7 ,
12) are generalized Bott manifolds, where V ( P ) denotesthe number of vertices of P as before and F ( P ) denotes the number of facets of P . ( P ) F ( P ) P H ∗ Diff ID5 5 1 1 1 1476 8 5 5 5 25, 138, 139, 144, 1456 9 4 3 3 44, { , } , 1467 11 3 3 3 24, 127, 1287 12 19 16 16 { , } , 31, 35, 42, 49, 66, { , } , 109117, { , } , 132, 133, 135, 140, 143, 1977 13 6 6 6 40, 41, 60, 64, 69, 1378 15 10 7 7 26, { , } , 45, 48, { , } , { , } , 1248 16 28 23 23 { , } , 75, { , } , { , } { , } , 33, 34, 37, 38, 47, 5993, 94, 104, { , } , 115, 1268 17 7 6 6 or 7 36, (50, 57), 58, 61, 65, 1108 18 2 2 2 53, 559 18 4 4 4 27, 46, 119, 1229 20 17 10 10 71, { , , } , { , } , 79, { , }{ , , } , 84, { , } , 102, 1209 21 4 4 4 51, 52, 56, 899 23 1 1 1 629 24 1 1 1 5410 24 8 6 6 { , } , { , } , 80, 85, 101, 12110 25 1 1 1 9810 30 1 1 1 6311 30 1 1 1 9912 36 1 1 1 100total 124 102 102 or 103 Table 6.
Diffeomorphism classification of toric Fano 4-foldsOur approach to obtain the table above is the same as the case of dimension 3 butthe analysis of the cohomology rings in dimension 4 becomes much more complicated andthere are many more cases to investigate.When V ( P ) = 5 , ,
12 or ( V ( P ) , F ( P )) = (9 , , (9 , , (10 , , (10 , V ( P ) , F ( P )) = (10 ,
24) is treated in Section 3. Indeed, toric Fano 4-folds with IDnumbers 72, 78, 80, 85, 86, 87, 101, 121 are respectively W , W , W , W , W , W , W , W in Subsection 3.4. Therefore, it suffices to investigate the remaining cases. We shall carryout this task one by one in this section. Convention .(1) The vertices of a smooth Fano 4-polytope P are shown in the database by Øbroand we number them as 1, 2, ... in the order shown in the database.(2) The first four vertices of P are the standard basis of Z , so we omit them andwrite the vertices from 5th in the tables below.(3) Minimal nonfaces of P are described using the numbering of the vertices of P .(4) I denotes the ideal of the cohomology ring H ∗ ( X q ) and its minimal generators aredescribed in the tables.(5) s.v.e., c.v.e. and 4-v.e. in the tables are up to sign unless the coefficient is Z / X p ∼ = X q means that X p is diffeomorphic to X q .(7) H ∗ ( X p ) ≇ H ∗ ( X q ) means that the cohomology rings are not isomorphic (as gradedrings).(8) The degree two cohomology elements corresponding to 5th, 6th, 7th, 8th, 9thvertices are respectively denoted by x, y, z, u, v . .1. The case where V ( P ) = 6 . We take two cases according to the values of F ( P ).5.1.1. ( V ( P ) , F ( P )) = (6 , . In this case, there are five smooth Fano 4-polytopes andthey are all combinatorially equivalent to a direct sum of a 3-simplex and a 1-simplex, sothe corresponding toric Fano 4-folds are generalized Bott manifolds. Using the data inTable 7, we obtain Table 8 which shows that the five cohomology rings are not isomorphicto each other.
ID vertices of P from 5th minimal nonfaces25 ( − , − , − , , (0 , , , − − , , , , (0 , − , − , −
1) 2346, 15139 ( − , − , − , , (0 , , , −
1) 1235, 46144 ( − , , , , (1 , − , − , −
1) 2346, 15145 ( − , , , , (0 , − , − , −
1) 2346, 15
Table 7.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (6 , ID I s.v.e. s.v.e. Z / Z / Z / x , y ( y − x ) ∅ ∅ x y x , y ( y − x ) ∅ ∅ x ∅ x , y ( y − x ) ∅ y x ( x − y ) , y y y y x , y y y ( x, y ) Table 8.
Ideals and invariants when ( V ( P ) , F ( P )) = (6 , V ( P ) , F ( P )) = (6 , . In this case, there are four smooth Fano 4-polytopes andthey are all combinatorially equivalent to a direct sum of two 2-simplices, so the corre-sponding toric Fano 4-folds are generalized Bott manifolds.
ID vertices of P from 5th minimal nonfaces44 ( − , − , , , (0 , , − , −
1) 125, 34670 ( − , − , , , (0 , , − , −
1) 125, 346141 ( − , − , , , (0 , , − , −
1) 125, 346146 ( − , − , , , (0 , , − , −
1) 125, 346
Table 9.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (6 , X ∼ = X , (5.1)see Subsection 5.5. On the other hand, using the data in Table 9, we obtain Table 10which shows that the three cohomology rings are not isomorphic to each other. ID I c.v.e. c.v.e. Z / x , y (2 x − y ) x x, y x , y ( x − y ) x x x , y x, y Table 10.
Ideals and invariants when ( V ( P ) , F ( P )) = (6 , The case where V ( P ) = 7 . .2.1. ( V ( P ) , F ( P )) = (7 , . In this case, there are three smooth Fano 4-polytopes andthey are all combinatorially equivalent. Using the data in Table 11, we obtain Table 12which shows that the three cohomology rings are not isomorphic to each other.
