On a Morelli type expression of cohomology classes of toric varieties
aa r X i v : . [ m a t h . A T ] J u l ON A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSESOF TORIC VARIETIES
AKIO HATTORI
Abstract.
Let X be a complete Q -factorial toric variety of dimension n and ∆ the fanin a lattice N associated to X . ∆ is necessarily simplicial. For each cone σ of ∆ therecorresponds an orbit closure V ( σ ) of the action of complex torus on X . The homologyclasses { [ V ( σ )] | dim σ = k } form a set of specified generators of H n − k ( X, Q ). It isshown that, given α ∈ H n − k ( X, Q ), there is a canonical way to express α as a linearcombination of the [ V ( σ )] with coefficients in the field of rational functions of degree 0on the Grassmann manifold G n − k +1 ( N Q ) of ( n − k + 1)-planes in N Q . This generalizesMorelli’s formula [10] for α the ( n − k )-th component of the Todd homology class of thevariety X . Morelli’s proof uses Baum-Bott’s residue formula for holomorphic foliationsapplied to the action of complex torus on X whereas our proof is entirely combinatorialso that it tends to more general situations. Introduction
Let X be a toric variety of dimension n and ∆ X the fan associated to X . ∆ X is acollection of rational convex cones in N R = N ⊗ R where N is a lattice of rank n . For each k -dimensional cone σ in ∆ X , let V ( σ ) be the corresponding orbit closure of dimension n − k and [ V ( σ )] ∈ A n − k ( X ) be its Chow class. Then the Todd class T n − k ( X ) of X canbe written in the form(1) T n − k ( X ) = X σ ∈ ∆ X , dim σ = k µ k ( σ )[ V ( σ )] . However, since the [ V ( σ )] are not linealy independent, the coefficients µ k ( σ ) ∈ Q are notdetermined uniquely. Danilov [2] asks if µ k ( σ ) can be chosen so that it depends only onthe cone σ not depending on a particular fan in which it lies.The equality (1) has a close connection with the number P ) of lattice points con-tained in a convex lattice polytope P in M R where M is a dual lattice of N . For apositive integer ν the number νP ) is expanded as a polynomial in ν (called Ehrhartpolynomial): νP ) = X k a k ( P ) ν n − k . A convex lattice polytope P in M R determines a complete toric variety X and aninvariant Cartier divisor D on X . There is a one-to-one correspondence between the cells { σ } of ∆ X and the faces { P σ } of P . Then the coefficient a k ( P ) has an expression(2) a k ( P ) = X dim σ = k µ k ( σ ) vol P σ with the same µ k ( σ ) as in (1). Mathematics Subject Classification.
We shall restrict ourselves to the case where X is non-singular. Put D i = [ V ( σ i )] forthe one dimensional cone σ i , and let x i ∈ H ( X ) denotes the Poincar´e dual of D i . Thedivisor D is written in the form D = P i d i D i with positive integers d i . Put ξ = P i d i x i .It is known that a k ( P ) = Z X e ξ T k ( X ) and vol P σ = Z X e ξ x σ , where T k ( X ) ∈ H k ( X ) Q = H k ( X ) ⊗ Q is the k -th component of the Todd cohomologyclass, the Poincar´e dual of T n − k ( X ), and x σ ∈ H k ( X ) is the Poincar´e dual of [ V ( σ )].The cohomology class x σ can also be written as x σ = Q j x j where the product runs oversuch j that σ j is an edge of σ . Then the equality (2) can be rewritten as(3) Z X e ξ T k ( X ) = X dim σ = k µ k ( σ ) Z X e ξ x σ . The reader is referred to [4] Section 5.3 for details and Note 17 there for references.In his paper [10] Morelli gave an answer to Danilov’s question. Let Rat( G n − k +1 ( N Q ))) denote the field of rational functions of degree 0 on the Grassmann manifold of ( n − k +1)-planes in N Q . For a cone σ of dimension k in N R he associates a rational function µ k ( σ ) ∈ Rat( G n − k +1 ( N Q ))) . With this µ k ( σ ), the right hand side of (1) belongs toRat( G n − k +1 ( N Q ))) ⊗ Q A n − k ( X ) Q , and the equality (1) means that the rational function with values in A n − k ( X ) Q in theright hand side is in fact a constant function equal to T n − k ( X ) in A n − k ( X ) Q . In otherwords this means that X σ ∈ ∆ X , dim σ = k µ k ( σ )( E )[ V ( σ )] = T n − k ( X )for any generic ( n − k + 1)-plane E in N Q .Morelli gives an explicit formula for µ k ( σ ) when the toric variety is non-singular usingBaum-Bott’s residue formula for singular foliations [1] applied to the action of ( C ∗ ) n on X . He then shows that the function µ k ( σ ) is additive with respect to non-singularsubdivisions of the cone σ . This fact leads to (1) in its general form.One can ask a similar question about general classes other than the Todd class whetherit is possible to define µ ( x, σ ) ∈ Rat( G n − k +1 ( N Q ))) for x ∈ A n − k ( X ) in a canonical wayto satisfy(4) x = X σ ∈ ∆ X , dim σ = k µ ( x, σ )[ V ( σ )] . When X is non-singular one can expect that µ ( x, σ ) satisfies a formula analogous to (3)(5) Z X e ξ x = X dim σ = k µ ( x, σ ) Z X e ξ x σ for any cohomology class ξ = P i d i x i . In this sense the formula does not explicitlyrefer to convex polytopes. Fulton and Sturmfels [5] used Minkowski weights to describeintersection theory of toric varieties. For complete non-singular varieties or Q -factorialvarieties X the Minkowski weight γ x : H n − k ) ( X ) → Q corresponding to x ∈ H k ( X ) isdefined by γ x ( y ) = R X xy . Thus, if the d i are considered as variables in ξ , the formula (5)is considered as describing γ x as a linear combination of the Minkowski weights of γ x σ . N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 3
