Cohomology of toric line bundles via simplicial Alexander duality
aa r X i v : . [ m a t h . AG ] F e b COHOMOLOGY OF TORIC LINE BUNDLES VIA SIMPLICIALALEXANDER DUALITY
SHIN-YAO JOW
Abstract.
We give a rigorous mathematical proof for the validity of the toricsheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen,B. Jurke, T. Rahn, and H. Roschy ( arXiv:1003.5217 ). We actually prove not onlythe original algorithm but also a speed-up version of it. Our proof is independentfrom (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T.Rahn ( arXiv:1006.2392 ), and has several advantages such as being shorter andcleaner and can also settle the additional conjecture on “Serre duality for Bettinumbers” which was raised but unresolved in arXiv:1006.2392 . Introduction
Recently in [BJRR10] a new algorithm for computing the cohomology groups ofline bundles on a toric variety was conjectured, which the authors observed to bemore efficient than previously known algorithms such as those described in [Ful93, § § X be a d -dimensional toric variety over Date : 23 December 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Toric variety; cohomology of line bundle; local cohomology; Alexanderduality. a field K associated to a fan ∆ in N ∼ = Z d . We denote by ∆(1) the set of one-dimensional cones (i.e. rays) in ∆, and assume that ∆(1) spans N R = N ⊗ Z R . TheCox ring of X ([Aud91], [Mus94], [Cox95]) is the polynomial ring S in the set ofvariables indexed by the rays in ∆: S = K [ x ρ | ρ ∈ ∆(1)] . One can consider on S the usual multigrading on a polynomial ring, which in thiscase is the grading by Z ∆(1) , the free abelian group with a basis indexed by the rays(of course in S only the N ∆(1) -graded pieces are nonzero, but later we will also needto consider localizations of S which invert some of the variables). There is also animportant squarefree monomial ideal B of S , called the irrelevant ideal, defined asfollows. For each cone σ in the fan ∆, let σ (1) = { ρ ∈ ∆(1) | ρ ⊂ σ } . Then B = h Y ρ/ ∈ σ (1) x ρ | σ ∈ ∆ i . The relations between X and ( S, B ) are very similar to those between a projectivespace and its homogeneous coordinate ring with the usual irrelevant ideal. For exam-ple it was shown in [Cox95] that X is a quotient of the complement of the algebraicsubset defined by B in the affine space Spec S . Moreover, under this quotient map,the image of the coordinate hyperplane ( x ρ = 0) ⊂ Spec S is the torus-invariant primedivisor D ρ ⊂ X corresponding to the ray ρ . In view of this, it is natural to endow S with another coarser grading by the class group of X : one simply sets the degree of amonomial Q ρ ∈ ∆(1) x a ρ ρ to be the divisor class [ P ρ ∈ ∆(1) a ρ D ρ ] in Cl( X ). In the specialcase when X is a projective space, this is simply the grading by the total degree. If F is a Cl( X )-graded module over S , then an associated quasi-coherent sheaf e F on X can be constructed in a way similar to the projective space case, and it is also truethat every coherent sheaf on X is of the form e F for some finitely generated F ([Cox95,Proposition 3.3] for simplicial X ; [EMS00, Theorem 2.1]). Using ˇCech cohomology,it was shown [EMS00, Proposition 2.3] that for i ≥