ID vertices of P from 5th minimal nonfaces24 ( − , − , − , , (0 , , , − , (0 , , , −
1) 1235, 1256, 37, 46, 47127 ( − , , , , (1 , , , − , ( − , − , − ,
0) 1237, 2347, 15, 46, 56128 ( − , , , , ( − , , , , (2 , − , − , −
1) 2347, 2357, 15, 16, 46
Table 11.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (7 , ID I s.v.e. c.v.e. Z / z ( x − y ) , y ( y + z − x ) , z ( z − x ) , x , x y ∅ y + z x ( x + z ) , y , xy, z , z ( y − x ) y x ( x + y − z ) , y ( y − z ) , y ( z − x ) , z , xz ∅ ∅ Table 12.
Ideals and invariants when ( V ( P ) , F ( P )) = (7 , V ( P ) , F ( P )) = (7 , . In this case, there are 19 smooth Fano 4-polytopes andthey are all combinatorially equivalent to a direct sum of 2-simplex and two 1-simplices,so the corresponding toric Fano 4-folds are generalized Bott manifolds.
ID vertices of P from 5th minimal nonfaces30 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 4731 ( − , − , , , (0 , , , − , (0 , , − ,
0) 125, 37, 4635 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 4742 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 4743 ( − , − , , , (0 , , , − , (0 , , − , −
1) 125, 37, 4649 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 4766 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 4768 ( − , − , , , (0 , , − , , (0 , , − , −
1) 125, 36, 4797 ( − , , , , (0 , − , , , (0 , , − , −
1) 347, 15, 26109 ( − , , , , (0 , − , , , (0 , , − , −
1) 347, 15, 26117 ( − , , , , (0 , , , − , (0 , − , − ,
0) 237, 15, 46129 ( − , , , , (0 , − , − , , (0 , , , −
1) 236, 15, 47132 ( − , , , , (0 , − , , , (0 , , − , −
1) 347, 15, 26133 ( − , , , , (0 , − , , , (0 , , − , −
1) 347, 15, 26134 ( − , , , , (0 , , , − , (1 , − , − ,
0) 237, 15, 46135 ( − , , , , (0 , , , − , (0 , − , − ,
0) 237, 15, 46136 ( − , , , , (0 , , , − , (0 , − , − , −
1) 237, 15, 46140 ( − , − , , , (0 , , − , , (0 , , , −
1) 125, 36, 47143 ( − , , , , (0 , − , , , (0 , , − , −
1) 347, 15, 26
Table 13.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (7 , X ∼ = X , X ∼ = X , X ∼ = X , (5.2)see Subsection 5.5. Using the data in Table 13, we obtain the following table. I s.v.e. s.v.e. Z / Z / Z / Z / x , y , z (2 x + y − z ) y y all ( x, y ) x, y, z,y + z x , y (2 x − y ) , z ( y − z ) ∅ y x ( x − y ) , y , z (2 x − z ) y ( y, z ) all y x , y , z (2 x − z ) y ( y, z ) all ( x, y )49 x , y ( x − y ) , z ( x + y − z ) ∅ ∅ ( x, y ) x x x , y ( x − y ) , z ( x − z ) ∅ ∅ all68(134) x , y ( x − y − z ) , z ( x − z ) ∅ ∅ ( x, z ) x x x , y , z ( x + y − z ) x, y ( x, y ) all ( x, y ) x, y, x + y,x + y + z x , y , z ( x − z )( y − z ) x, y ( x, y ) ( x, y )117 x , y ( x − y ) , z ( y − z ) x, x − y x ( x, y )129(136) x , y , z ( x + y − z ) x x ( x, y ) ( x, y )132 x , y ( y − z ) , z ( x − z ) x x ( x, z ) x x , y , z ( x − z ) x, y ( x, y ) all ( x, y ) x, y, x + yx + z x , y ( x − y ) , z x, x − y x all140 x , y , z ( x − z ) y y all ( x, y ) x, y x , y , z x, y ( x, y ) all all Table 14.
Ideals and invariants when ( V ( P ) , F ( P )) = (7 , H ∗ ( X ) and H ∗ ( X ) are both x up to sign. Therefore, if there isan isomorphism H ∗ ( X ) → H ∗ ( X ), then it induces an isomorphism H ∗ ( X ) / ( x ) → H ∗ ( X ) / ( x ). However, Table 15 shows that this does not occur, so H ∗ ( X ) = H ∗ ( X ). ID I + ( x ) c.v.e. Z / x , y ( x − y ) , z ( x + y − z ) all68(134) x , y ( x − y − z ) , z ( x − z ) ( x, z ) Table 15.
Distinguishment between H ∗ ( X ) and H ∗ ( X )The s.v.e. of H ∗ ( X ) and H ∗ ( X ) are both x, y up to sign and the transposition of x and y induces an automorphism of H ∗ ( X ) since the ideal I of H ∗ ( X ) is invariantunder the transposition. Therefore, if there is an isomorphism F : H ∗ ( X ) → H ∗ ( X ),then we may assume that F ( x ) = ± x , so that F induces an isomorphism : H ∗ ( X ) / ( x ) → H ∗ ( X ) / ( x ). However, Table 16 shows that this does not occur, so H ∗ ( X ) = H ∗ ( X ). ID I + ( x ) c.v.e. Z / y , z ( y − z ) y y , z all Table 16.
Distinguishment between H ∗ ( X ) and H ∗ ( X )5.2.3. ( V ( P ) , F ( P )) = (7 , . In this case, there are six smooth Fano 4-polytopes. Weshall prove that these six cohomology rings are not isomorphic to each other.
