Displaying the cohomology of toric line bundles
Klaus Altmann, David Ploog

TL;DR
This paper provides a concise and direct proof of a simplified formula for computing the cohomology of toric line bundles, replacing complex topological methods with a set-theoretic difference approach.
Contribution
It offers a straightforward proof of a known cohomology formula for toric line bundles, simplifying the computational process.
Findings
Simplified cohomology calculation using set-theoretic difference.
Provides a direct proof of the formula from prior work.
Enhances understanding of toric line bundle cohomology.
Abstract
There is a standard method to calculate the cohomology of torus-invariant sheaves on a toric variety via the simplicial cohomology of associated subsets of the space of 1-parameter subgroups of the torus. For a line bundle represented by a formal difference of polyhedra in the character space , [ABKW18] contains a simpler formula for the cohomology of , replacing by the set-theoretic difference . Here, we provide a short and direct proof of this formula.
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Displaying the cohomology of toric line bundles
Klaus Altmann
Institut für Mathematik, FU Berlin, Arnimallee 3, 14195 Berlin, Germany
and
David Ploog
Fachbereich Mathematik,Welfengarten 1, 30167 Hannover, Germany
Abstract.
There is a standard method to calculate the cohomology of torus-invariant sheaves on a toric variety via the simplicial cohomology of associated subsets of the space of 1-parameter subgroups of the torus. For a line bundle represented by a formal difference of polyhedra in the character space , [ABKW] contains a simpler formula for the cohomology of , replacing by the set-theoretic difference . Here, we provide a short and direct proof of this formula.
MSC 2010: 14M25; 52B20, 14C20, 14F05.
Key words: toric variety, Cartier divisor, line bundle, sheaf cohomology, lattice, polytope.
1. Introduction
1.1. Toric varieties
Let be an algebraically closed field. We consider the algebraic torus where is a free abelian group of finite rank and denotes the semigroup ring of . Then can be recovered from as its character group. The dual is the group of 1-parameter subgroups in . By definition, we have a natural perfect pairing
[TABLE]
The theory of toric varieties deals with partial compactifications such that the group law of extends to an algebraic action of on the -variety . By a cone in , we mean a subset which is finitely generated and convex () and rational (). Each affine toric variety is obtained from a pointed cone , i.e. , via
[TABLE]
Each face provides a natural open embedding . In particular, corresponds to the open subset . Gluing affine toric varieties along such open subsets leads to general (non-affine) toric varieties determined by a polyhedral fan in :
[TABLE]
A fan is a finite set of pointed cones in closed under taking faces and such that for any two cones . Then the affine varieties turn into affine, open charts .
The main advantage of toric varieties is that the -action provides a fine -grading on all algebraic structures functorially associated to , allowing a combinatorial description. Examples are the cohomology groups of -invariant sheaves or the modules describing the infinitesimal deformations and obstructions of toric varieties. The latest general reference for the theory of toric varieties is the book [CLS11].
1.2. Cohomology of -invariant Weil divisors
Let us assume that the support of a given fan is a full-dimensional, not necessarily pointed cone, e.g. . We will identify the rays, i.e. the one-dimensional cones , with their primitive generators . They give rise to the one-codimensional -orbits , and their closures
[TABLE]
are precisely the -invariant (or “toric”) prime divisors. Hence, an arbitrary toric Weil divisor has the form
[TABLE]
One of the salient features of toric varieties is that these special divisors generate the full class group. This is reflected by the famous exact sequence
[TABLE]
whose first map is the dual of the natural map , , sending standard basis vectors to primitive ray generators. During the last five decades, it has been one of the basic results in toric geometry to express the -eigenspaces of the sheaf cohomology of on as the singular cohomology of certain subsets of the vector space , where . In [Dem70, Prop. 2.6], [KKMSD73, I.3], [Dan78, §7], [Oda88, (2.2)], [Ful93, (3.5)], or in [CLS11, (9.1)] one defines, for all characters these sets as
[TABLE]
The basic formula allured to above is, with coefficients on the right-hand side
[TABLE]
Recall that the -st reduced cohomology is \widetilde{\operatorname{H}}^{-1}(V)=\left\{\begin{array}[]{ll}0&\mbox{if }V\neq\emptyset\\ k&\mbox{if }V=\emptyset\end{array}\right..
1.3. Cohomology of -invariant Cartier divisors
While the previous formula is very useful, its main ingredient is the rather technically defined subset . In [ABKW, III.3], a Fourier transformation argument was used to replace these subsets by much more natural subsets of the dual vector space , at least for Cartier divisors in quasi-projective toric varieties. This makes it possible to literally display the cohomology of the line bundles representing the divisors:
In this setting, a nef Cartier divisor is represented by a convex lattice polyhedron in , see Subsection (2.3). The lattice points of the polyhedron correspond to global sections of , yielding .