The purpose of the present paper is to establish the formula (5) by showing an explicitformula for µ ( x, σ ) when X is a Q -factorial complete toric variety, that is, when ∆ is acomplete simplicial fan. Moreover our proof is based on a simple combinatorial argumentwhich can be generalized to the case of multi-fans introduced by Masuda [8]. Topologicallythe formula concerns equivariant cohomology classes on so-called torus orbifolds (see [6]).This would suggest that actions of compact tori equipped with some nice conditions admittopological residue formulas similar to Baum-Bott’ formula.In Section 2 the definition of µ ( x, σ ) will be given and main results will be stated. Intopological language, Theorem 2.1 states that (5) holds for any complete Q -factorial toricvarieties X . In Corollary 2.2 the explicit formula µ k for the Todd class x = T ( X ) isgiven, generalizing that of Morelli. In Corollary 2.3 it will be shown that (4) holds for acomplete Q -factorial toric varieties X .Section 3 is devoted to the proof of Theorem 2.1 and Corollary 2.3. Corollary 2.2 willbe proved in a generalized form in Section 5. The results in Section 2 are generalizedto complete multi-fans in Section 5 after some generalities about multi-fans and multi-polytopes are introduced in Section 4. Theorem 5.1, Corollary 5.2 and Corollary 5.3are the corresponding generalized results. Roughly speaking Theorem 5.1 and Corollary5.2 hold for (compact) torus orbifolds in general whereas Corollary 5.3 holds for torusorbifolds whose cohomology ring is generated by its degree two part like complete Q -factorial toric varieties. 2. Statement of main results
It is convenient to describe a fan ∆ in a form suited for combinatorial manipulation.We assume that the fan ∆ is simplicial, that is, every k -dimensional cone of ∆ has exactly k edges (one dimensional cones). Let Σ (1) denote the set of one dimensional cones. Thenthe set of cones of ∆ forms a simplicial complex Σ with vertex set Σ (1) . The set of ( k − k -dimensional cones, will be denoted by Σ ( k ) . The cone corresponding toa simplex J ∈ Σ will be denoted by C ( J ) and the fan ∆ is denoted by ∆ = (Σ , C ). Notethat Σ (0) consists of a unique element o corresponding to the empty set as a subset ofΣ (1) and C ( o ) = 0. (In [6] Σ with Σ (0) added is called augmented simplicial set.)For each i ∈ Σ (1) let v i be the primitive vector in N ∩ C ( i ). We put V = { v i } i ∈ Σ (1) and V J = { v j | j ∈ J } for J ∈ Σ. Let N J, V be the sublattice generated by V J and N J theminimal primitive (saturated) sublattice containing N J, V . The quotient group N J /N J, V is denoted by H J . If σ = C ( J ), H J is the multiplicity along the normal direction at ageneric point of V ( σ ) in the toric variety (orbifold) X ∆ corresponding to ∆. The fan ∆is non-singular if and only if H I = 0 for all I ∈ Σ ( n ) , n = rank ∆. In this case H J = 0for all J ∈ ∆ since H J is contained in H I if J ⊂ I .We denote the torus N R /N by T . N can be identified with Hom( S , T ), and the dual M = N ∗ with Hom( T, S ). Since Hom( T, S ) is identified with H ( T ) = H ( BT ) = H T ( pt ), M is identified with H T ( pt ).The Stanley-Reisner ring of the simplicial set Σ is denoted by H ∗ T (∆). It is the quotientring of the polynomial ring Z [ x i | i ∈ Σ (1) ] by the ideal generated by { x K = Q i ∈ K x i | K ⊂ Σ (1) , K / ∈ Σ } . It is considered as a ring over H ∗ T ( pt ) (regarded as embedded in H ∗ T (∆)) by the formula(6) u = X i ∈ Σ (1) h u, v i i x i . AKIO HATTORI
When ∆ is the fan ∆ X associated to a complete Q -factorial toric variety X , H ∗ T (∆ X ) Q = H ∗ T (∆ X ) ⊗ Q can be identified with the equivariant cohomology ring of X with respectto the action of compact torus T acting on X (see [8]).For each J ∈ Σ let { u Jj } be the basis of N ∗ J, V . N ∗ J, V contains N ∗ J . In particular N ∗ I, V contains N ∗ = M for I ∈ Σ ( n ) and will be considered as embedded in M Q = H T ( pt ) Q .Define ι ∗ I : H T (∆) Q → M Q by(7) ι ∗ I X i ∈ Σ (1) d i x i = X i ∈ I d i u Ii .ι ∗ I extends to H ∗ T (∆) Q → H ∗ T ( pt ) Q . It is an H ∗ T ( pt ) Q -module map, since ι ∗ I ( u ) = u for u ∈ H ∗ T ( pt ) Q . Let S be the multiplicative set in H ∗ T ( pt ) Q generated by non-zero elements in H T ( pt ) Q .The push-forward π ∗ : H ∗ T (∆) Q → S − H ∗ T ( pt ) Q is defined by(8) π ∗ ( x ) = X I ∈ Σ ( n ) ι ∗ I ( x ) | H I | Q i ∈ I u Ii . It is an H ∗ T ( pt ) Q -module map, and lowers the degrees by 2 n . It is known [6] that, if ∆ isa complete simplicial fan, then the image of π ∗ lies in H ∗ T ( pt ) Q .Assume that ∆ is complete. Let p ∗ : H ∗ T (∆) Q → Q be the composition of π ∗ : H ∗ T (∆) Q → H ∗ T ( pt ) Q and H ∗ T ( pt ) Q → H T ( pt ) Q = Q . Note that p ∗ induces R ∆ : H ∗ (∆) Q → Q as noted in [6] where H ∗ (∆) Q is the quotient of H ∗ T (∆) Q by the ideal generated by H + T ( pt ) Q . Note that H ∗ (∆) Q is defined independently of V . If ¯ x denotes the image of x ∈ H ∗ T (∆) Q in H ∗ (∆) Q , then R ∆ ¯ x = p ∗ ( x ).If X = X ∆ is the complete toric variety associated to ∆, then H ∗ (∆) Q is identifiedwith H ∗ ( X ) Q , π ∗ with the push-forward H ∗ T ( X ) Q → H ∗ T ( pt ) Q and R ∆ with the ordinaryintegral R X (see [6]).Assume that 1 ≤ k . For J ∈ Σ ( k ) let M J be the annihilator of N J and ω J ∈ V n − k M be the element determined by M J with an orentaiton, namely ω J = u ∧ . . . ∧ u n − k withan oriented basis { u , . . . , u n − k } of M J . Define f J ( x i ) ∈ V n − k +1 M Q by f J ( x i ) = ι ∗ I ( x i ) ∧ ω J with J ⊂ I ∈ Σ ( n ) .f J ( x i ) is well-defined independently of I containing J . Let S ∗ ( V n − k +1 M Q ) be the sym-metric algebra over V n − k +1 M Q . f J : H T (∆) Q → V n − k +1 M Q extends to f J : H ∗ T (∆) Q → S ∗ ( V n − k +1 M Q ). For x = Q i x α i i ∈ H kT (∆) Q we put f J ( x ) = ( f J ( x i )) α i . The definition of f J depends on the orientations chosen, but f J ( x ) f J ( x J ) does not. It belongsto the fraction field of the symmetric algebra S ∗ ( V n − k +1 M Q ) and has degree 0. Hence itcan be considered as an element of Rat ( P ( V n − k +1 N Q )) , the field of rataional functionsof degree 0 on P ( V n − k +1 N Q ). Let ν ∗ : Rat ( P ( V n − k +1 N Q )) → Rat ( G n − k +1 ( N Q )) bethe induced homomorphism of the Pl¨ucker embedding ν : G n − k +1 ( N Q ) → P ( V n − k +1 N Q ).The image ν ∗ ( f J ( x ) f J ( x J ) ) will be denoted by µ ( x, J ).Our first main result is stated in the following N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 5
Theorem 2.1.