23D vertices of P from 5th minimal nonfaces40 ( − , − , , , (0 , , , − , (0 , , − , −
1) 125, 156, 237, 347, 4641 ( − , − , , , (1 , , − , − , (0 , , , −
1) 125, 127, 346, 356, 4760 ( − , − , , , (0 , , − , , (0 , , − , −
1) 125, 156, 247, 347, 3664 ( − , − , , , (1 , , − , − , ( − , − , ,
0) 125, 127, 346, 347, 5669 ( − , − , , , (0 , , − , − , (0 , , − , −
1) 125, 156, 346, 347, 27137 ( − , , , , (1 , − , , − , ( − , , − ,
0) 137, 246, 256, 347, 15
Table 17.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (7 , ID I c.v.e. c.v.e. Z / x , x y, z ( x − y ) , z ( x − z ) , y (2 x − y − z ) x x, y + z x ( x − y ) , y ( x − y ) , y ( y + z ) , xy , z (2 x − y − z ) ∅ x , x y, z ( x − y )( x − z ) , z ( x − z ) , y ( x − y − z ) x x x ( x + z ) , z ( x − y + z ) , y , z ( x − y ) , xy y y, z x , x y, y ( x − y − z ) , z , z ( x − y ) x, z x, z z , y , xy , z ( x − y ) , x ( x − y + z ) y, z y, z Table 18.
Ideals and invariants when ( V ( P ) , F ( P )) = (7 , H ∗ ( X ) and H ∗ ( X ) are respectively x aand y up to sign. There-fore, if there is an isomorphism H ∗ ( X ) → H ∗ ( X ), then it induces an isomorphism H ∗ ( X ) / ( x ) → H ∗ ( X ) / ( y ). However, this does not occur because H ∗ ( X ) / ( x ) = Z [ y, z ] / ( yz , z , y ( y + z )) ,H ∗ ( X ) / ( y ) = Z [ x, z ] / ( x ( x + z ) , z ( x + z ) , x z ) , and the degree sequences of these ideals are different. Therefore H ∗ ( X ) = H ∗ ( X ).Similarly, the c.v.e. of H ∗ ( X ) and H ∗ ( X ) are respectively x, z and y, z up tosign. Therefore, if there is an isomorphism H ∗ ( X ) → H ∗ ( X ), then it induces anisomorphism H ∗ ( X ) / ( x, z ) → H ∗ ( X ) / ( y, z ). However, this does not occur because H ∗ ( X ) / ( x, z ) = Z [ y ] / ( y ) , H ∗ ( X ) / ( y, z ) = Z [ x ] / ( x ) , and these quotient rings are not isomorphic. Therefore H ∗ ( X ) = H ∗ ( X ).5.3. The case where V ( P ) = 8 . V ( P ) , F ( P )) = (8 , . In this case, there are ten smooth Fano 4-polytopes andthey are all combinatorially equivalent to a direct sum of a 5-gon and a 2-simplex.
24D vertices of P from 5th minimal nonfaces26 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , − ,
0) 125, 36, 38, 47, 48, 6728 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , , −
1) 125, 36, 38, 47, 48, 6732 ( − , − , , , (0 , , , − , (0 , , − , , (0 , , , −
1) 125, 37, 38, 46, 48, 6745 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , − ,
0) 125, 36, 38, 47, 48, 6748 ( − , − , , , (0 , , − , , (0 , , − , , (0 , , , −
1) 125, 36, 37, 47, 48, 6867 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , − , −
1) 125, 36, 38, 47, 48, 67118 ( − , , , , (1 , , , − , ( − , , , , (0 , − , − ,
1) 238, 15, 17, 46, 47, 56123 ( − , , , , (1 , , , − , ( − , , , , (1 , − , − ,
0) 238, 15, 17, 46, 47, 56124 ( − , , , , (1 , , , − , ( − , , , , (0 , − , − ,
0) 238, 15, 17, 46, 47, 56125 ( − , , , , (1 , , , − , ( − , , , , (1 , − , − , −
1) 238, 15, 17, 46, 47, 56
Table 19.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) = (8 , X ∼ = X , X ∼ = X , X ∼ = X , (5.3)see Subsection 5.5. We obtain the following table from Table 19. ID I s.v.e. s.v.e. Z / x , y ( y + u ) , u ( y − z + u ) ,z (2 x − z ) , u (2 x − u ) , yz ∅ ( z, u )28(32) x , y , u ( y − z ) ,z (2 x − z − u ) , u (2 x − u ) , yz y ( y, u )45 x , y ( x − y − u ) , u (2 x − u ) ,x ( x − z ) , u ( x + y − z ) , yz ∅ u x , y ( x − y − z ) , z ( x − y − z ) ,z ( x + y − u ) , u ( x − u ) , yu ∅ ∅ x x , y ( x − y − u ) , u ( x − y − u ) ,z ( x − z − u ) , u ( y − z ) , yz ∅ ∅ x u , x ( x + z − u ) , z ( z − u ) ,y , z ( x − y ) , xy y y u , x ( x + z ) , z , y , z ( x − y ) , xy ∞ Table 20.
Ideals and invariants when ( V ( P ) , F ( P )) = (8 , H ∗ ( X ) and H ∗ ( X )are both x up to sign. Therefore, if there is an isomorphism H ∗ ( X ) → H ∗ ( X ), then itinduces an isomorphism H ∗ ( X ) / ( x ) → H ∗ ( X ) / ( x ). However, Table 21 shows thatthis does not occur, so H ∗ ( X ) = H ∗ ( X ). ID I + ( x ) c.v.e. Z / x , y ( x − y − z ) , z ( x − y − z ) , z ( x + y − u ) , u ( x − u ) , yu ( x, y, z, u )67(118) x , ( y ( x − y − u ) , u ( x − y − u ) , z ( x − z − u ) , u ( x − z − u ) , yz ( x, y + u, z + u ) Table 21.