1, there is an isomorphism ofCl( X )-graded S -modules M α ∈ Cl( X ) H i (cid:16) ] F ( α ) (cid:17) ∼ = H i +1 B ( F ) , where F ( α ) is the shifted module F ( α ) β = F α + β , and H i +1 B ( F ) is the ( i + 1) th localcohomology of F with support on B [MS05, Definition 13.1]. This isomorphism isthe starting point for studying sheaf cohomology on a toric variety both in [EMS00]and in this paper. We will focus on the case F = S , since ] S ( α ) ∼ = O X ( D ) if D is adivisor with divisor class α , so H i +1 B ( S ) already encodes the i th sheaf cohomology ofall line bundles on X . Following [EMS00], we use the notation H i ∗ ( O X ) to denote the OHOMOLOGY OF TORIC LINE BUNDLES VIA SIMPLICIAL ALEXANDER DUALITY 3 left-hand side of the above isomorphism when F = S . So we have(1) H i ∗ ( O X ) = M α ∈ Cl( X ) H i (cid:16) ] S ( α ) (cid:17) ∼ = H i +1 B ( S ) . The local cohomology H i +1 B ( S ) can be computed by a ˇCech complex whose termsinvolve only localizations of S inverting some of the variables ([MS05, Theorem 13.7];see also Section 3 of this paper). Consequently H i +1 B ( S ), and hence H i ∗ ( O X ), enjoy thefiner Z ∆(1) -grading. It is known that each of the graded pieces H i ∗ ( O X ) p , p ∈ Z ∆(1) ,depends only on which coordinates of p are negative ([EMS00, Theorem 2.4]; see alsoProposition 3.1). More precisely, if one definesneg( p ) = { ρ ∈ ∆(1) | the ρ coordinate of p is negative } , then H i ∗ ( O X ) p is canonically isomorphic to H i ∗ ( O X ) q for any p, q ∈ Z ∆(1) such thatneg( p ) = neg( q ). Hence it makes sense to define H i ∗ ( O X ) I for a subset I ⊂ ∆(1), bysetting it to be H i ∗ ( O X ) p for any p ∈ Z ∆(1) such that neg( p ) = I . One then wantsto know which subsets I give rise to nonzero graded pieces H i ∗ ( O X ) I , and how thesegraded pieces can be computed. Our main theorem provides a simple combinatorialanswer to this question when X is a simplicial projective toric variety. To state it,it will be convenient to define two collections of subsets of ∆(1): first following thenotation in [BJRR10] we defineSR = n ˆ J ⊂ ∆(1) (cid:12)(cid:12)(cid:12) ˆ J does not span a cone in ∆, but every propersubset of ˆ J spans a cone in ∆. o . The squarefree monomials of the form Q ρ ∈ ˆ J x ρ , ˆ J ∈ SR are precisely the minimalgenerators of the so-called Stanley-Reisner ideal of ∆. Then we define U SR to be thecollection of all subsets of ∆(1) which can be expressed as a union ˆ J ∪ · · · ∪ ˆ J m forsome ˆ J , . . . , ˆ J m in SR. Theorem 1.1.
Let X be a simplicial projective toric variety of dimension d associatedto a fan ∆ . Let I be a subset of ∆(1) and let i ≥ be a positive integer. Then (a) H i ∗ ( O X ) I = 0 unless I ∈ U SR . (b) Let ˆ I denote the complement of I in ∆(1) . If i = d , then H i ∗ ( O X ) I is naturallydual to H d − i ∗ ( O X ) ˆ I (as K -vector spaces). Combined with part (a) this implies inparticular that H i ∗ ( O X ) I = 0 unless both I and ˆ I are in U SR . (c) If I ∈ U SR , let ˆ J , . . . , ˆ J m be all of the elements in SR that are contained in I . Define Λ I to be the following abstract simplicial complex on the vertex set { , . . . , m } : Λ I = (cid:26) K ⊂ { , . . . , m } (cid:12)(cid:12)(cid:12)(cid:12) [ k ∈ K ˆ J k = I (cid:27) . SHIN-YAO JOW
Then there is a natural isomorphism H i ∗ ( O X ) I ∼ = e H | I |− i − (Λ I ) , where the right-hand side is the reduced homology of Λ I (with coefficients in K ). We remark that part (b) is the “Serre duality for Betti numbers” conjectured in thelast section of [RR10], while part (c) is essentially the “remnant” cohomology H i ( Q )in [BJRR10]. As an immediate corollary we obtain the following speed-up version of the algorithmin [BJRR10]:
Corollary 1.2.
