Toric Vector Bundles and Parliaments of Polytopes

We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and we relate edges to jets. Using the polytopes, we also exhibit toric vector bundles that are ample but not globally generated, and toric vector bundles that are ample and globally generated but not very ample. 1. OVERVIEW OF RESULTS The importance and prevalence of toric varieties stems from their calculability and their close relation to polyhedral objects. The challenge is to emulate this success and enlarge the class of varieties with both features. Rather than contemplating spherical varieties or all T -varieties, we extend the theory of toric varieties by studying torus-equivariant vector bundles and their projective bundles. Motivated by the ensuing simplifications in the toric dictionary between line bundles and polyhedra, we concentrate on vector bundles over a smooth complete toric variety. The goal of this paper is to give explicit polyhedral interpretations for properties of these vector bundles. To accomplish this goal, we fix a smooth complete toric variety X , over C, associated to the fan Σ. Let M denote the character lattice of the dense torus in X and write v1,v2, . . . ,vn ∈HomZ(M,Z) for the unique minimal generators of the rays in Σ. A toric vector bundle on X is a torus-equivariant locally-free OX -module E of finite rank r. The celebrated Klyachko classification proves that E corresponds to a finite-dimensional vector space E ∼= Cr equipped with compatible decreasing filtrations E ⊇ ·· · ⊇ E i( j) ⊇ E i( j+1) ⊇ ·· · ⊇ 0 where 1 6 i 6 n and j ∈ Z; see Section 2. This collection of linear subspaces embeds into the lattice of flats for a distinguished matroid M(E ). For each element e in the ground set of the matroid M(E ), we introduce the convex polytope Pe := { u ∈M⊗ZR : 〈u,vi〉6max ( j ∈ Z : e ∈ E i( j) ) for all 16 i6 n } . The set of all such polytopes Pe is called the parliament of polytopes for E ; see Section 3. Although the defining half-spaces for the polytopes Pe together with the elements e in the ground set of M(E ) encode the filtrations, the polytopes themself may be empty; compare with Remark 3.6. The following result gives the first substantive connection between the parliament of polytopes and the toric vector bundle. Proposition 1.1. The lattice points in the polytopes of the parliament for E correspond to the torus-equivariant generators for the space of global sections of E . Example 3.5 recovers the polytope associated to a toric line bundle on X . However, when the rank of E is greater than 1, Example 3.8 demonstrates that the lattice points in the polytopes of the parliament need not yield a basis for the space of global sections. This highlights the key difference between higher-rank toric vector bundles and toric line bundles: toric vector bundles depend on 2010 Mathematics Subject Classification. 14M25; 14J60, 51M20. 1 ar X iv :1 40 9. 31 09 v3 [ m at h. A G ] 1 F eb 2 01 7 2 S. DI ROCCO, K. JABBUSCH, AND G.G. SMITH both the combinatorics of the polytopes Pe and the properties of the elements e in the ground set of the matroid M(E ). For line bundles, we may overlook the elements indexing the polytope because linear algebra in a one-dimensional vector space is trivial. Our criterion for deciding whether a toric vector bundle is globally generated underscores this distinction. To outline this criterion, consider a maximal cone σ ∈ Σ. The restriction of the toric vector bundle E to the affine open toric variety Uσ splits equivariantly as a direct sum of toric line bundles. Since toric line bundles on Uσ correspond to lattice points in M, we obtain a multiset u(σ)⊂M of associated characters for each maximal cone σ ∈ Σ; see Section 2. With this notation, we have our second result. Theorem 1.2. A toric vector bundle is globally generated if and only if, for all maximal cones σ ∈ Σ, the associated characters in u(σ) are vertices of polytopes in the parliament and the elements indexing these polytopes form a basis in the matroid M(E ). Example 4.4 demonstrates that global generation is not simply a property of the individual polytopes in the parliament, and Example 5.5 shows that the higher-cohomology groups of a globally-generated ample toric vector bundle may be nonzero. The parliament of polytopes for E gives new insights into the projective bundle P(E ). This is particularly relevant for the positivity properties of E defined by the corresponding attribute for the tautological