Distinguishment between H ∗ ( X ) and H ∗ ( X )5.3.2. ( V ( P ) , F ( P )) = (8 , . In this case, there are 28 smooth Fano 4-polytopes andthere are two combinatorial types among them. Indeed, 13 polytopes among them have thesame combinatorial type as a cross-polytope as shown in Table 22. The corresponding toricFano 4-folds are Bott manifolds. The cohomology rings associated to these two differentcombinatorial types are not isomorphic to each other because the degree sequences of their deals are different, see Tables 23 and 25. More generally, it is known that a toric manifoldwhich has the same cohomology ring as a Bott manifold is indeed a Bott manifold ([18]). ID vertices of P from 5th minimal nonfaces74 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 4875 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − ,
0) 15, 26, 38, 4783 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 4895 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 4896 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − , −
1) 15, 26, 38, 47105 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − ,
0) 15, 26, 38, 47106 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 48108 ( − , , , , (0 , − , , , (0 , , − , , (0 , , − , −
1) 15, 26, 37, 48112 ( − , , , , (0 , , , − , (0 , − , , , (0 , , − ,
0) 15, 27, 38, 46114 ( − , , , , (0 , , , − , (0 , − , , − , (0 , , − ,
0) 15, 27, 38, 46130 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 48131 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − , −
1) 15, 26, 38, 47142 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 26, 37, 48
Table 22.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) =(8 , X ∼ = X , X ∼ = X , X ∼ = X , (5.4)see Subsection 5.5. Using the data in Table 22, we obtain the following table. ID I s.v.e. 4-v.e. Z / x , y , z , u ( x + y + z − u ) x, y, z all75 x , y , u ( z − u ) , z ( x + y − z ) x, y all83(108) x , y ( y − z ) , z , u ( x + y − u ) x, z, z − y ( x, y, z )95(131) x , y , z , u ( x + y − u ) x, y, z all105 x , y ( y − z ) , u ( y − u ) , z ( x − z ) x, x − z ( x, y, z )106 x , y , z ( y − z ) , u ( x − u ) x, y, x − u, y − z x , z , u ( y − u ) , y ( x − y ) x, z, x − y all114 x , z , u ( y + z − u ) , y ( x − y − z ) x, z ( x, y, z )130 x , y , z , u ( x − u ) x, y, z, x − u x , y , z , u x, y, z, u Table 23.
Ideals and invariants when ( V ( P ) , F ( P )) = (8 , F : H ∗ ( X ) → H ∗ ( X ). Since the s.v.e. of H ∗ ( X ) and H ∗ ( X ) are x, y, z and any permutation of x, y, z is an automorphism of H ∗ ( X ), we may assume that F ( x ) = ± x , F ( y ) = ± y , F ( z ) = ± z . Therefore, F inducesan isomorphism H ∗ ( X ) / ( x, y ) ∼ = H ∗ ( X ) / ( x, y ). However, this does not occur because H ∗ ( X ) / ( x, y ) = Z [ z, u ] / ( z , u ( u − z )) , H ∗ ( X ) = Z [ z, u ] / ( z , u )and these two rings are not isomorphic (e.g. their s.v.e. are different). Therefore, H ∗ ( X ) = H ∗ ( X ).Next, we shall treat the other case where P is not combinatorially equivalent to across-polytope. There are 15 smooth Fano 4-polytopes in this case as shown in Table 24.
26D vertices of P from 5th minimal nonfaces29 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , , −
1) 125, 157, 28, 36, 47, 4833 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , , −
1) 125, 157, 28, 36, 47, 4834 ( − , − , , , (0 , , − , , (1 , , , − , (0 , , , −
1) 125, 257, 18, 36, 47, 4837 ( − , − , , , (0 , , , − , (0 , , − , , (0 , , , −
1) 125, 156, 28, 37, 46, 4838 ( − , − , , , (0 , , , − , (0 , , , − , (0 , , − , −
1) 125, 156, 27, 38, 46, 4739 ( − , − , , , (0 , , , − , (0 , , , − , (0 , , − , −
1) 125, 156, 27, 38, 46, 4747 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , , −
1) 125, 157, 28, 36, 47, 4859 ( − , − , , , (0 , , − , , (0 , , − , , (0 , , , −
1) 125, 156, 27, 36, 37, 4893 ( − , , , , (0 , − , , , (0 , , , − , (0 , − , − ,
0) 238, 348, 15, 26, 47, 6794 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − , −
1) 238, 348, 15, 26, 47, 67104 ( − , , , , (0 , − , , , (0 , , − , , (0 , − , , −
1) 248, 348, 15, 26, 37, 67111 ( − , , , , (0 , , , − , (0 , − , − , , (0 , , , −
1) 237, 267, 15, 38, 46, 48115 ( − , , , , (0 , , , − , (0 , , , − , (1 , − , − ,
0) 238, 268, 15, 37, 46, 47116 ( − , , , , (0 , , , − , (0 , , , − , (0 , − , − ,
0) 238, 268, 15, 37, 46, 47126 ( − , , , , (1 , , , − , (0 , − , , , ( − , , − ,
0) 138, 348, 15, 27, 46, 56
Table 24.
Vertices and minimal nonfaces of P with ( V ( P ) , F ( P )) =(8 , X ∼ = X , X ∼ = X , (5.5)see Subsection 5.5. Using the data in Table 24, we obtain the following table. ID I s.v.e. s.v.e. Z / Z / Z / x , x z, u ( x − z ) , y , z (2 x + y − z − u ) , u ( x + y − u ) y y ( x, y, z + u ) ( x, y )33 x ( x − y ) , x z, u ( x − y − z ) , y ,z (2 x − z − u ) , u (2 x − z − u ) y ( y, z + u ) all y x ( x − y ) , xz ( x − y ) , u ( x − z ) ,y , z (2 x − z − u ) , u ( x − u ) y ( y, z + u ) ( x, y, z + u )37 x , x y, u ( x − y ) , z ,y (2 x − y − u ) , u ( x − u ) z ( y + u, z ) all ( x, y, z )38 x ( x − u ) , x y, z ( x − y − u ) , u ,y (2 x − y − z − u ) , z ( x − z ) u u ( x, y + z, u ) u x , x z, u ( x − z ) , y ( x − y ) ,z ( x + y − z − u ) , u ( y − u ) ∅ ∅ ( x, y, z ) x x , x y, z ( x − y ) , y ( x − y − z ) ,z , u ( x − u ) z z all93 u ( y − z + u ) , u ( x − u ) ,x , y ( y + u ) , z ( x − z ) , yz x, x − z x ( x, z, u ) ( x, z )94 u ( y − z ) , u ( x − u ) , x ,y , z ( x − z − u ) , yz x, y ( x, y ) all ( x, y )104 u ( x − u ) , u ( y − z )( x − u ) ,x , y ( y + u ) , z , yz x, z ( x, z ) ( x, z, u )111(116) z , yz , x , u ( y − z ) ,y ( x − y + z − u ) , u ( x − u ) x, x − u x ( x, z, u ) ( x, z, u )115 u , yu , x ( x − u ) , z ( y − u ) ,y ( x − y − z ) , z ( x − y − z ) ∅ ∅ ( x, y, u ) u u , u ( x − y ) , x ( x + u ) ,z , y , xy y, z ( y, z ) all ( y, z, u ) Table 25.