Let ∆ be a complete simplicial fan in a lattice N of rank n and x ∈ H kT (∆) Q . For any ξ ∈ H T (∆) Q we have (9) p ∗ ( e ξ x ) = X J ∈ Σ ( k ) µ ( x, J ) p ∗ ( e ξ x J ) in Rat ( G n − k +1 ( N Q )) . We say that ξ = P i d i x i ∈ H T (∆) is T -Cartier if ι ∗ I ( ξ ) belongs to M = H T ( pt ) for all I ∈ Σ ( n ) . When ∆ and ξ come from a convex lattice polytope P , ξ is T -Cartier if andonly if P is a lattice polytope. In this case we have p ∗ ( e ξ x J ) = vol P J | H J | , where P J is the face of P corresponding to J ( P J = P σ with σ = C ( J )), cf. e.g. [4].Furthermore it will be shown in Section 4 that there is an element T T (∆) ∈ H ∗ T (∆) Q such that p ∗ ( e νξ T T (∆)) = νP ) = n X k =0 a k ( P ) ν n − k . Applying Theorem 2.1 to x = T T (∆) k and the above ξ the following Corollary will beobtained generalizing Morelli’s formula. Corollary 2.2.
Let P be a convex simple lattice polytope, ∆ the associated completesimplicial fan. Then we have a k ( P ) = X J ∈ Σ ( k ) µ k ( J ) vol P J with µ k ( J ) = 1 | H J | X h ∈ H J ν ∗ Y j ∈ J − χ ( u Jj , h ) e − f J ( x j ) ! in Rat( G n − k +1 ( N C )) , where χ ( u, h ) = e π √− h u,v ( h ) i for u ∈ N ∗ J, V and v ( h ) ∈ N J is a liftof h ∈ H J to N J . As another immadiate corollay of Theorem 2.1 we obtain
Corollary 2.3.
Let ∆ be a complete simplicial fan and x ∈ H kT (∆) Q . Then ¯ x = X J ∈ Σ ( k ) µ ( x, J )¯ x J in Rat ( G n − k +1 ( N Q )) ⊗ Q H kT (∆) Q . Proofs of Theorem 2.1 and Corollary 2.3 will be given in the next section.3.
Proof of Theorem 2.1 and Corollary 2.3
For a primitive sublattice E of N of rank n − k +1 let w E ∈ V n − k +1 N be a representativeof ν ( E ) ∈ P ( V n − k +1 N Q ). The equality (9) is equivalent to the condition that p ∗ ( e ξ x ) = X J ∈ Σ ( k ) f J ( x ) f J ( x J ) ( w E ) p ∗ ( e ξ x J ) holds for every generic E. Let E be a generic primitive sublattice in N of rank n − k + 1. The intersection E ∩ N J has rank one for each J ∈ Σ ( k ) . Take a non-zero vector v E,J in E ∩ N J . (One can choose v E,J to be the unique primitive vector contained in E ∩ C ( J ). But any non-zero vectorwill suffice for the later use.) For x ∈ H kT (∆) and J ∈ Σ ( k ) the value of ι ∗ I ( x ) evaluated AKIO HATTORI on v E,J for I ∈ Σ ( n ) containing J depends only on ι ∗ J ( x ) so that it will be denoted by ι ∗ J ( x )( v E,J ). Similarly we shall simply write h u Jj , v E,J i instead of h u Ij , v E,J i . Lemma 3.1.
Put f Jj = u Jj ∧ ω J . Then a h f Jj , w E i = h u Jj , v E,J i , where a is a non-zero constant depending only on v E,J .Proof.
Take an oriented basis u , . . . , u n − k of M J . Take also a basis w , . . . , w n − k +1 of E and write v E,J = P l c l w l . Then, since h u i , v E,J i = 0, n − k +1 X l =1 c l h u i , w l i = 0 , for i = 1 , . . . , n − k. The matrix ( a il ) = ( h u i , w l i ) has rank n − k and we get( c , . . . , c n − k +1 ) = a ( A , . . . , A n − k +1 ) , a = 0 , where A l = ( − l − det a . . . c a l . . . a n − k +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a n − k . . . [ a n − k l . . . a n − k n − k +1 . Then h u Jj , v E,J i = n − k +1 X l =1 c l h u Jj , w l i = a n − k +1 X l =1 h u Jj , w l i A l = a det h u Jj , w i . . . h u Jj , w n − k +1 ih u , w i . . . h u , w n − k +1 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h u n − k , w i . . . h u n − k , w n − k +1 i = a h f Jj , w E i where f Jj = u Jj ∧ u · · · ∧ u n − k and w E = w ∧ · · · ∧ w n − k +1 . (cid:3) Remark 3.1.
Let X be a non-singular complete toric variety of dimension n and ∆ theassociated fan. Let T = T n be the compact torus acting on X . E ∩ N J determines asubcircle T E,J of T . Then T E,J pointwise fixes an invariant complex submanifold X J .Hence it acts on the normal vector space at each generic point in X J . Then the numbers h u Jj , v E,J i are weights of this action.Lemma 3.1 implies that f J ( x ) f J ( x J ) ( w E ) = ι ∗ J ( x ) Q j ∈ J u Jj ( v E,J ) . Then the equality (9) in Theorem holds if and only if(10) p ∗ ( e ξ x ) = X J ∈ Σ ( k ) ι ∗ J ( x ) Q j ∈ J u Jj ( v E,J ) p ∗ ( e ξ x J ) N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 7 holds for every generic E .The following lemma is easy to prove, cf. e.g. [6] Lemma 8.1. Lemma 3.2.
