EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE THIRTEEN: TORIC VARIETIES

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE THIRTEEN: TORIC VARIETIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 Let X be a complete nonsingula...
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EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE THIRTEEN: TORIC VARIETIES WILLIAM FULTON NOTES BY DAVE ANDERSON

1 Let X be a complete nonsingular toric variety. In this lecture, we will descibe HT∗ X. First we recall some basic notions about toric varieties. Let T be an n-dimensional torus with character group M , and let N = HomZ (M, Z) be the dual lattice. Then X = X(Σ), for a complete nonsingular fan Σ. That is, Σ is a collection of cones σ in NR = N ⊗Z R such that two cones meet along a face of each; each cone must be generated by part of a basis for N (the nonsingular condition), and the union of the cones is all of NR (the completeness condition). The toric variety X is covered by open affines Uσ = Spec C[σ ∨ ∩ M ], where σ ∨ = {u | hu, vi ≥ 0 for all v ∈ σ}. In fact, Uσ ∼ = Ck × (C∗ )n−k , where k = dim σ, and the n-dimensional cones suffice to cover. Also, U{0} = Spec C[M ] = T , and Uσ ∩ Uτ = Uσ∩τ . Write χu ∈ C[M ] for the element corresponding to u ∈ M . Each cone τ determines a T -invariant subvariety V (τ ) ⊂ X, which is closed and nonsingular, of codimension equal to dim τ . On open affines, this is given by V (τ ) ∩ Uσ = Spec C[τ ⊥ ∩ σ ∨ ∩ M ], with the containment in Uσ given by C[σ ∨ ∩ M ] → C[τ ⊥ ∩ σ ∨ ∩ M ], with χu 7→ χu if u ∈ τ ⊥ and χu 7→ 0 otherwise. (This is a homomorphism because τ ⊥ ∩ σ ∨ is a face of σ ∨ .) Then V (τ ) is a nonsingular toric variety, for the torus with character group τ ⊥ ∩ M ; it corresponds to a fan in N/Nτ , where Nτ is the sublattice generated by τ . The T -fixed points of X are pσ = V (σ) for dim σ = n. X is projective if and only if there is a lattice polytope P ⊂ MR , with vertices in M , such that Σ is the normal fan to P . That is, to each face F of P , the corresponding cone in Σ is σF = {v | hu′ , vi ≥ hu, vi for all u′ ∈ P, u ∈ F }. Date: April 30, 2007. 1

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This correspondence reverses dimensions: dim σF = codim F . Example 1.1. The standard n-dimensional simplex corresponds to Pn . An n-cube corresponds to (P1 )n . Figures... For X projective, choose a general vector v ∈ NR , giving an ordering of the vertices u1 , . . . , uN (so that hu1 , vi < · · · huN , vi), and thus an ordering of the n-dimensional cones σ1 , . . . , σN . For 1 ≤ i ≤ N , let \ τi = σi ∩ σj , j>i dim(σj ∩σi )=n−1

so τ1 = {0}, τN = σN , and τp ⊆ τq implies p ≤ q. (Such an ordering of cones is called a shelling of the fan.) This gives a cellular decomposition of X, with closures of cells being V (τ1 ), . . . , V (τN ), so [V (τ1 )], . . . , [V (τN )] forms a basis for H ∗ X. It follows that [V (τ1 )]T , . . . , [V (τN )]T form a basis for HT∗ X. If X is not projective, one can always find a refinement Σ′ of Σ (by subdividing cones), giving a surjective, birational, T -equivariant morphism π : X ′ → X, with X ′ projective and nonsingular. Under π, V (τ ′ ) maps to V (τ ), where τ is the smallest cone containing τ ′ ; this is birational if they have the same dimension. Since π∗ ◦ π ∗ = id on H ∗ X or HT∗ X, one sees the following: Lemma 1.2. For X a complete nonsingular toric variety, H ∗ X is generated by the classes [V (τ )] over Z, and HT∗ X is generated by [V (τ )]T over Λ. Also, ∼ HT∗ X ⊗Λ Z − → H ∗ X. We will see that H ∗ X and HT∗ X are always free of rank N , the number of n-dimensional cones. Question 1.3. Is there always a basis of [V (τ )]’s for H ∗ X? If not, an old combinatorial conjecture on shellability is false. For any cones σ and τ , if they span a cone γ, then V (σ) ∩ V (τ ) = V (γ); if dim γ = dim σ + dim τ , the intersection is transversal, so [V (σ)] · [V (τ )] = [V (γ)] and [V (σ)]T · [V (τ )]T = [V (γ)]T . If σ and τ are contained in a cone of Σ, then V (σ) ∩ V (τ ) = ∅, and the corresponding products are zero. Let D1 , . . . , Dd be the T -invariant divisors, with Di = V (τi ) for rays τi ; let vi ∈ N be the minimal generator of the ray τi . For u ∈ M , with corresponding rational function χu , X div(χu ) = hu, vi iDi .