Ideals and invariants when ( V ( P ) , F ( P )) = (8 , H ∗ ( X ) and H ∗ ( X ) are respectively and u up to sign. Therefore, if there is an isomorphism H ∗ ( X ) → H ∗ ( X ), then itinduces an isomorphism H ∗ ( X ) / ( x ) → H ∗ ( X ) / ( u ). However, Table 26 shows thatthis does not occur, so H ∗ ( X ) = H ∗ ( X ). ID I + ((c.v.e.) ) c.v.e. Z / x , u ( x − z ) , y ( x − y ) , z ( x + y − z − u ) , u ( y − u ) ( x, y, z + u )115 u , x ( x − u ) , z ( y − u ) , y ( x − y − z ) , z ( x − y − z ) ( x, u ) Table 26.
Distinguishment between H ∗ ( X ) and H ∗ ( X )5.3.3. ( V ( P ) , F ( P )) = (8 , . There are seven smooth Fano 4-polytopes in this case andthere are two combinatorial types among them: one is ID numbers 36 ,
65 and the otherone is ID numbers 50 , , , , Z / X and X do not satisfy the condition in Lemma 2.3 up to unimod-ular equivalence, so they are not weakly equivariantly diffeomorphic with respect to therestricted ( S ) -actions. However, their cohomology rings are isomorphic to each other.Indeed, the map ( x, y, z, u ) → ( − x + 2 u, − y + u, u, − z ) (5.6)gives an isomorphism H ∗ ( X ) → H ∗ ( X ). It does not preserve their total Chern classes(even their first Chern classes) but it does preserve their total Pontryagin classes (in fact,the first and second Pontryagin classes because of dimensional reason) which are given by p ( X ) = (1 + x ) (1 + ( x − y − z + u ) )(1 + ( x − y ) )(1 + ( x − z ) )(1 + y )(1 + z )(1 + u ) ,p ( X ) = (1 + x ) (1 + ( x − y + u ) )(1 + ( x − y ) )(1 + ( x − z ) )(1 + y )(1 + z )(1 + u ) , see Remark 2.2. ID vertices of P from 5th minimal nonfaces36 ( − , − , , , (0 , , , − , (0 , − , − , , (0 , , , −
1) 28, 46, 48, 125, 156, 158, 237, 347, 36750 ( − , − , , , (0 , , − , , (0 , , , − , (0 , − , ,
0) 28, 36, 47, 125, 156, 157, 34857 ( − , − , , , (0 , , − , , (0 , − , , , (0 , , , −
1) 27, 36, 48, 125, 156, 158, 34758 ( − , − , , , (0 , , − , , (0 , − , , , (0 , , − , −
1) 27, 36, 48, 125, 156, 158, 34761 ( − , − , , , (1 , , − , − , ( − , , , , (0 , − , ,
0) 17, 28, 56, 125, 346, 347, 34865 ( − , − , , , (1 , , − , − , ( − , − , , , (0 , , − , −
1) 56, 58, 67, 125, 127, 128, 346, 347, 348110 ( − , , , , (0 , , , − , (1 , , − , , ( − , − , ,
0) 15, 37, 46, 128, 238, 248, 567
Table 27. ( V ( P ) , F ( P )) = (8 , I c.v.e. Z /
236 ( x − y + z ) u, (2 x − y + z − u ) y, ( x − u ) u, x ( x + z ), x + z, ( y + z, u ) x y, x u, ( x + z ) z , z ( x − u ) , z y xy, xu, yz, ( x + z ) x, ( x + z ) z , z + u, y + u, x + z ( y − z ) u, ( x − y − u ) y, ( x − u ) z, ( y + u ) u
50 ( x − y − z + u ) u, ( x − y ) y, ( x − z ) z , ( y, z ) , x + ux ( x + u ) , x y, x z, ( x − y )( x − z ) u
57 ( x − y + z ) z, ( x − y ) y, ( x − u ) u , ( y, u ) , x + zx ( x + z ) , x y, x u, ( x − y )( x − u ) z z ( x − y + z − u ) , y ( x − y − u ) , u ( x − u ), z, u, y + ux ( x + z ) , x y, x u, z ( x − y )( x − u )61 z ( x − y + z ) , u ( x − y + u ) , xy , ( y, z, u ) x ( x + z )( x + u ) , y , z , u
110 ( x − z + u ) x, ( y − z ) z, ( x − y ) y , z, uu , u ( y − z ) , u ( x − y ) , xyz Table 28.