line bundle OP(E )(1). For instance, we may picture the restriction of E to a torus-invariant curve in X as the normalized distances between appropriately matched characters associated to E ; see Section 4. Hence, Theorem 2.1 in [HMP] allows us to quickly recognize ample and nef toric vector bundles. Exploiting our polyhedral interpretations, Example 5.3 exhibits a toric vector bundle F on P2 that is ample but not globally generated, and Example 6.4 exhibits a toric vector bundle H on P2 that is ample and globally generated but not very ample. Better still, Proposition 5.4 and Remark 6.8 prove that F and H have the minimal rank among all toric vector bundles on Pd with the given traits. Beyond answering Question 7.5 in [HMP], these examples reinforce the conventional wisdom that versions of positivity that coincide for line bundles diverge for higher-rank vector bundles. The discrete geometry within the parliament of polytopes nevertheless captures the positivity of jets. In contrast with the conventional wisdom, several forms of higher-order positivity are equivalent for toric vector bundles. A vector bundle E separates `-jets for ` ∈ N if, for every closed point x ∈ X with maximal ideal mx ⊆ OX , the natural map H0(X ,E )→ H0(X ,E ⊗OX OX/m x ) is surjective; see Section 6. As an enhancement of Theorem 1.2, Theorem 6.2 establishes that a toric vector bundle E separates `-jets if and only if certain edges in the polytopes of the parliament have normalized length at least `. This leads to the following equivalences. Theorem 1.3. A toric vector bundle E separates `-jets if and only if it is `-jet ample. Moreover, a toric vector bundle E separates 1-jets if and only if it is very ample. Unlike arbitrary vector bundles on a smooth projective variety, these versions of positivity coincide for toric vector bundles. Specializing to line bundles, we recover the main theorems in [DiR]. We also obtain a polyhedral characterization for very ampleness; see Corollary 6.7. TORIC VECTOR BUNDLES AND PARLIAMENTS OF POLYTOPES 3 Future directions. The introduction of the parliament of polytopes for a toric vector bundle suggests some new research projects. The most straightforward advances would provide polyhedral interpretations for other properties of toric vector bundles. For example, we suspect that a toric vector bundle is big if and only if some Minkowski sum of the polytopes in the parliament is fulldimensional. For a globally-generated toric vector bundle E , the complete linear series of OP(E )(1) maps the projective bundle P(E ) into projective space. Can one characterize the homogeneous equations of the image in terms of combinatorial commutative algebra? If so, then one expects a description of the initial ideals via regular triangulations; compare with Section 8 in [Stu]. Since there exists ample, but not globally generated, line bundles on varieties of the form P(E ), this class of varieties makes an interesting testing ground for Fujita’s conjecture; see Conjecture 10.4.1 in [La2]. More ambitiously, for an ample toric vector bundle E , one could even ask for an effective polyhedral bound on m ∈ N such that Symm(E ) is globally generated or very ample. Finally, we wonder if there are natural topological hypotheses on the parliament of polytopes which imply that all of the higher-cohomology groups vanish. Conventions. Throughout the document, N denotes the nonnegative integers and X is a smooth complete toric variety over the complex numbers C. The linear subspace generated by the vectors e1,e2, . . . ,em in a C-vector space is denoted by span(e1,e2, . . . ,em), and the polyhedral cone generated by the vectors v1,v2, . . . ,vm in a R-vector space is denoted by pos(v1,v2, . . . ,vm). Acknowledgements. We thank Alex Fink, Milena Hering, Nathan Ilten, Diane Maclagan, Bernt Ivar Utstøl Nødland, Sam Payne, Vic Reiner, Mike Roth, and Frank Sottile for helpful conversations. We especially thank an anonymous referee for wonderfully constructive feedback and for suggesting Example 3.7. The first author was partially supported by the Vetenskapsrådet grants NT:2010-5563 and NT:2014-4736, the second was partially supported by the Göran Gustafsson Stiftelse, and the third was partially supported by NSERC. 