Given a line bundle L on X and an integer < i < d , we have h i ( X, L ) = X I (cid:26) p ∈ Z ∆(1) (cid:12)(cid:12)(cid:12)(cid:12) neg( p ) = I and h X ρ ∈ ∆(1) p ρ D ρ i = L (cid:27) · dim e H | I |− i − (Λ I ) , where the sum is over all I ⊂ ∆(1) such that both I and ˆ I are in U SR . (In the originalalgorithm in [BJRR10] the sum is over all I in U SR .) We remark that from a theoretic point of view, the efficiency of this algorithm seemsto come primarily from the vanishing in Theorem 1.1 (a)(b), which greatly reducesthe amount of homological computation one needs to perform. We refer the readersto the Introduction in [RR10] for more details on implementation of the algorithmand how it compares with other previously known algorithms.The organization of the rest of this article is as follows: Section 2 contains sometopological preliminaries we will need in our proof, including our primary tool the sim-plicial Alexander duality (Corollary 2.3) and the topological source of the vanishingin Theorem 1.1 (a) (Lemma 2.5). In Section 3 we review the method for computingthe local cohomology H iB ( S ) using the natural cellular resolution of S/B supportedon the moment polytope of X , as done in [MS04, § Acknowledgements
The author would like to thank Ezra Miller for several useful comments on the firstdraft of this article. In our notations the “remnant” cohomology H i ( Q ) is really the (reduced) relative homology ofthe full simplex on { , . . . , m } modulo Λ I , hence is essentially the same as the reduced homology ofΛ I since the full simplex is contractible. The cases i = 0 or d are known to be easy. When i = 0 the cohomology can be computed bycounting lattice points in a certain polytope: see for example [Ful93, p.66]. The case i = d can bereduced to the case i = 0 by the usual Serre duality. OHOMOLOGY OF TORIC LINE BUNDLES VIA SIMPLICIAL ALEXANDER DUALITY 5 Combinatorial topological preliminaries
In this section we collect some results from simplicial topology which will comeinto the proof of the main theorem, most notably the simplicial Alexander duality.Throughout this section Γ will denote an (abstract) simplicial complex on a finitevertex set V , i.e. Γ is a collection of subsets of V , such that if σ ∈ Γ and τ ⊂ σ then τ ∈ Γ. Each σ ∈ Γ is called a simplex or a face of Γ. The underlying topological spaceof Γ is denoted by k Γ k . Also recall that the link of a face σ in Γ is the subcomplexlink Γ σ = { τ ∈ Γ | τ ∪ σ ∈ Γ and τ ∩ σ = ∅ } . Given any subset σ ⊂ V (not necessarily a face of Γ), the notation ˆ σ will denoteits complement ˆ σ = V \ σ , and the notation Γ ≤ σ will denote the simplicial complexconsisting of every face of Γ that is contained in σ :Γ ≤ σ = { τ ∈ Γ | τ ⊂ σ } . Definition 2.1.
The
Alexander dual of Γ, denoted by Γ ∗ , is the following simplicialcomplex on the vertex set V : Γ ∗ = { σ ⊂ V | ˆ σ / ∈ Γ } . Theorem 2.2 (Simplicial Alexander duality) . For every integer j there is an iso-morphism e H j (Γ ∗ ) ∼ = e H | V |− − j (Γ) . A self-contained proof from first principles was given in [BT09], and a proof usinghomological algebra can be found in [MS05, Section 5.1]. The connection to thetopological Alexander duality e H j ( S n − \ A ) ∼ = e H n − − j ( A ) is that if | V | = n , thenthe simplicial complex consisting of all proper subsets of V has an underlying spacehomeomorphic to S n − , and it contains the closed subcomplex Γ whose underlyingspace plays the role of A . Furthermore it can be shown that S n − \ k Γ k is homotopyequivalent to k Γ ∗ k : see Proposition 2.27 of the book [BP02]. What we will use in factis the following equivalent version given in Proposition 2.29 of that same book: Corollary 2.3 (Simplicial Alexander duality—alternative version) . If σ ∈ Γ ∗ , then e H j (link Γ ∗ σ ) ∼ = e H | V |− − j −| σ | (Γ ≤ ˆ σ ) . Proof.
This follows from Theorem 2.2 because link Γ ∗ σ is the Alexander dual of Γ ≤ ˆ σ when both are viewed as simplicial complexes on the vertex set ˆ σ . (On the other handsetting σ = ∅ recovers Theorem 2.2, so the two statements are actually equivalent.) (cid:3) We will also need the following two simple lemmas:
Lemma 2.4.