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Equivariantly, χu is a rational section of the line bundle Lu corresponding to the character u, so X hu, vi i[Di ]T u = cT1 (Lu ) = [div(χu )]T = in HT∗ X. Note that [Di1 ] · · · [Dir ] = [V (τ )] if vi1 , . . . , vir span a cone τ , and the product is 0 otherwise; the same is true for equivariant products. 2 Let X1 , . . . , Xd be variables, one for each ray. In Z[X] = Z[X1 , . . . , Xd ], we have two ideals: (i) I is generated by all monomials Xi1 · · · Xir such that vi1 , . . . , vir do not span a cone of Σ. It suffices to take minimal such sets, so that any proper subset does span a cone. The ring Z[X]/I is called the Stanley-Reisner ring; it appears in combinatorics. P (ii) J is generated by all elements di=1 hu, vi iXi , for u ∈ M . It suffices to let u run through a basis for M . We have (∗)

Z[X]/(I + J) → H ∗ X,

where the map is given by Xi 7→ [Di ]. We have seen that I and J map to 0, so this is well-defined. It is surjective since [V (τ )] = [Di1 ] · · · [Dir ] if vi1 , . . . , vir span τ . In fact, (∗) is an isomorphism, as was proved by Jurkiewicz in the projective case, and by Danilov in general [Jur80, Dan78]. We will recover this result. In Λ[X] = Λ[X1 , . . . , Xd ], we have two ideals: (i) I ′ , with the same generators as I, i.e., monomials Xi1 · · · Xir such that vi1 , . . . , vir do not P span a cone in Σ. ′ (ii) J , with generators hu, vi iXi − u, for all u in M (or a basis of M ). We have (∗T )

Λ[X]/(I ′ + J ′ ) → HT∗ X,

by Xi 7→ [Di ]T . Again, we have seen that I ′ and J ′ map to 0. Similarly, this map is surjective. We will prove that (∗T ) is also an isomorphism. All this will follow from the construction of a complex often used in toric geometry (see for example Danilov, Lunts, etc.). (refs) For each cone τ , let vi1 , . . . , vik be its generators, and set Z[τ ] := Z[Xi1 , . . . , Xik ] = Z[X]/(Xj | vj 6∈ τ ). Consider this as a Z-module, and also as a Z[X]/I-module. Set M Ck = Z[τ ]. dim τ =k

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For a face γ of τ , there is a canonical surjection Z[τ ] → Z[γ]. Define d : Ck → Ck−1 by taking Z[τ ] to the sum of those Z[γ] for facets γ of τ : Let vi1 , . . . , vik be the generators of τ , with i1 < · · · < ik , and let γ be generated by vi1 , . . . , vˆip , . . . , vik ; then dk is (−1)p times the canonical surjection Z[τ ] → Z[γ]. Lemma 2.1. This gives an exact sequence of Z[X]/I-modules (1)

d

d

n 1 0 → Z[X]/I → Cn −→ Cn−1 → · · · −→ C0 → 0.

Proof. The map dk is a homomorphism of graded modules over Z[X], decomposing into a direct sum with one piece for each monomial X1m1 · · · Xdmd . All components vanish unless the set of vi with mi > 0 span a cone λ in Σ. Each Ck contributes a copy of Z for each τ that contains λ. The resulting complex is the one computing the reduced homology of a simplicial sphere in N/Nλ .  Lemma 2.2. The canonical homomorphism Z[X]/I → Λ[X]/(I ′ + J ′ ) is an isomorphism.