2. BACKGROUND ON TORIC VECTOR BUNDLES In this section, we collect the needed definitions and notation for toric varieties and vector bundles. Let X be a smooth complete d-dimensional toric variety, over C, determined by the strongly convex rational polyhedral fan Σ in N⊗ZR∼= Rd , where N is a lattice of rank d. The dual lattice is M := HomZ(N,Z), and the dense algebraic torus acting on X is T := SpecC[M]. For σ ∈ Σ, the corresponding affine toric variety is Uσ := SpecC[σ∨∩M], where σ∨ denotes the dual cone. The j-dimensional cones of Σ form the set Σ( j). For each maximal cone σ ∈ Σ(d), the corresponding T -fixed point is xσ ∈ X . We order the 1-dimensional cones Σ(1) (also known as rays) and, for 1 6 i 6 n, we write vi ∈ N for the unique minimal generator of the i-th ray. The i-th ray also corresponds to the irreducible T -invariant divisor Di on X , and the divisors D1,D2, . . . ,Dn generate the group DivT (X)∼=Zn of T -invariant divisors. Since X is complete, there is a short exact sequence 0−→M div −−→ DivT (X)−→ Pic(X)−→ 0 4 S. DI ROCCO, K. JABBUSCH, AND G.G. SMITH where divu := 〈u,v1〉D1 + 〈u,v2〉D2 + · · ·+ 〈u,vn〉Dn and the second map is the projection from the group of divisors to the Picard group. The invertible sheaf or line bundle associated to a divisor D ∈ DivT (X) is denoted by OX(D). For more information on toric varieties, see [CLS] or [Ful]. A toric vector bundle is a locally-free OX -module E of finite rank r equipped with a T -action that is compatible with the T -action on X . In other words, there exists a T -action on the variety V(E ) := Spec(SymE ) such that the projection map π : V(E )→ X is T -equivariant and T acts linearly on the fibres. For all σ ∈ Σ, there is also an induced T -action on the C-vector spaces of sections H(Uσ ,E ), where Uσ is the corresponding affine toric variety. Given a lattice point u ∈M, the trivial line bundle OX(divu) has a canonical T -equivariant structure. Explicitly, for all σ ∈ Σ, we have H0 ( Uσ ,OX(divu) ) = ⊕ u′∈σ∨∩M C ·χu ′−u ⊂ T , where χu,χu are the characters associated to the lattice points u′,u ∈ M; the identity in this semigroup is χ−u. As in [HMP], we follow the standard convention in invariant theory for the action of the group on the ring of functions, even though the opposite sign convention is more common in the toric literature. Every toric line bundle on the affine toric variety Uσ is T -equivariantly isomorphic to a line bundle OX(divu)|Uσ , where the class u of the lattice point u in Mσ := M/(σ⊥∩M) is uniquely determined. In addition, any toric vector bundle on an affine toric variety splits T -equivariantly as a direct sum of toric line bundles whose underlying line bundles are trivial; see Proposition 2.2 in [Pa1]. Hence, for all σ ∈ Σ, there is a unique multiset u(σ)⊂Mσ such that E |Uσ ∼= ⊕ u∈u(σ)OX(divu)|Uσ , where u ∈M is any lift of u. If σ is a maximal cone, then the multiset u(σ)⊂M is uniquely determined by the toric vector bundle E and the d-dimensional cone σ . We call the multisets u(σ), for all σ ∈ Σ(d), the associated characters of the toric vector bundle E . Toric vector bundles are classified in Theorem 0.1.1 of [Kl1] by canonical filtrations. To summarize this classification, let E be the fibre of E over the identity of the torus T , so E is a C-vector space isomorphic to Cr. The category of toric vector bundles on X is naturally equivalent to the category of finite-dimensional C-vector spaces E with separated exhaustive decreasing filtrations {E i( j)} j∈Z, for all 16 i6 n, that satisfy the compatibility condition: (CC) For each maximal cone σ ∈ Σ(d), there exists a decomposition E = ⊕ u∈u(σ)Lu such that E i( j) = ∑〈u,vi〉> j Lu. This compatibility condition is equivalent to the T -equivariant splitting into a direct sum of toric line bundles on the affine open toric variety Uσ , for all σ ∈ Σ(d); see Theorem 1.3.2 in [Kl2]. Indirectly, the decreasing filtrations provide the gluing data needed to assemble these direct sums into a toric vector bundle. The filtrations being separated and exhaustive, for each 16 i6 n, means that E i( j) = 0 for all j 0 and E i( j) = E ∼=Cr for all j 0, so each filtration contains only finitely many distinct linear subspaces. Hence, for a fixed i, we may conveniently describe the filtration {E i( j)} j∈Z via a labelled basis e1,e2, . . . ,er for E ∼= Cr, where each vector ek ∈ E is labelled by an integer and the linear subspace E i( j) is simply the span of the basis vectors with labels greater than TORIC VECTOR BUNDLES AND PARLIAMENTS OF POLYTOPES 5 or equal to j. For a self-contained exposition of this classification, we recommend Subsection 2.3 in [Pa1]; Subsection 2.4 in [Pa1] also provides a brief historical summary. Given a toric vector