For any subset σ ⊂ V , it holds that k Γ ≤ ˆ σ k is a deformation retract of k Γ k \ k Γ ≤ σ k . SHIN-YAO JOW
Proof.
Let Γ σ, ˆ σ be the set consisting of every face of Γ which is neither contained in σ nor in ˆ σ . Each τ ∈ Γ σ, ˆ σ can be written as a proper disjoint union τ = τ σ ⊔ τ ˆ σ , where τ σ = τ ∩ σ and τ ˆ σ = τ ∩ ˆ σ . Observing that k Γ k \ k Γ ≤ σ k = k Γ ≤ ˆ σ k ∪ (cid:18) [ τ ∈ Γ σ, ˆ σ k τ k \ k τ σ k (cid:19) , the lemma thus follows since each k τ k\k τ σ k can be deformation retracted to k τ ˆ σ k . (cid:3) Lemma 2.5.
Given σ ∈ Γ , let τ , . . . , τ m ∈ Γ be the maximal faces of Γ containing σ . If τ ∩ · · · ∩ τ m % σ , then k link Γ σ k is contractible.Proof. The maximal faces of link Γ σ are precisely τ \ σ, . . . , τ m \ σ , and by assumptionthey have a nonempty intersection. Hence the lemma follows. (cid:3) Toric local cohomology
Let X be a d -dimensional simplicial projective toric variety associated to a fan ∆in N ∼ = Z d . Let ∆( i ) denote the set of i -dimensional cones in ∆. Recall from theIntroduction that the Cox ring of X is the polynomial ring S = K [ x ρ | ρ ∈ ∆(1)], andthe irrelevant ideal is B = h Q ρ/ ∈ σ (1) x ρ | σ ∈ ∆ i . In this section we review the methodfor computing the local cohomology H iB ( S ) using the natural cellular resolution of S/B supported on the moment polytope of X . Our main reference is [MS05] (see also[MS04, §
3] or [Mil00, Example 6.6]).A common way to compute local cohomology is to use the usual ˇCech complex[MS05, Definition 13.5]. This is often not the most efficient complex one can use. Infact any free resolution F • of S/B gives rise to a generalized ˇCech complex ˇ C •F [MS05,Definition 13.28], which can be used to compute H iB ( S ) [MS05, Theorem 13.31]. Theusual ˇCech complex came from the Taylor resolution [MS05, § F • is the naturalcellular resolution supported on the moment polytope of X , which turns out to beminimal [MS05, § § X . In the theory of projectivetoric varieties this is a well-known polytope in the dual space of N , but since we areonly concerned about its combinatorial structure, it suffices to know that the set ofall of its faces is in an inclusion-reversing bijection with ∆. Hence we can identify theset of i -dimensional faces of the moment polytope with the set ∆( d − i ), and labeleach face σ with the monomial x ˆ σ def = Q ρ/ ∈ σ (1) x ρ . The moment polytope together withthis labeling fits the definition of a labeled cell complex [MS05, Definition 4.2], and OHOMOLOGY OF TORIC LINE BUNDLES VIA SIMPLICIAL ALEXANDER DUALITY 7 gives rise to the following cellular free resolution of
S/B : F • : 0 ←− S ←− M σ ∈ ∆( d ) S x ˆ σ ←− M σ ∈ ∆( d − S x ˆ σ ←− · · · ←− M σ ∈ ∆(0) S x ˆ σ ←− , where the homomorphism from a direct summand S x ˆ σ of F i to a direct summand S x ˆ τ of F i − is 0 if τ σ , and is the inclusion map times ± τ ⊃ σ . The sign isdetermined by arbitrary but fixed orientations for the cones in ∆. The correspondinggeneralized ˇCech complex ˇ C •F is by definitionˇ C •F : 0 −→ S −→ M σ ∈ ∆( d ) S h x ˆ σ i −→ M σ ∈ ∆( d − S h x ˆ σ i −→ · · · −→ M σ ∈ ∆(0) S h x ˆ σ i −→ , where the homomorphism between two direct summands S [ x ˆ τ ] and S [ x ˆ σ ] for eachcodimension one σ ⊂ τ is again the inclusion map times the sign determined byorientations.By [MS05, Theorem 13.31] we have H iB ( S ) = H i ( ˇ C •F ). Since ˇ C •F is Z ∆(1) -graded, so is H iB ( S ). In fact given any p ∈ Z ∆(1) we can describe the graded piece H iB ( S ) p explicitlyas follows. Let I ⊂ ∆(1) be the set of places where p has negative coordinates, i.e. I = neg( p ) def = { ρ ∈ ∆(1) | the ρ coordinate of p is negative } . Let ˆ I = ∆(1) \ I . Then the graded piece S [ x ˆ σ ] p is 0 if σ (1) ˆ I , and is Kx p if σ (1) ⊂ ˆ I , where x p denotes the monomial whose exponent vector is p . Taking thegraded piece of degree p of every term in ˇ C •F and then computing the cohomology, oneobtains the following description of H iB ( S ) p , which is essentially the same as [EMS00,Theorem 2.7] or [MS04, Proposition 3.2]: Proposition 3.1.
