EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE THIRTEEN: TORIC VARIETIES

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1 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 = Hom Z (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 N R = 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 N R (the completeness condition). The toric variety X is covered by open affines U σ = SpecC[σ M], where σ = {u u,v 0 for all v σ}. In fact, U σ = C k (C ) n k, where k = dimσ, and the n-dimensional cones suffice to cover. Also, U {0} = SpecC[M] = T, and U σ U τ = U σ τ. Write χ u 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 dimτ. On open affines, this is given by V (τ) U σ = Spec C[τ σ M], with the containment in U σ given by C[σ M] C[τ σ M], with χ u χ u if u τ and χ u 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/N τ, where N τ is the sublattice generated by τ. The T-fixed points of X are p σ = V (σ) for dimσ = n. X is projective if and only if there is a lattice polytope P M R, 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 σ F = {v u,v u,v for all u P, u F }. Date: April 30,

2 2 13 TORIC VARIETIES This correspondence reverses dimensions: dim σ F = codim F. Example 1.1. The standard n-dimensional simplex corresponds to P n. An n-cube corresponds to (P 1 ) n. Figures... For X projective, choose a general vector v N R, giving an ordering of the vertices u 1,...,u N (so that u 1,v < u N,v ), and thus an ordering of the n-dimensional cones σ 1,...,σ N. For 1 i N, let τ i = σ i σ j, j>i dim(σ j σ i )=n 1 so τ 1 = {0}, τ N = σ N, and τ p τ q 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 (τ 1 ),...,V (τ N ), so [V (τ 1 )],...,[V (τ N )] forms a basis for H X. It follows that [V (τ 1 )] T,...,[V (τ N )] 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 H T X is generated by [V (τ)]t over Λ. Also, H T 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 D 1,...,D d be the T-invariant divisors, with D i = V (τ i ) for rays τ i ; let v i N be the minimal generator of the ray τ i. For u M, with corresponding rational function χ u, div(χ u ) = u,v i D i.

3 EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY 3 Equivariantly, χ u is a rational section of the line bundle L u corresponding to the character u, so u = c T 1 (L u) = [div(χ u )] T = u,v i [D i ] T in HT X. Note that [D i1 ] [D ir ] = [V (τ)] if v i1,...,v ir span a cone τ, and the product is 0 otherwise; the same is true for equivariant products. Let X 1,...,X d be variables, one for each ray. In Z[X] = Z[X 1,...,X d ], we have two ideals: 2 (i) I is generated by all monomials X i1 X ir such that v i1,...,v ir 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. (ii) J is generated by all elements d i=1 u,v i X i, 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 X i [D i ]. We have seen that I and J map to 0, so this is well-defined. It is surjective since [V (τ)] = [D i1 ] [D ir ] if v i1,...,v ir 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[X 1,...,X d ], we have two ideals: (i) I, with the same generators as I, i.e., monomials X i1 X ir such that v i1,...,v ir do not span a cone in Σ. (ii) J, with generators u,v i X i u, for all u in M (or a basis of M). We have ( T ) Λ[X]/(I + J ) H TX, by X i [D i ] 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 v i1,...,v ik be its generators, and set Z[τ] := Z[X i1,...,x ik ] = Z[X]/(X j v j τ). Consider this as a Z-module, and also as a Z[X]/I-module. Set C k = Z[τ]. dim τ=k

4 4 13 TORIC VARIETIES For a face γ of τ, there is a canonical surjection Z[τ] Z[γ]. Define d : C k C k 1 by taking Z[τ] to the sum of those Z[γ] for facets γ of τ: Let v i1,...,v ik be the generators of τ, with i 1 < < i k, and let γ be generated by v i1,..., ˆv ip,...,v ik ; then d k 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 0 Z[X]/I C n n Cn 1 d1 C 0 0. Proof. The map d k is a homomorphism of graded modules over Z[X], decomposing into a direct sum with one piece for each monomial X m 1 1 X m d d. All components vanish unless the set of v i with m i > 0 span a cone λ in Σ. Each C k 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/N λ. Lemma 2.2. The canonical homomorphism is an isomorphism. Z[X]/I Λ[X]/(I + J ) Proof. Let u 1,...,u n be a basis for M. The elements Z(u j ) = i u j,v i X i u j form a regular sequence in Λ[X] (since Λ = Z[u 1,...,u n ]), with quotient Λ[X]/J. In particular, Λ Λ[X] Z[X]/I takes u M to u,v i X i. 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 v i1,...,v ik, choose v(k + 1),...,v(n) to complete a basis of N. Let u 1,...,u n be the dual basis of M. Then Z[τ] = Λ/(u k+1,...,u n ) as a Λ-module, so the projective dimension of C k is pd Λ C k = n k. It follows by induction that pd Λ (ker(c k C k 1 )) n k. Therefore pd Λ Z[X]/I = 0. By the (easier) graded version of the Quillen- Suslin theorem, Z[X]/I is free. Now consider the beginning of (1): 0 Z[X]/I C n C n 1. C n is free over Λ on N generators, since Λ = Z[σ] for n-dimensional cones σ. C n 1 is a torsion Λ-module. Thus Z[X]/I is free on N generators. Exercise 2.4. The Hilbert series rk Z (Z[X]/I) m t m is equal to m=0 n i=0 ( 1) n i a i (1 t) i.

5 EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY 5 (We will not need this, however.) Consider the diagram 0 Z[X]/I C n d n C n 1 ϕ restr HTX HTX T, where ϕ takes Z[σ] to HT (p σ) = Λ as follows: if v i1,...,v in span σ, let u 1,...,u n be the dual basis in M, and let ϕ be the isomorphism Z[σ] = Z[X i1,...,x in ] Λ given by X ij u j. The left vertical map is the composition Z[X]/I Λ[X]/(I + J ) HT X, taking X i to [D i ] T. Exercise 2.5. Show that this diagram commutes. (The restriction to HT (p σ) factors through HT (U σ), and U σ = C n, with T acting by weights u 1,...,u n. If v i σ, with i = i j, then [D i ] T restricts to u j indeed, D i restricts to the jth coordinate hyperplane in U σ = C n, so its equivariant class restricts to c T 1 (L u j ) = u j. If v i σ, then [D i ] T 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/MΛ = Z, we have (Λ[X]/(I + J )) Λ Z = Z[X]/(I + J), and H T X Λ Z H X, so ( ) follows. Also, we have the following descriptions: H TX = Z[X]/I = ker(d n ) = {(f σ ), f σ Z[σ] = Λ f σ τ = f σ τ if τ is a facet of σ and σ } = {piecewise polynomial functions on N R }, where piecewise polynomial means continuous functions on N R 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. H 2 T X = {piecewise linear functions} = Div T M. Remark 2.7. If the fan Σ is only simplicial (so the generators of each cone form part of basis for N R, 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.

6 6 13 TORIC VARIETIES References [Bri97] M. Brion, The structure of the polytope algebra, Tohoku Math. J. (2) 49 (1997), no. 1, [Bri-Ver] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), [Dan78] V. Danilov, The geometry of toric varieties, Russ. Math. Surveys 33 (1978), [Ful93] W. Fulton, Introduction to Toric Varieties, Princeton [Jur80] J. Jurkiewicz, Chow ring of projective nonsingular torus embedding, Colloq. Math. 43 (1980), no. 2,