The vector space H kT (∆) Q is spanned by elements of the form u · · · u k x J k , J k ∈ Σ ( k − k ) , u i ∈ M Q , with ≤ k ≤ k − .Note. For x = u · · · u k x J k , J k ∈ Σ ( k − k ) , with k ≥ p ∗ ( e ξ x ) = 0 . In view of this lemma we proceed by induction on k .For x = x J with J ∈ Σ ( k ) , the left hand side of (10) is equal to p ∗ ( e ξ x J ). Since i ∗ J ( x ) = 0 unless J = J and i ∗ J ( x ) / Q j ∈ J u Jj = 1 for J = J , the right hand side is alsoequal to p ∗ ( e ξ x J ). Hence (10) holds with x of the form x = x J for J ∈ Σ ( k ) .Assuming that (10) holds for x of the form u · · · u k x J k with J k ∈ Σ ( k − k ) , we shallprove that it also holds for x = u · · · u k u k +1 x J k with J k +1 ∈ Σ ( k − ( k +1)) . Put K = J k +1 .Case a). u k +1 belongs to M K Q , that is, h u k +1 , v i i = 0 for all i ∈ K . In this case u k +1 = X i ∈ Σ (1) \ K h u k +1 , v i i x i since h u k +1 , v i i = 0 for all i ∈ K . For i K , x i x J k is either of the form x J i with J i ∈ Σ ( k − k ) or equal to 0. Thus, for x = u · · · u k x i x J k with i K , the equality (10)holds by induction assumption, and it also holds for x = u · · · u k u k +1 x J k by linearity.Case b). General case. We need the following Lemma 3.3.
For K ∈ Σ ( k − k ) with k ≥ , the composition homomorphism M K Q ⊂ M Q → E ∗ Q is surjective. The proof will be given later. By this lemma, there exists an element u ∈ M K Q suchthat h u k +1 , v E,J i = h u, v E,J i for all J ∈ Σ ( k ) . Note that h ι ∗ J ( u ) , v E,J i = h u, v E,J i for any u ∈ M Q . Then, in (10) for x = u · · · u k u k +1 x J k with J k +1 ∈ Σ ( k − ( k +1)) , we have ι ∗ J ( x )( v E,J ) = ( k Y i =1 h u i , v E,J i ) h u k +1 , v E,J i = ( k Y i =1 h u i , v E,J i ) h u, v E,J i . Hence if we put x ′ = u · · · u k ux J k , the right hand side of (10) is equal to X J ∈ Σ ( k ) ι ∗ J ( x ′ ) Q j ∈ J u Jj ( v E,J ) p ∗ ( e ξ x J ) . This last expression is equal to p ∗ ( e ξ x ′ ) since x ′ belongs to Case a). Furthermore p ∗ ( e ξ x ′ ) = 0 and p ∗ ( e ξ x ) = 0 by Note after Lemma 3.2. Thus both side of (10) for AKIO HATTORI x = u · · · u k u k +1 x J k are equal to 0. This completes the proof of Theorem 2.1 exceptfor the proof of Lemma 3.3.Proof of Lemma 3.3.Take a simplex I ∈ Σ ( n ) which contains K and a simplex K ′ ∈ Σ ( k − such that K ⊂ K ′ ⊂ I . Such a K ′ exists since k − k ≤ k −
1. Then there are exactly n − k + 1simplices J , . . . , J n − k +1 ∈ Σ ( k ) such that K ′ ⊂ J i ⊂ I . It is easy to see that thevectors v E,J , . . . , v E,J n − k +1 are linearly independent so that they span E Q . Moreover M K ′ Q detects these vectors, that is, M K ′ Q → M Q → E ∗ Q is surjective. Since M ′ K ⊂ M K ⊂ M , M K Q → E ∗ Q is surjective.Proof of Corollary 2.3.Fix a generic sublattice E and put x ′ = P J ∈ Σ ( k ) µ ( x, J ) x J . Then p ∗ ( e ξ x ′ ) = X J ∈ Σ ( k ) µ ( x, J ) p ∗ ( e ξ x J ) = p ∗ ( e ξ x )by Theorem 2.1. It follows that p ∗ ( e ξ ( x ′ − x )) = 0. Thus, in order to prove Corollary 2.3,it suffices to show that p ∗ ( e ξ y ) = 0 , ∀ ξ ∈ H T (∆) Q , implies that y belongs to the ideal J generated by H T ( pt ) Q . Since H ∗ (∆) Q = H ∗ T (∆) Q / J ∼ = H ∗ ( X ∆ ) Q is a Poincar´e dualityspace generated by H (∆) Q , p ∗ ( e ξ y ) = 0 implies that p ∗ ( y ) = 0, i.e. y ∈ J .4. Multi-fans and multi-polytopes
The notion of multi-fan and multi-polytope were introduced in [8]. In this article weshall be concerned only with simplicial multi-fans. See [8, 6, 7] for details.Let N be a lattice of rank n . A simplicial multi-fan in N is a triple ∆ = (Σ , C, w )where Σ = F nk =0 Σ ( k ) is an (augmented) simplicial complex, C is a map from Σ ( k ) intothe set of k -dimensional strongly convex rational polyhedral cones in the vector space N R = N ⊗ R for each k , and w is a map Σ ( n ) → Z . Σ (0) consists of a single element o = the empty set. (The definition in [8] and [6] requires additional restriction on w .)We assume that any J ∈ Σ is contained in some I ∈ Σ ( n ) and Σ ( n ) is not empty.The map C is required to satisfy the following condition; if J ∈ Σ is a face of I ∈ Σ, then C ( J ) is a face of C ( I ), and for any I , the map C restricted on Σ( I ) = { J ∈ Σ | J ⊂ I } is an isomorphism of ordered sets onto the set of faces of C ( I ). It follows that C ( I ) isnecessarily a simplicial cone and C ( o ) = 0. A simplicial fan is considered as a simplicialmulti-fan such that the map C on Σ is injective and w ≡ K ∈ Σ we set Σ K = { J ∈ Σ | K ⊂ J } . It inherits the partial ordering from Σ and becomes a simplicial set where Σ ( j ) K ⊂ Σ ( j + | K | ) . K is the unique element in Σ (0) K . Let N K be the minimal primitive sublattice of N containing N ∩ C ( K ), and N K the quotient lattice of N by N K . For J ∈ Σ K we define C K ( J ) to be the cone C ( J ) projected on N K ⊗ R . We define a function w : Σ ( n −| K | ) K ⊂ Σ ( n ) → Z to be the restrictions of w to Σ ( n −| K | ) K . The triple ∆ K = (Σ K , C K , w ) is a multi-fan in N K and is called the projected multi-fan with respect to K ∈ Σ. For K = o , the projectedmulti-fan ∆ o is nothing but ∆ itself. N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 9
A vector v ∈ N R will be called generic if v does not lie on any linear subspacespanned by a cone in C (Σ) of dimesnsion less than n . For a generic vector v we set d v = P v ∈ C ( I ) w ( I ), where the sum is understood to be zero if there is no such I . Definition.