Let P be the abstract simplicial complex on the vertex set ∆(1) given by P = { σ (1) | σ ∈ ∆ } . Then for every p ∈ Z ∆(1) and i ≥ there is anisomorphism H iB ( S ) p ∼ = e H d − i ( P ≤ ˆ I ) , where I = neg( p ) . In particular H iB ( S ) p only depends on neg( p ) . Proof of Theorem
We now present the proof of our main theorem.
Proof of Theorem 1.1.
By (1) we have H i ∗ ( O X ) I ∼ = H i +1 B ( S ) I , and by Proposition 3.1we have H i +1 B ( S ) I ∼ = e H d − − i ( P ≤ ˆ I ). Lemma 2.4 then implies that e H d − − i ( P ≤ ˆ I ) ∼ = e H d − − i ( kPk \ kP ≤ I k ) . Since kPk is homeomorphic to S d − , the topological Alexander duality gives e H d − − i ( kPk \ kP ≤ I k ) ∼ = e H i − ( P ≤ I ) . SHIN-YAO JOW
Now there are two possible cases for I : Case 1: ˆ I / ∈ P ∗ . This is equivalent to I ∈ P , which implies that P ≤ I is the fullsimplex on I , so e H i − ( P ≤ I ) = 0. Case 2: ˆ I ∈ P ∗ . In this case we can apply Corollary 2.3 to get e H i − ( P ≤ I ) ∼ = e H | I |− − i (link P ∗ ˆ I ) . Let J , . . . , J m be the maximal faces of P ∗ containing ˆ I . There are two furthersubcases: either J ∩ · · · ∩ J m % ˆ I or J ∩ · · · ∩ J m = ˆ I . If J ∩ · · · ∩ J m % ˆ I then e H | I |− − i (link P ∗ ˆ I ) = 0 by Lemma 2.5. Hence H i ∗ ( O X ) I = 0 unless J ∩ · · · ∩ J m = ˆ I , or equivalently I = ˆ J ∪ · · · ∪ ˆ J m . This proves part (a) sinceSR = { ˆ J | J is a maximal face of P ∗ } . Now if J ∩ · · · ∩ J m = ˆ I , then J \ ˆ I, . . . , J m \ ˆ I are precisely the maximalfaces of link P ∗ ˆ I , and the nerve they form (in the sense of [Gr¨u70]) is preciselyΛ I . Hence e H | I |− − i (link P ∗ ˆ I ) ∼ = e H | I |− − i (Λ I )by [Gr¨u70, Theorem 10], which proves part (c).Finally to see part (b), note that the series of isomorphisms in the beginning part ofthe proof showed that H i ∗ ( O X ) I ∼ = e H d − − i ( P ≤ ˆ I ) ∼ = e H i − ( P ≤ I ) . So H i ∗ ( O X ) I ∼ = e H i − ( P ≤ I ), and H d − i ∗ ( O X ) ˆ I ∼ = e H i − ( P ≤ ˆˆ I ) = e H i − ( P ≤ I ) if i = d , hencethey are dual to each other. (cid:3) References [Aud91] M. Audin,
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Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
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