Ideals and invariants when ( V ( P ) , F ( P )) = (8 , V ( P ) , F ( P )) = (8 , . In this case, there are two smooth Fano polytopes as shownin Table 29. Their cohomology rings can be distinguished by c.v.e. over Z / ID vertices of P from 5th minimal nonfaces53 ( − , − , , , (0 , , − , , (1 , , , − , ( − , − , ,
0) 36, 47, 125, 128, 138, 156, 248, 257, 348, 56755 ( − , − , , , (0 , , − , , (1 , , , − , (0 , , − , −
1) 36, 47, 125, 128, 138, 156, 248, 257, 348, 567
Table 29. ( V ( P ) , F ( P )) = (8 , ID I c.v.e. Z /
353 ( x − y ) y, ( x − z ) z, ( x + u ) x, u , ( x − y ) u , u ( x + u ) xy, ( x − z ) u , ( x + u ) xz, ( x − y )( x − z ) u, xyz
55 ( x − y − u ) y, ( x − z − u ) z, x , ( x − y )( x − z ) u, u , ( x, u ) x y, u ( x − z ) , x z, u ( x − y ) , xyz Table 30.
Ideals and invariants when ( V ( P ) , F ( P )) = (8 , The case where V ( P ) = 9 . V ( P ) , F ( P )) = (9 , . In this case, there are four smooth Fano 4-polytopes asshown in Table 31 and they are all combinatorially equivalent to a direct sum of a 2-simplex and a 6-gon. Using the data in Table 31, we obtain Table 32 which shows thatthe four cohomology rings are not isomorphic to each other.
ID vertices of P from 5th minimal nonfaces27 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , − , , (0 , , , −
1) 36, 38, 39, 47, 48, 49, 67, 69, 78, 12546 ( − , − , , , (0 , , − , , (0 , , , − , (0 , , − , , (0 , , , −
1) 36, 38, 39, 47, 48, 49, 67, 69, 78, 125119 ( − , , , , (1 , , , − , ( − , , , , (0 , − , − , , (0 , , , −
1) 15, 17, 19, 46, 47, 49, 56, 59, 67, 238122 ( − , , , , (1 , , , − , ( − , , , , (0 , , , − , (0 , − , − ,
0) 15, 17, 18, 46, 47, 48, 56, 58, 67, 239
Table 31. ( V ( P ) , F ( P )) = (9 , I s.v.e. Z / Z / y ( y + u ) , u ( y + u ) , v ( z − u ) , z (2 x − z − v ) , ( y + u, z + v, u + v ) ( y + u, x ) u (2 x + y − v ) , v (2 x − z − v ) , yz, yv, zu, x y ( x − y − u ) , u ( x − y − u ) , v ( x + z − u ) , z ( x − z − v ) , u + v xu ( x + y − v ) , v ( x − z − v ) , yz, yv, zu, x x ( x + z ) , z ( x + z ) , v ( y − z ) , y ( y − u + v ) , x + z ( x + z, u ) z ( x + u − v ) , v ( y − u + v ) , xy, xv, yz, u x ( x + z ) , z ( x + z ) , u ( y − z ) , y ( y + u ) , ( x + z, y + u, z + u ) all z ( x − u ) , u ( y + u ) , xy, xu, yz, v Table 32.
Ideals and invariants when ( V ( P ) , F ( P )) = (9 , V ( P ) , F ( P )) = (9 , . In this case, there are 17 smooth Fano 4-polytopes asshown in Table 33 and they are all combinatorially equivalent to a direct sum of two1-simplices and a 5-gon.
ID vertices of P from 5th minimal nonfaces71 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , − , (0 , , − ,
0) 15, 26, 37, 39, 48, 49, 7873 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , − , (0 , , , −
1) 15, 26, 37, 39, 48, 49, 7876 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − , , (0 , , , −
1) 15, 26, 38, 39, 47, 49, 7877 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , − , (0 , − , ,
0) 15, 26, 29, 37, 48, 49, 6879 ( − , , , , (0 , − , , , (0 , , − , , (1 , , , − , ( − , , ,
0) 15, 19, 26, 37, 48, 49, 5881 ( − , , , , (0 , − , , , (0 , , − , , (1 , , , − , (0 , , , −
1) 15, 19, 26, 37, 48, 49, 5882 ( − , , , , (0 , − , , , (0 , , − , , (0 , − , , , (0 , , , −
1) 15, 26, 28, 37, 48, 49, 6984 ( − , , , , (0 , − , , , (0 , , , − , (0 , − , , , (0 , , − ,
0) 15, 26, 28, 39, 47, 48, 6788 ( − , , , , (0 , − , , , (0 , , , − , (0 , − , , , (0 , , − , −
1) 15, 26, 28, 39, 47, 48, 6790 ( − , , , , (0 , − , , , (0 , , , − , (0 , , − , , (0 , , , −
1) 15, 26, 29, 38, 47, 49, 6791 ( − , , , , (0 , − , , , (0 , , , − , (0 , , , − , (0 , , − , −
1) 15, 26, 28, 39, 47, 48, 6792 ( − , , , , (0 , − , , , (0 , , , − , (0 , , , − , (0 , , − , −
1) 15, 26, 28, 39, 47, 48, 67102 ( − , , , , (0 , − , , , (0 , , − , , (0 , − , , , (0 , , , −
1) 15, 26, 28, 37, 38, 49, 67103 ( − , , , , (0 , − , , , (0 , , − , , (0 , − , , , (0 , , − , −
1) 15, 26, 28, 37, 38, 49, 67107 ( − , , , , (0 , − , , , (0 , , − , , (0 , , , − , (0 , , − , −
1) 15, 26, 37, 39, 48, 49, 78113 ( − , , , , (0 , , , − , (0 , − , , , (0 , , − , , (0 , , , −
1) 15, 27, 38, 39, 46, 49, 68120 ( − , , , , (1 , , , − , ( − , , , , (0 , − , , , (0 , , − ,
0) 15, 17, 28, 39, 46, 47, 56
Table 33. ( V ( P ) , F ( P )) = (9 , X ∼ = X ∼ = X , X ∼ = X , X ∼ = X ,X ∼ = X ∼ = X , X ∼ = X , (5.7)see Subsection 5.5. Using the data in Table 33, we obtain the following table. I s.v.e. s.v.e. Z / Z / x , y , z ( z + v ) , v ( z − u + v ) ,u ( x + y − u ) , v ( x + y − v ) , zu x, y x , y , z , v ( z − u ) ,u ( x + y − u − v ) , v ( x + y − v ) , zu x, y, z x , y ( y − z + v ) , v ( x + z − v ) ,z , u ( x − u ) , v ( x + y − u ) , yu x, z, x − u ( x, z ) ( x, z, u, v )79 x ( x + v ) , v ( v − y ) , y ( y − z ) ,z , u ( y − u ) , v ( x + y − u ) , xu z, z − y x , v ( x − u ) , y ( y − z ) , z ,u ( y − u − v ) , v ( y − v ) , xu x, z, z − y ( x, z ) ( x, y, z, v )82(91, 107) x , y ( y − z + u ) , u ( y − z + u ) ,z , u ( x + y − v ) , v ( x − v ) , yv x, z, x − v,z − y − u ( x, z ) ( x, y + u, z, v )84 x , y ( y + u ) , u ( x − u ) , v ,u ( x + y − z ) , z ( x − z ) , yz x, v, x − u,x − z ( x, v ) all90(113) x , y , v ( y − z ) , u ,v ( x − v ) , z ( x − z − v ) , yz x, y, u,x − v x , y ( y + u ) , u , z ,u ( y − z ) , v ( x − v ) , yz ∞ ( x, z, u )120 x ( x + z ) , z , u , v ,y , z ( x − y ) , xy ∞ ( y, z, u, v ) Table 34.