A simplicial multi-fan ∆ = (Σ , C, w ) is called pre-complete if the integer d v is independent of generic vectors v . In this case this integer will be called the degree of∆ and will be denoted by deg(∆). It is also called the Todd genus of ∆ and is denotedby Td[∆]. A pre-complete multi-fan ∆ is said to be complete if the projected multi-fan∆ K is pre-complete for every K ∈ Σ.A multi-fan is complete if and only if the projected multi-fan ∆ J is pre-complete forevery J ∈ Σ ( n − .Like a toric variety gives rise to a fan, a torus orbifold gives rise to a complete simplicialmulti-fan, though this correspondence is not one to one. A torus orbifold is a closedoriented orbifold with an action of a torus of half the dimension of the orbifold itself withnon-empty fixed point set and with some additional conditions on the isotopy groups(see [6]). Most typical non-toric examples are given in [3]. Cobordism invariants of torusorbifolds are encoded in the associated multi-fans.In the sequel we shall often consider a set V consisting of non-zero edge vectors v i foreach i ∈ Σ (1) such that v i ∈ N ∩ C ( i ). We do not require v i to be primitive. This hasmeaning for torus orbifolds (see [6]). For any K ∈ Σ put V K = { v i } i ∈ K . Let N K, V be thesublattice of N K generated by V K . The quotient group N K /N K, V is denoted by H K, V .Let ∆ = (Σ , C, w ) be a simplicial multi-fan in a lattice N . We define the equivariantcohomology H ∗ T (∆) of a multi-fan ∆ as the Stanley-Reisner ring of the simplicial complexΣ as in Section 1.Let V = { v i } i ∈ Σ (1) be a set of prescribed edge vectors as before. Let { u Ji } i ∈ K be thebasis of N ∗ K, V dual to V K . We define a homomorphism M = N ∗ = H T ( pt ) → H T (∆)by the same formula (6) as in the case of fans. Since this definition depends on the set V , the H ∗ T ( pt )-module structure of H ∗ T (∆) also depends on V . To emphasize this factwe shall use the notation H ∗ T (∆ , V ). When all the v i are taken primitive, the notation H ∗ T (∆) is used.For I ∈ Σ ( n ) the map ι ∗ I : H ∗ T (∆ , V ) Q → H ∗ T ( pt ) Q is defined by (7) as in the case offans. On the other hand the definition of the push-forward is altered from (8) to π ∗ ( x ) = X I ∈ Σ ( n ) w ( I ) ι ∗ I ( x ) | H I | Q i ∈ I u Ii , cf. [6]. If ∆ is complete the image of π ∗ lies in H ∗ T ( pt ) Q as in the case of fans, and themap p ∗ : H ∗ T (∆ , V ) Q → Q is also defined in a similar way.Let K ∈ Σ ( k ) and let ∆ K = (Σ K , C K , w K ) be the projected multi-fan. The link Lk K of K in Σ is a simplicial complex consisting of simplices J such that K ∪ J ∈ Σ and K ∩ J = ∅ . It will be denoted by Σ ′ K in the sequel. There is an isomorphism from Σ ′ K to Σ K sending J ∈ Σ ′ ( l ) K to K ∪ J ∈ Σ ( l ) K . We consider the polynomial ring R K generatedby { x i | i ∈ K ∪ Σ ′ (1) K } and the ideal I K generated by monomials x J = Q i ∈ J x i such that J / ∈ Σ( K ) ∗ Σ ′ K where Σ( K ) ∗ Σ ′ K is the join of Σ( K ) and Σ ′ K . We define the equivariantcohomology H ∗ T (∆ K ) of ∆ K with respect to the torus T as the quotient ring R K / I K .If V is a set of prescribed edge vectors, H T ( pt ) is regarded as a submodule of H T (∆ K ) bya formula similar to (6). This defines an H ∗ T ( pt )-module structure on H ∗ T (∆ K ) which willbe denoted by H ∗ T (∆ K , V ) to specify the dependence on V . The projection H ∗ T (∆ , V ) → H ∗ T (∆ K , V ) is defined by sending x i to x i for i ∈ K ∪ Σ ′ (1) K and putting x i = 0 for i / ∈ K ∪ Σ ′ (1) K . The restriction homomorphism ι ∗ I : H ∗ T (∆ K , V ) Q → H ∗ T ( pt ) Q for I ∈ Σ ( n − k ) K and the push-forward π ∗ : H ∗ T (∆ K , V ) Q → S − H ∗ T ( pt ) Q are also defined in a similar wayas before.Given ξ = P i ∈ K ∪ Σ ′ (1) K d i x i ∈ H T (∆ K , V ) R , d i ∈ R , let A ∗ K be the affine subspacein the space M R defined by h u, v i i = d i for i ∈ K . Then we introduce a collection F K = { F i | i ∈ Σ ′ (1) K } of affine hyperplanes in A ∗ K by setting F i = { u | u ∈ A ∗ K , h u, v i i = d i } . The pair P K ( ξ ) = (∆ K , F K ) will be called a multi-polytope associated with ξ ; see [7]. Incase K = o ∈ Σ (0) , P K ( ξ ) is simply denoted by P ( ξ ).For ξ = P i ∈ Σ (1) d i x i and K ∈ Σ ( k ) put ξ K = P i ∈ K ∪ Σ ′ (1) K d i x i and P ( ξ ) K = P K ( ξ K ). Itwill be called the face of P ( ξ ) corresponding to K .For I ∈ Σ ( n − k ) K , i.e. I ∈ Σ ( n ) with I ⊃ K , we put u I = ∩ i ∈ I F i = ∩ i ∈ I \ K F i ∩ A ∗ K ∈ A ∗ K .Note that u I is equal to ι ∗ I ( ξ ). The dual vector space ( N K R ) ∗ of N K R is canonically identifiedwith the subspace M K R of M R = H T ( pt ) R . It is parallel to A ∗ K , and u Ii lies in M K R for I ∈ Σ ( n − k ) K and i ∈ I \ K . A vector v ∈ N K R is called generic if h u Ii , v i 6 = 0 for any I ∈ Σ ( n − k ) K and i ∈ I \ K . The image in N K R of a generic vector in N R is generic. We takea generic vector v ∈ N K R , and define( − I := ( − { j ∈ I \ K |h u Ij ,v i > } and ( u Ii ) + := ( u Ii if h u Ii , v i > − u Ii if h u Ii , v i < . for I ∈ Σ ( n − k ) K and i ∈ I \ K . We denote by C ∗ K ( I ) + the cone in A ∗ K spanned by the( u Ii ) + , i ∈ I \ K, with apex at u I , and by φ I its characteristic function. With theseunderstood, we define a function DH P K ( ξ ) on A ∗ K \ ∪ i F i byDH P K ( ξ ) = X I ∈ Σ ( n − k ) K ( − I w ( I ) φ I . As in [7] we call this function the
Duistermaat-Heckman function associated with P K ( ξ ).When K = o , DH P ( ξ ) is defined on M R \ ∪ i F i .Suppose that ∆ is a simplicial fan. If all the d i are positive and the set P = { u ∈ M R | h u, v i i ≤ d i } is a convex polytope, then DH P ( ξ ) equals 1 on the interior of P and 0 on other componentsof M R \ ∪ i F i .The following theorem is fundamental in the sequel, cf. [7] Theorem 2.3 and [6] Corol-lary 7.4. Theorem 4.1.