P Proof. Let u1 , . . . , un be a basis for M . The elements Z(uj ) = i huj , vi iXi − uj form a regular sequence in Λ[X] (since Λ = Z[u1 , . . . , un ]), with quotient Λ[X]/J ′ .  P In particular, Λ → Λ[X] → Z[X]/I takes u ∈ M to hu, vi iXi . Therefore the exact sequence (1) is an exact sequence of Λ-modules. Proposition 2.3. Z[X]/I ∼ = Λ[X]/(I ′ + J ′ ) is free over Λ of rank N , the number of n-dimensional cones. Proof. For a cone τ spanned by vi1 , . . . , vik , choose v(k + 1), . . . , v(n) to complete a basis of N . Let u1 , . . . , un be the dual basis of M . Then Z[τ ] ∼ = Λ/(uk+1 , . . . , un ) as a Λ-module, so the projective dimension of Ck is pdΛ Ck = n − k. It follows by induction that pdΛ (ker(Ck → Ck−1 )) ≤ n − k. Therefore pdΛ Z[X]/I = 0. By the (easier) graded version of the QuillenSuslin theorem, Z[X]/I is free. Now consider the beginning of (1): 0 → Z[X]/I → Cn → Cn−1 . Cn is free over Λ on N generators, since Λ ∼ = Z[σ] for n-dimensional cones σ. Cn−1 is a torsion Λ-module. Thus Z[X]/I is free on N generators.  Exercise 2.4. The Hilbert series ∞ X rkZ (Z[X]/I)m tm m=0

is equal to n X (−1)n−i ai i=0

(1 − t)i

.

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(We will not need this, however.) Consider the diagram 0

- Cn

- Z[X]/I

dn

Cn−1

ϕ ?

restr -

?

HT∗ X T , HT∗ X where ϕ takes Z[σ] to HT∗ (pσ ) = Λ as follows: if vi1 , . . . , vin span σ, let u1 , . . . , un be the dual basis in M , and let ϕ be the isomorphism Z[σ] = Z[Xi1 , . . . , Xin ] → Λ given by Xij 7→ uj . The left vertical map is the composition Z[X]/I → Λ[X]/(I ′ + J ′ ) → HT∗ X, taking Xi to [Di ]T . Exercise 2.5. Show that this diagram commutes. (The restriction to HT∗ (pσ ) factors through HT∗ (Uσ ), and Uσ ∼ = Cn , with T acting by weights u1 , . . . , un . If vi ∈ σ, with i = ij , then [Di ]T restricts to uj — indeed, Di restricts to the jth coordinate hyperplane in Uσ = Cn , so its equivariant class restricts to cT1 (Luj ) = uj . If vi 6∈ σ, then [Di ]T 7→ 0.) We have seen that the left vertical map is surjective; it follows that it is an isomorphism, proving (∗T ). Tensoring over Λ with Z, and noting Λ/M Λ = Z, we have (Λ[X]/(I ′ + J ′ )) ⊗Λ Z = Z[X]/(I + J), ∼

and HT∗ X ⊗Λ Z − → H ∗ X, so (∗) follows. Also, we have the following descriptions: HT∗ X = = = =

Z[X]/I ker(dn ) {(fσ ), fσ ∈ Z[σ] ∼ = Λ | fσ |τ = fσ′ |τ if τ is a facet of σ and σ ′ } {piecewise polynomial functions on NR },

where “piecewise polynomial” means continuous functions on NR defined by a polynomial in Λ on each maximal cone σ [Bri97]. This is the GKM theorem for toric varieties (with Z coefficients). Example 2.6. HT2 X = {piecewise linear functions} = DivT M . Remark 2.7. If the fan Σ is only simplicial (so the generators of each cone form part of basis for NR , but not necessarily for N ), then all the statements here remain true if Z is replaced by Q. (There may also be some multiplicities in products: V (σ) · V (τ ) = m · V (γ).) The ring of piecewise polynomial functions on the support |Σ| can be defined for any fan Σ, so it is natural to ask what geometric significance this has, for an arbitrary toric variety X = X(Σ). The answer was given by S. Payne: It is the equivariant operational Chow cohomology, A∗T X. There are also descriptions of (ordinary and equivariant) intersection homology groups for singular toric varieties.

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References [Bri97] M. Brion, “The structure of the polytope algebra,” Tohoku Math. J. (2) 49 (1997), no. 1, 1–32. [Bri-Ver] M. Brion and M. Vergne, “An equivariant Riemann-Roch theorem for complete, simplicial toric varieties,” J. Reine Angew. Math. 482 (1997), 67–92. [Dan78] V. Danilov, “The geometry of toric varieties,” Russ. Math. Surveys 33 (1978), 97–154. [Ful93] W. Fulton, Introduction to Toric Varieties, Princeton 1993. [Jur80] J. Jurkiewicz, “Chow ring of projective nonsingular torus embedding,” Colloq. Math. 43 (1980), no. 2, 261–270.