bundle E , the filtrations {E i( j)} j∈Z have a couple different geometric interpretations. For all cones σ ∈ Σ and all lattice points u ∈M, evaluating sections at the identity of the torus T gives an injective map H(Uσ ,E )u ↪→ E. The image of this map is the linear subspace Eσ u ⊆ E. Following Subsection 4.2 in [Pa2], we define a linear subspace Ev( j)⊆ E for all v ∈ N and all j ∈ Z. Since X is complete, there exists a unique cone σ ∈ Σ containing the lattice point v in its relative interior. Set Ev( j) := ∑〈u,v〉> j Eσ u . For any lattice point v ∈ N, the family of linear subspaces {Ev( j)} j∈Z give a separated exhaustive decreasing filtration of E. When the lattice point v equals vi for some 16 i6 n, we obtain the filtration {E i( j)} j∈Z. For the second interpretation of the filtrations, consider a cone σ ∈ Σ and suppose that we have E |Uσ ∼= ⊕ u∈u(σ)OX(divu)|Uσ . If the linear subspace Lu ⊆ E is the fibre of OX(divu) over the identity of the torus T , then we obtain a decomposition E = ⊕ u∈u(σ)Lu. Hence, the linear subspace Eσ u′ is spanned by the linear subspaces Lu for which u−u ′ ∈ σ∨ and Ev( j) = ⊕ 〈u,v〉> j Lu. For each maximal cone σ ∈ Σ, there exists a subset u(σ)⊂M and a decomposition E = ⊕ u∈u(σ)Eu such that, for all v ∈ σ and for all j ∈ Z, we have Ev( j) = ⊕ 〈u,v〉> j Eu. It follows that Eu = ⊕ u∈u(σ)Lu, so dimEu equals the multiplicity of u in the multiset u(σ) and u(σ) is the underlying set of elements in u(σ). 3. GLOBAL SECTIONS AND LATTICE POLYTOPES This section introduces explicit T -equivariant generators for the global sections of the toric vector bundle that correspond to the lattice points in a collection of polytopes. Each toric line bundle L on X corresponds to a rational convex polytope in M⊗ZR. We generalize this correspondence by associating a finite collection of convex polytopes to a toric vector bundle E . The polytopes in this collection are indexed by the elements in the ground set of a matroid associated to E . To describe this matroid, we first observe that the toric vector bundle E determines the finite poset L(E ), consisting of all the linear subspaces V := ⋂n i=1 E i( ji)⊆ E, where ( j1, j2, . . . , jn) ∈ Zn, ordered by inclusion. Since the filtrations {E i( j)} j∈Z are separated, exhaustive, and decreasing, we see that 0 ∈ L(E ), E ∈ L(E ), and L(E ) is closed under intersection. Hence, the pair ( L(E ),∩ ) forms a meet-semilattice. The next result shows that L(E ) embeds into the lattice of flats for a distinguished representable matroid. Proposition 3.1. For a toric vector bundle E , there exists a unique matroid M(E ), representable over C, such that (M1) the poset L(E ) is isomorphic to a meet-subsemilattice in the lattice of flats, (M2) among all matroids satisfying (M1), the number of elements in the ground set is minimal, and (M3) among all matroids satisfying (M1) and (M2), the number of circuits is minimal. In the language of linear subspace arrangements (and ordering the subspaces by reversed inclusion), Proposition 3.1 is equivalent to Theorem I.4.9 in [Zie]. 6 S. DI ROCCO, K. JABBUSCH, AND G.G. SMITH Proof. We verify that Algorithm 3.2 returns a representable matroid M with the desired conditions. By construction, each linear subspace V in L(E ) is generated by a subset of vectors in the ground Algorithm 3.2 (Construction of the representable matroid associated to a toric vector bundle). Input: The poset L(E ) of linear subspaces associated to the toric vector bundle E . Output: The canonical matroid M(E ) associated to E . Set r to be the dimension of the largest linear subspace E in L(E ); Initialize G to be a set consisting of a basis vector for each one-dimensional subspace in L(E ); For each integer k from 2 to r do For each k-dimensional linear subspace V in L(E ) do Set G′ to be the subset of elements in G that lie in V ; If the linear subspace span(G′) is a proper subspace in V then Append to G a basis for a complementary subspace to span(G′) in V ; Return the linear matroid defined by the vectors in G. set of the matroid M. The subset of the ground set consisting of all elements contained in V is the flat FV in M corresponding to V . It follows that span(FV ) =V , rank(FV ) = dim(V ), and the induced injective map from the poset L(E ) into lattice of flats for M is compatible with intersections. Thus, the matroid M satisfies the condition (M1). For any matroid, the lattice of flats is relatively complemented; see Proposition 3.4.4 in [Whi]. It follows that, for any