Let ∆ be a complete simplicial multi-fan. Let ξ = P i ∈ K ∪ Σ ′ K (1) d i x i ∈ H T (∆ K , V ) be as above with all d i integers and put ξ + = P i ( d i + ǫ ) x i with < ǫ < .Then (11) X u ∈ A ∗ K ∩ M DH P K ( ξ + ) ( u ) t u = X I ∈ Σ ( n − k ) K w ( I ) | H I, V | X h ∈ H I, V χ I ( ι ∗ I ( ξ ) , h ) t ι ∗ I ( ξ ) Q i ∈ I \ K (1 − χ I ( u Ii , h ) − t − u Ii ) , where χ I ( u, h ) for u ∈ N ∗ I, V is defined as in Corollary 2.2. N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 11
Note.
The left hand side of (11) is considered as an element in the group ring of M over R or the character ring R ( T ) ⊗ R considered as the Laurent polynomial ring in t = ( t , . . . , t n ). The equality shows that the right hand side, which is a rational fucnctionof t , belongs to R ( T ). ξ = P i d i x i ∈ H T (∆ , V ) is called T -Cartier if ι ∗ I ( ξ ) ∈ M for all I ∈ Σ ( n ) . This conditionis equivalent to u I ∈ M for all I ∈ Σ ( n ) . In this case P ( ξ ) is said lattice multi-polytope.If ξ is T -Cartier, then χ I ( ι ∗ I ( ξ ) , h ) ≡
1. Hence the above formula (11) for DH P K ( ξ K + ) reduces in this case to(12) X u ∈ A ∗ K ∩ M DH P K ( ξ K + ) ( u ) t u = X I ∈ Σ ( n − k ) K w ( I ) | H I, V | X h ∈ H I, V t ι ∗ I ( ξ K ) Q i ∈ I \ K (1 − χ I ( u Ii , h ) − t − u Ii ) . Let H ∗∗ T ( ) denote the completed equivariant cohomology ring. The Chern characterch sends R ( T ) ⊗ R to H ∗∗ T ( pt ) R by ch( t u ) = e u . The image of (12) by ch is given by(13) X u ∈ A ∗ K ∩ M DH P K ( ξ K + ) ( u ) e u = X I ∈ Σ ( n − k ) K w ( I ) | H I, V | X h ∈ H I, V e ι ∗ I ( ξ K ) Q i ∈ I \ K (1 − χ I ( u Ii , h ) − e − u Ii ) . Assume that ξ = P i d i x i ∈ H T (∆ , V ) is T -Cartier. The number P ( ξ ) K ) is definedby P ( ξ ) K ) = X u ∈ A ∗ K ∩ M DH P K ( ξ K + ) ( u ) . It is obtained from (13) by setting u = 0, that is, it is equal to the image of (13) by H ∗∗ T ( pt ) R → H T ( pt ) Q .The equivariant Todd class T T (∆ , V ) is defined in such a way that π ∗ ( e ξ T T (∆ , V )) = X u ∈ M DH P ( ξ + ) ( u ) e u for ξ T -Cartier. In order to give the definition we need some notations.For simplicity identify the set Σ (1) with { , , . . . , m } and consider a homomorphism η : R m = R Σ (1) → N R sending a = ( a , a , . . . , a m ) to P i ∈ Σ (1) a i v i . For K ∈ Σ ( k ) we define˜ G K, V = { a | η ( a ) ∈ N and a j = 0 for j K } and define G K, V to be the image of ˜ G K, V in ˜ T = R m / Z m . It will be written G K forsimplicity. The homomorphism η restricted on ˜ G K, V induces an isomorphism η K : G K ∼ = H K, V ⊂ T = N R /N. Put G ∆ = [ I ∈ Σ ( n ) G I ⊂ ˜ T and DG ∆ = [ I ∈ Σ ( n ) G I × G I ⊂ G ∆ × G ∆ . Let v ( g ) = a = ( a , a , . . . , a m ) ∈ R m be a representative of g ∈ ˜ T . The factor a i willbe denoted by v i ( g ). It is determined modulo integers. If g ∈ G I , then v i ( g ) is necessarilya rational number. Define a homomorphism χ i : ˜ T → C ∗ by χ i ( g ) = e π √− v i ( g ) Let g ∈ G I and h = η I ( g ) ∈ H I, V . Then η ( v ( g )) ∈ N I is a representative of h in N I which will be denoted by v ( h ). Then, for g ∈ G I and i ∈ I , v i ( g ) ≡ h u Ii , v ( h ) i mod Z , and χ i ( g ) = e π √− h u Ii ,v ( h ) i = χ I ( u Ii , h ) . Let ∆ be a complete simplicial multi-fan. Define T T (∆ , V ) = X g ∈ G ∆ Y i ∈ Σ (1) x i − χ i ( g ) e − x i ∈ H ∗∗ T (∆ , V ) Q . Proposition 4.2.