Ideals and invariants when ( V ( P ) , F ( P )) = (9 , Z / H ∗ ( X ) and H ∗ ( X )are both ( x, z ). Therefore, if there is an isomorphism H ∗ ( X ) → H ∗ ( X ), then it inducesan isomorphism ( H ∗ ( X ) ⊗ Z / / ( x, z ) → ( H ∗ ( X ) ⊗ Z / / ( x, z ). However, Table 35shows that this does not occur because the degree sequences of the ideals are different, so H ∗ ( X ) ≇ H ∗ ( X ). ID I ⊗ Z / x, z )77(88) y ( y + v ) , v , u , v ( y + u ) , yu uv, y , u ( y + u ) , v ( y + v ) Table 35.
Distinguishment between H ∗ ( X ) and H ∗ ( X )5.4.3. ( V ( P ) , F ( P )) = (9 , . In this case, there are four smooth Fano 4-polytopes andthere are two combinatorial types among them. Indeed, the combinatorial type of ID 52is different from the others as is seen from Table 36.
ID vertices of P from 5th minimal nonfaces51 ( − , − , , , (0 , , − , , (0 , , , − , (0 , − , , , (0 , , − ,
0) 28, 29, 36, 39, 47, 68, 125, 156, 157, 159, 34852 ( − , − , , , (0 , , − , , (1 , , , − , (0 , , − , , (0 , , , −
1) 19, 28, 36, 38, 47, 49, 125, 156, 257, 56756 ( − , − , , , (0 , , − , , (0 , − , , , (0 , , − , , (0 , , , −
1) 27, 28, 36, 38, 49, 67, 125, 156, 158, 159, 34789 ( − , , , , (0 , − , , , (0 , , , − , (0 , − , , , (1 , , − , −
1) 15, 26, 28, 47, 48, 67, 178, 239, 349, 359, 369
Table 36. ( V ( P ) , F ( P )) = (9 , H ∗ ( X ) can be distin-guished from the other three cohomology rings by the degree sequences of the ideals. Onthe other hand, Table 38 shows that the three cohomology rings are not isomorphic toeach other. I u ( x − z + u ) , v ( x − y − z + u ) , y ( x − y − v ) , v ( x − y − v ) ,z ( x − z ) , yu, x ( x + u ) , x y, x z, x v, u ( x − v )( x − z )52 v ( x − z ) , u ( x − y ) , y ( x − y − u ) , u , z ( x − z − v ) ,v , x , x y, x z, xyz z ( x + z ) , u ( u + z ) , y ( x − y − u ) , u ( x − y − u ) , v ( x − v ) ,yz, x ( x + z ) , x y, x u, x v, z ( x − u )( x − v )89 x ( x − v ) , y ( y + u ) , u ( y − z + u ) , z ( x − z − v ) , u ( x − u − v ) ,yz, zu ( x − v ) , v ( z − u ) , v ( z + v ) , v x, v y Table 37.
Ideals when ( V ( P ) , F ( P )) = (9 , ID s.v.e. Z / Z c.v.e. Z / Z c.v.e. Z / Z ∅ ( z, y + v ) x + u, x + u + v, z, y + z, y + v, y + z + v
52 ( u, v )56 ∅ ( x + z, y + u, v )89 ∅ ( x, y + u ) x, x + z + u, y + u, z + v, u + v Table 38. ( V ( P ) , F ( P )) = (9 , Diffeomorphism.