linear subspace V in L(E ) and any matroid satisfying condition (M1), there exists a flat F ′ such that the join of FV and F ′ is FE and the meet of FV and F ′ is F{0}. By iterating from the smallest to the largest linear subspaces in L(E ), the Algorithm 3.2 finds a minimal set of complementary subspaces for L(E ). Adjoining these to L(E ), we obtain a new meet-semilattice L′ such that the complementary subspaces are minimal among the nonzero subspaces, and every linear subspace is generated by some collection of minimal nonzero subspaces. Using the terminology from Section 3.4 in [Whi], we see that the atoms in L′ are the one-dimension linear subspaces in L(E ) together with the adjoined complementary subspaces. Moreover, L′ is the minimal atomistic meet-semilattice containing L(E ). Finally, we claim that the matroid M is the free expansion of L′; see Proposition 10.2.3 in [Whi]. By construction, the ground set of M consists of a basis for each atom in L′, so the number of elements in the ground set of M equals the number of elements in the ground set of the free expansion of L′. Moreover, the conditional statement in Algorithm 3.2 implies that a flat D in the matroid M is dependent if and only if there exists a linear subspace W ∈ L′ such that |D∩FW |> dim(W ). We conclude that M is the free expansion of L′. Therefore, Proposition 10.2.2 and Proposition 10.2.6 in [Whi] establish that the matroid M satisfies conditions (M2) and (M3) respectively. Remark 3.3. Since E ∈ L(E ), Algorithm 3.2 shows that the number of elements in the ground set of the matroid M(E ) is at least the rank r of E . To have equality, there must be a basis for E such that every linear subspace in L(E ) is a direct sum of coordinate subspaces. Hence, the number of TORIC VECTOR BUNDLES AND PARLIAMENTS OF POLYTOPES 7 elements in the ground set of the matroid M(E ) equals r if and only if the toric vector bundle E splits T -equivariantly into a direct sum of toric line bundles. For each maximal cone σ ∈ Σ(d), the compatibility condition (CC) is equivalent to saying that the subposet of L(E ) consisting of the linear subspaces ⋂ vi∈σ E i( ji), where ji ∈ Z, is a distributive lattice; see Remark 2.2.2 in [Kl1]. Equivalently, the matroid M(E ) contains a compatible basis Bσ such that each component E i( j), for vi ∈ σ and j ∈ Z, is a direct sum of the corresponding coordinate subspaces. Example 4.4 demonstrates that, for a given maximal cone σ , there may be more that basis in M(E ) with this property. Remark 3.4. For an element e in the ground set of the matroid M(E ) and a linear subspace V ∈ L(E ), the relation e ∈ V depends only on the matroid M(E ) and not on the choice of a representation for M(E ). Nevertheless, Algorithm 3.2 does produce a particular representation for M(E ). This is analogous to a minimal free presentation for a finitely generated graded module over a polynomial ring: the ranks of the free modules are intrinsic invariants, but the matrix representing the map depends on the choice of bases; compare with Section 1B in [Eis]. For each element e in the ground set of the matroid M(E ), the associated convex polytope is Pe := { u ∈M⊗ZR : 〈u,vi〉6max ( j ∈ Z : e ∈ E i( j) ) for all 16 i6 n } . Using a traditional term of venery (namely, the collective noun for owls), we call the collection of all such polytopes Pe the parliament of polytopes for the toric vector bundle E . The number of polytopes in the parliament for E is at least the rank of E and equals the rank of E precisely when E splits into a direct sum of toric line bundles; see Remark 3.3. Extending the classic theorem [CLS, Theorem 4.3.3] for line bundles on a toric variety, we have the following interpretation for the lattice points in a parliament of polytopes. Proposition 1.1. The lattice points in the polytopes of the parliament for E correspond to the T -equivariant generators for the space of global sections of E : H0(X ,E )∼= ∑ e span(e⊗χ−u : u ∈ Pe∩M)⊂ E⊗C T , where the sum is over all elements e in the ground set of the matroid M(E ). Proof of Proposition 1.1. The T -action on the space of global sections yields a decomposition into isotypical components H0(X ,E )u, where u ∈M. The regular T -eigenfunction χ−u is an element of H0(X ,E )u and we have H0(X ,E ) = ⊕ u∈M H 0(X ,E )u. Since X is complete, at most finitely many of the isotypical components are nonzero. Following Corollary 4.1.3 in [Kl1], evaluation at the identity of the torus T gives a canonical isomorphism