Let ∆ be a complete simplicial multi-fan. Assume that ξ ∈ H T (∆ , V ) is T -Cartier. Then π ∗ ( e ξ T T (∆ , V )) = X u ∈ M DH P ( ξ + ) ( u ) e u . Consequently p ∗ ( e ξ T T (∆ , V )) = P ( ξ )) . Proof. (cf. [6] Section 8). Let g ∈ G ∆ and I ∈ Σ ( n ) . If g / ∈ G I , then there is an element i / ∈ I such that χ i ( g ) = 1; so x i − χ i ( g ) e − x i = (1 − χ i ( g )) − x i + higher degree termsfor such i . Hence i ∗ I ( x i − χ i ( g ) e − xi ) = 0. Therefore, only elements g in G I contribute to ι ∗ I ( T T (∆ , V )). Now suppose g ∈ G I . Then χ i ( g ) = 1 for i / ∈ I , so ι ∗ I ( x i − χ i ( g ) e − xi ) = 1 forsuch i . Finally, since ι ∗ I ( x i ) = u Ii for i ∈ I , we have ι ∗ I ( T T (∆ , V )) = X g ∈ G I Y i ∈ I u Ii − χ i ( g ) e − u Ii . This together with (13) shows that π ∗ ( e ξ T T (∆ , V )) = π ∗ e ξ X g ∈ G ∆ m Y i =1 x i − χ i ( g ) e − x i ! = X I ∈ Σ ( n ) w ( I ) e ι ∗ I ( ξ ) | H I, V | X g ∈ G I Q i ∈ I (1 − χ i ( g ) e − u Ii )= X u ∈ M DH P ( ξ + ) ( u ) e u . (cid:3) More generally, for K ∈ Σ ( k ) , define T T (∆ , V ) K by T T (∆ , V ) K = X g ∈ G ∆ K Y i ∈ Σ ′ (1) K x i − χ i ( g ) e − x i ∈ H ∗∗ T (∆ , V ) Q . Then the same proof as for Propsition 4.2 yields
Proposition 4.3.
Let ∆ be a complete simplicial multi-fan. Assume that ξ ∈ H T (∆ , V ) is T -Cartier. Then π ∗ ( e ξ x K T T (∆ , V ) K ) = X u ∈ A ∗ K ∩ M DH P K ( ξ K + ) ( u ) e u . for K ∈ Σ ( k ) , where x K = Q i ∈ K x i . Consequently p ∗ ( e ξ x K T T (∆ , V ) K ) = P ( ξ ) K ) . N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 13
The lattice M ∩ A ∗ K defines a volume element dV K on A ∗ K . For ξ = P i ∈ K ∪ Σ ′ K (1) d i x i ∈ H T (∆ K , V ) R , the volume vol P K ( ξ ) of P K ( ξ ) is defined byvol P K ( ξ ) = Z A ∗ K DH P K ( ξ ) dV ∗ K . Proposition 4.4.
For ξ = P i ∈ Σ (1) d i x i ∈ H T (∆ , V ) R | H K, V | vol P ( ξ ) K = p ∗ ( e ξ x K ) . Proof.
We shall give a proof only for the case where ξ is T -Cartier. The general case canbe reduced to this case, cf. [6], Lemma 8.6. By Proposition 4.3 P ( ξ ) K ) = p ∗ ( e ξ x K T T (∆ , V ) K ) . The highest degree term with respect to { d i } in the right hand side is nothing butvol P ( ξ ) K and is equal to p ∗ (cid:18) ξ n − k ( n − k )! x K (cid:19) X g ∈ G ∆ K Y i ∈ Σ ′ (1) K x i − χ i ( g ) e − x i , where the suffix 0 means taking 0-th degree term. But Y i ∈ Σ ′ (1) K x i − χ i ( g ) e − x i = ( , g ∈ G K , g / ∈ G K . Hence vol P ( ξ ) K = | G K | p ∗ (cid:18) ξ n − k ( n − k )! x K (cid:19) = | H K, V | p ∗ ( e ξ x K ) . (cid:3) Generalization
The definition of f J ( x ) for simplcial fans can be also applied for simplicial multi-fans.Consequently µ ( x, J ) ∈ Rat ( G n − k +1 ( N Q )) is also defined. Theorem 2.1 is generalized inthe following form. Theorem 5.1.
Let ∆ be a complete simplicial multi-fan and x ∈ H kT (∆ , V ) Q . For any ξ ∈ H T (∆) Q we have p ∗ ( e ξ x ) = X J ∈ Σ ( k ) µ ( x, J ) p ∗ ( e ξ x J ) in Rat ( G n − k +1 ( N Q )) . Lemma 3.1, Lemma 3.2, Lemma 3.3 all hold in this new setting. Hence the proofs inSection 2 literally apply to prove Theorem 5.1.As to Corollary 2.2 its generalization takes the following form.
Corollary 5.2.
Let ∆ be a complete simplicial multi-fan in a lattice of rank n . Assumethat ξ ∈ H T (∆ , V ) is T -Cartier. Set P ( νξ )) = n X k =0 a k ( ξ ) ν n − k . Then we have a k ( ξ ) = X J ∈ Σ ( k ) µ k ( J ) vol P ( ξ ) J with µ k ( J ) = 1 | H J, V | ν ∗ X h ∈ H J, V Y j ∈ J − χ ( u Jj , h ) e − f J ( x j ) in Rat( G n − k +1 ( N C )) .Note. It can be proved without difficulty that µ k ( J ) does not depend on the choice of V .Hence one has only consider the case where all the v i are primitive. Proof.
By Proposition 4.3 P ( νξ )) = p ∗ ( e νξ T T (∆ , V )) . Put x = ( T T (∆ , V )) k ∈ H kT (∆ , V ) Q . By (9) which is valid under the assumption ofTheorem 5.1 too and by Proposition 4.4 a k ( ξ ) = X J ∈ Σ ( k ) ν ∗ (cid:18) f J ( x ) f J ( x J ) (cid:19) vol P ( ξ ) J | H J, V | . Thus it suffices to show that f J ( x ) f J ( x J ) = X h ∈ H J, V Y j ∈ J − χ ( u Jj , h ) e − f J ( x j ) , or f J ( x ) = X h ∈ H J, V Y j ∈ J f J ( x j )1 − χ ( u Jj , h ) e − f J ( x j ) k . Let g ∈ G ∆ . If g G J , then there is an element i J such that χ i ( g ) = 1, and, forsuch i , f J ( x i − χ i ( g ) e − x i ) = f J ((1 − χ i ( g )) − x i + higher degree terms) = 0 , since f J ( x i ) = 0. Thus f J Y i ∈ Σ (1) x i − χ i ( g ) e − x i = 0for g G J .If g ∈ G J , then χ i ( g ) = 1 for i J . Thus f J (cid:18) x i − χ i ( g ) e − x i (cid:19) = f J (1 + 12 x i + higher degree terms) = 1for g ∈ G J , i J . It follows that f J X g ∈ G ∆ Y i ∈ Σ (1) x i − χ i ( g ) e − x i = X g ∈ G J Y i ∈ J f J ( x i )1 − χ i ( g ) e − f J ( x i ) . N A MORELLI TYPE EXPRESSION OF COHOMOLOGY CLASSES OF TORIC VARIETIES 15
This implies f J ( T T (∆ , V ) k ) = X h ∈ H J, V Y j ∈ J f J ( x j )1 − χ J ( u Jj , h ) e − f J ( x j ) k . (cid:3) As to Corollary 5.3 we need to put an additional condition on the multi-fan ∆. Asimplicial complex Σ is said to be Q -Gorenstein ∗ if e H i (Lk J ) Q = ( Q , i = dim Lk J , i < dim Lk J for all J ∈ Σ ( k ) , ≤ k ≤ n . It is equivalent to say that the realization | Σ | of Σ is a Q -homology manifold and has the same Q -homology with the sphere S n − . A completesimplicial multi-fan ∆ = (Σ , C, w ) is called Q -Gorenstein ∗ if Σ is Q -Gorenstein ∗ . Corollary 5.3.