In the previous subsections, we claimed some diffeomorphismsamong toric Fano 4-folds, i.e. (5.1), (5.2), (5.3), (5.4), (5.5), (5.7). In this subsection, weestablish them by showing that the condition in Lemma 2.3 is satisfied in each case.In the following, we express the vertices of a smooth Fano polytope P q with ID number q in terms of a matrix where each row shows a vertex of P q and the numbers written on theleft side of a matrix are the numbers of the vertices. For instance, the vertices of P arearranged as (1 , , , , ,
6) while the vertices of P are arranged as (1 , , , , ,
4) in theirmatrices. The correspondence (1 , , , , , → (1 , , , , ,
4) gives a bijection betweenthe vertices of P and P , which preserves the combinatorial structures of P and P .The multiplication by a 4 × •
70 and 141 − − − − − − − − − = − − − − •
30 and 43 − − − − − − − −
10 0 0 − − − − = − − −
11 1 0 −
20 0 − •
68 and 134 − − − − − − − − − = − − − − •
129 and 136 − − − − − − − −
10 0 0 − − − = − −
11 0 0 − − − •
28 and 32 − − − −
10 0 0 − −
10 0 0 − − − − − = − − − − −
10 0 − •
67 and 118 − − − −
10 0 − − − − − − − = − − − − •
123 and 125 − − − − − − − − − − − −
10 1 0 00 0 1 00 0 0 − = − −
11 0 0 − − − − •
74 and 96 − − − − − − − −
10 0 0 − − − − = − − −
11 0 0 −
10 1 0 −
10 0 − •
83 and 108 − − − − − − − − − − −
10 1 0 0 = − −
11 0 0 − − − •
95 and 131 − − − − − − − − − − − = − −
11 0 0 − − − •
29 and 39 − − − −
10 0 0 − − − − −
10 1 0 −
10 0 0 − − − − = − − −
11 1 0 −
20 0 − − •
111 and 116 − − − − − −
10 0 0 − − − − − − = − − − − − −
10 0 0 1 •
73 and 76 and 92 − − − −
10 0 0 − −
10 0 0 − − − − − = − − − − −
10 0 − − − − − −
10 0 0 − − − − = − − −
11 0 0 − − −
10 0 − •
77 and 88 − − − − − − − − − − − − −
10 0 1 00 0 0 − = − − −
11 0 0 −
10 1 0 −
10 1 − − •
81 and 103 − − − −
10 0 0 − − − − − − − − − = − −
11 0 0 − − − − •
82 and 91 and 107 − − − − − − − − −
10 1 0 −
10 0 0 − − −
10 0 1 00 0 0 − = − − −
11 0 0 −
10 1 0 −
10 1 − − − − − − − − − − = − − − − − − •
90 and 113 − − −
10 0 − − −
10 1 0 00 0 0 − − − − − = − − − − − − c -preserving cohomology ring isomorphism As we observed, cohomology rings do not distinguish toric Fano manifolds as varieties.Very recently, motivated by McDuff’s question on the uniqueness of toric actions on amonotone symplectic manifold, Y. Cho, E. Lee, S. Park and the third author made thefollowing conjecture and verified it for Fano Bott manifolds ([7]). onjecture ([7]) . If there is a cohomology ring isomorphism between toric Fano manifoldswhich preserves their first Chern classes, then they are isomorphic as varieties.
In this section, we prove the following theorem mentioned in Introduction, which givesfurther supporting evidence to the conjecture.
Theorem 6.1.
The conjecture above is true for toric Fano d -folds with d = 3 , or withPicard number ≥ d − . In order to prove the theorem above, it suffices to check that there is no c -preservingcohomology ring isomorphism between toric Fano d -folds which have isomorphic cohomol-ogy rings. If there is a c -preserving cohomology ring isomorphism between toric Fano d -folds X and Y , then c ( X ) d evaluated on the fundamental class of X , in other wordsthe degree ( − K X ) d of X , agrees with that of Y . We obtain the following tables from thedatabase of Øbro. They together with Tables 1 and 6 show that the degrees are differentfor toric Fano 3- or 4-folds which have isomorphic cohomology rings except one pair ID70 and 141. ID 11, 18 10, 13degree 52, 44 44, 40
Table 39.
Degrees of toric Fano 3-folds with isomorphic cohomology rings
ID 70, 141 30, 43 68, 134 129, 136 28, 32 67, 118 123, 125degree 513, 513 592, 400 432, 480 496, 400 478, 382 351, 447 415, 367ID 74, 96 83, 108 95, 131 29, 39 111, 116 50, 57 73, 76, 92degree 480, 352 448, 352 416, 352 463, 337 389, 347 417, 369 394, 330, 310ID 77, 88 81, 103 82, 91, 107 90, 113 72, 87 78, 86degree 405, 331 373, 325 341, 363, 229 352, 320 308, 268 298, 278
Table 40.
Degrees of toric Fano 4-folds with isomorphic cohomology ringsAs for ID 70 and 141, more detailed observation is necessary. It follows from Table 9and Remark 2.2 that H ∗ ( X ) = Z [ x, y ] / ( x , y ( x − y ) ) , c ( X ) = x + 3 yH ∗ ( X ) = Z [ x, y ] / ( x , y ( x − y )) , c ( X ) = 2 x + 3 y. (6.1)An elementary computation shows that an isomorphism H ∗ ( X ) → H ∗ ( X ) is givenby either ( x, y ) → ( x, x − y ) or ( x, y ) → ( − x, − x + y ) but both isomorphisms are not c -preserving. This completes the proof of the theorem when d = 3 , d -folds with Picard number ≥ d − Z d = Z × ( X P ) d − and Z d = Z × ( X P ) d − , where d is odd ≥ W d = W × ( X P ) d − and W d = W × ( X P ) d − , where d is even ≥ W d = W × ( X P ) d − and W d = W × ( X P ) d − , where d is even ≥ Z , Z ) = ( X , X ) , ( W , W ) = ( X , X ) , ( W , W ) = ( X , X ) s mentioned above Convention in Sections 4 and 5. The degree of a product of projectivevarieties X and Y of dimension p and q is (cid:0) p + qp (cid:1) times the product of degrees of X and Y , so it follows from Tables 39 and 40 that the three pairs above have different degreesrespectively. This completes the proof of the theorem. References [1] B. Assarf, M. Joswig and A. Paffenholz, Smooth Fano polytopes with many vertices,
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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] (M. Masuda) Department of Mathematics, Graduate School of Science, Osaka City Uni-versity, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address : [email protected]@sci.osaka-cu.ac.jp