Let ∆ be a Q -Gorenstein ∗ simplicial multi-fan and x ∈ H kT (∆) Q . Then ¯ x = X J ∈ Σ ( k ) µ ( x, J )¯ x J in Rat ( G n − k +1 ( N Q )) ⊗ Q H k (∆) Q . We shall show that H ∗ (∆) Q is a Poincar´e duality space and is generated by H (∆) Q if∆ is Q -Gorenstein ∗ . Then the proof of Corollary 2.3 can be applied in this case too.Our construction follows [3]. Assume that Σ is Q -Gorenstein ∗ .Let Σ ∗ be the dual complex of Σ. It is triangulated by the barycentric subdivision ofΣ. The set of dual cells are in one to one corresondence with F nk =1 Σ ( k ) . The dual cellcorreponding to K ∈ Σ ( k ) is denoted by K ∗ .Let P be the cone over Σ ∗ . P is itself a Q -homology cell since Σ is Q -Gorenstein ∗ . Itis considered as the dual cell o ∗ of o . The dual cell K ∗ of K ∈ Σ ( k ) has codimension k in P . For p ∈ P define D ( p ) to be the minimal dual cell containing p . The sublattice N K determindes a subtorus T K of T . We put T p = T K when D ( p ) = K ∗ .Then put ˜ P = T × P/ ∼ where the equivalence relation ∼ is defined by( g, p ) ∼ ( h, q ) ⇐⇒ p = q, gh − ∈ T p . ˜ P has a natural T -action. It is easy to see that ˜ P is an orientable Q -homology manifold.Theorem 4.8 of [3] says that the cohomology ring H ∗ T ( ˜ P ) Q is isomorphic to H ∗ T (∆) Q .Theorem 5.10 of [3] tells us that H ∗ ( ˜ P ) Q is isomorphic to H ∗ T ( ˜ P ) Q / J where J is theideal generated by the image of H ∗ T ( pt ) Q in H ∗ T ( ˜ P ) Q . It follows that H ∗ (∆) Q is isomorphicto H ∗ ( ˜ P ) Q . Since ˜ P is an orientable Q -homology manifold, H ∗ (∆) Q is a Poincar´e dualityspace generated by its degree two part.This finishes the proof of Corollary 5.3. Remark 5.1.
Let X be a torus orbifold and ∆ X the associated multi-fan. It is knownthat, if the cohomology ring H ∗ ( X ) Q is generated by H ( X ) Q , then H ∗ ( X ) Q is isomorphicto H ∗ (∆ X ) Q ([8] Proposition 3.4) and ∆ X is Q -Gorenstein ∗ ([9] Lemma 8.2 ). HenceCorollary 5.3 holds for x ∈ H T ( X ) Q and ¯ x ∈ H ∗ ( X ) Q . Remark 5.2.
When ∆ is the fan associated to a convex polytope P and ξ = D , theCartier divisor associated to P , we know (see, e.g. [4]) that µ ( o ) = 1 , a ( ξ ) = vol P ( ξ ) , µ ( i ) = 12 , a ( ξ ) = 12 X i ∈ Σ (1) vol P ( ξ ) i . This is also true for simplicial multi-fans and T -Cartier ξ .As to a n we have a n ( ξ ) = Td(∆) . In fact a n ( ξ ) = p ∗ ( T T (∆ , V )) = ( π ∗ ( T T (∆ , V ))) . Thus the above equality follows fromthe following rigidity property: Theorem 5.4.
Let ∆ be a complete simplicial multi-fan. Then π ∗ ( T T (∆ , V )) = ( π ∗ ( T T (∆ , V ))) = Td[∆] . See [6] Theorem 7.2 and its proof. Note that Td[∆] = 1 for any complete simplicialfan ∆.The explicit formula for π ∗ ( T T (∆ , V )) is given by π ∗ ( T T (∆ , V )) = X I ∈ Σ ( n ) w ( I ) | H I, V | X h ∈ H I, V Y i ∈ I − χ I ( u Ii , h ) e − u Ii . This does not depend on the choice of V and is in fact equal to Td[∆].Let ∆ be a (not necessarily complete) simplicial fan in a lattice of rank n . Set T d T (∆) = X I ∈ Σ ( n ) | H I | X h ∈ H I Y i ∈ I − χ I ( u Ii , h ) e − u Ii ∈ S − H ∗∗ T ( pt ) Q . For a simplex I let Σ( I ) be the simplicial complex consisting of all faces of I . For a fan∆( I ) = (Σ( I ) , C ), T d T (∆( I )) is denoted by T d T ( I ). Theorem 5.5.
T d T ( I ) is additive with respect to simplicial subdivisions of the cone C ( I ) .Namely, if ∆ is the fan determined by a simplicial subdivison of C ( I ) , then the followingequality holds T d T (∆) = T d T ( I ) . For the proof it is sufficient to assume that ∆( I ) and ∆ are non-singular. In such aform a proof is give in [10]. The following corollary ensures that µ k ( J ) can be defined forgeneral polyhedral cones as pointed out by Morelli in [10]. Corollary 5.6.
Let ∆( J ) = (Σ( J ) , C ) be a fan in a lattice N of rank n where J is asimplex of dimension k − . Then µ k ( J ) ∈ Rat( G n − k +1 ( N Q )) is additive with respect tosimplicial subdivisions of C ( J ) . References [1] P. Baum and R. Bott,
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Graduate School of Mathematical Science, University of Tokyo, Tokyo, Japan
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