An arbitrary Cartier divisor is a difference of two nef Cartier divisors, and then these two parts can be represented by convex lattice polyhedra respectively. The Minkowski sums among those polytopes correspond to the sums of the associated divisors, i.e. to the tensor products of their sheaves of sections. The divisor itself corresponds to the formal difference in the Grothendieck group of the semigroup of polyhedra with Minkowski addition, see Section 2 for more details. Now, the result of Theorem III.6 in [ABKW] is
Theorem**.**
\;\operatorname{H}^{i}\!\big{(}X,{\mathcal{O}}_{X}(D)\big{)}(m)=\widetilde{\operatorname{H}}^{i-1}\!\big{(}\Delta^{-}\setminus(\Delta^{+}-m)\big{)}.
The goal of the present paper is to provide a direct, straight-forward proof of this result (subsequently called Theorem 2), avoiding the application of the traditional formula and of the Fourier transformation argument linking to . This might become especially useful when trying to generalize Theorem 2 to the situation of Okounkov bodies or to -varieties of higher complexity. In both situations, the “-side” of the story does not exist or is at least not easily accessible.
1.4. An easy example
The first Hirzebruch surface \mathbb{F}_{1}=\mathbb{P}\big{(}{\mathcal{O}}_{\mathbb{P}^{1}}\oplus{\mathcal{O}}_{\mathbb{P}^{1}}(1)\big{)} is obtained by blowing up in one (torus-invariant) point. The fan of looks like
[TABLE]
The nef cone in is freely generated by the line bundles associated to the two polytopes
[TABLE]
See Subsection (2.3) for details on this correspondence, and Section 4 for the actual line bundles. Ample line bundles arise from tensor products of both generators; the two simplest examples are represented by the Minkowski sums
[TABLE]
Now, the pictures
[TABLE]
show that there are exactly three degrees such that the translates are contained in , i.e. that – but only the shift by divides into two connected components. Hence \operatorname{H}^{0}\!\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(2B-A)\big{)} is three-dimensional, supported in the degrees , and
[TABLE]
is one-dimensional, supported in the single degree . Similarily, one can spot the unique integral shift of into the interior of leading to a one-dimensional \operatorname{H}^{2}\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(A-4B)\big{)} supported in :
[TABLE]
Since , one can easily visualize Serre duality. For instance, the one-dimensional \operatorname{H}^{1}\!\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(A-2B)\big{)} leads to the one-dimensional \operatorname{H}^{1}\!\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(-2A)\big{)}^{\!\vee}, and the latter can be seen from Theorem 2 since the polytope is an interval of length 2 and thus has a unique interior lattice point.
1.5. An even easier example with non-trivial tail cone
The blowing up of the origin in is represented by the natural map of fans
[TABLE]
The negative of the exceptional divisor is ample, and the associated sheaf is represented by the polyhedron (see Subsection (2.1) for tail cones)
[TABLE]
Now, Theorem 2 and this picture
[TABLE]
show that \operatorname{H}^{1}\!\big{(}\widetilde{\mathbb{A}}^{2},\,{\mathcal{O}}_{\widetilde{\mathbb{A}}^{2}}(0E-(-2E))\big{)}=\operatorname{H}^{1}\!\big{(}\widetilde{\mathbb{A}}^{2},\,{\mathcal{O}}_{\widetilde{\mathbb{A}}^{2}}(2E)\big{)} is one-dimensional.
2. Cartier divisors and polyhedra
2.1. Lattice polytopes with prescribed normal fan
If is a lattice polyhedron, then we define its tail cone (also called its recession cone) as
[TABLE]
Now, we turn the tables. Once and for all, we fix a pointed cone and take it as the prescribed tail cone of all our polyhedra, i.e. we consider the set
[TABLE]
The most important case is , as is the set of compact lattice polyhedra. In any case, is a semigroup under Minkowski addition with neutral element .
Definition 1**.**
Let be a lattice polyhedron with . A fan in is called compatible with if it is a subdivision of the normal fan , i.e. if , and if the concave, piecewise linear function
[TABLE]
is linear on the cones of .
We turn the tables once more: Fixing a fan with support , the subset
[TABLE]
is a finitely generated subsemigroup. Let us now fix a polyhedron , i.e. a compatible pair . By definition, the function is linear on every cone . If is full-dimensional, this function is realized by a unique element
[TABLE]
This immediately implies that is a vertex and . Actually,
[TABLE]
The functions , are obviously additive. However, unless , some of them will coincide. Finally, if is not full-dimensional, then the lattice points still exist, but they are no longer uniquely determined. Instead, one obtains a well-defined function .
2.2. The Grothendieck groups
Since we have fixed the tail cone, the semigroup is cancellative. That is, can be canonically embedded into the group
[TABLE]
of all formal differences. Analogously, the subsemigroup leads to a finitely generated subgroup
[TABLE]
The additive function on extends to , however with
[TABLE]
now understood as the virtual vertices of the formal difference . Recall that in the quasi-projective case of , the set can also be described as
[TABLE]
In particular, in this case, the elements of can be written as by just using the “ample polyhedron” determining and some to form the negative part.
2.3. The nef divisor associated to a polyhedron
Assume that is a polyhedron with tail cone and that is a fan in compatible with . Consider again the toric variety with its open torus embedding . Denoting by the monomial associated to , we define an invertible sheaf on via
[TABLE]
where the gluing condition for on is obtained from
[TABLE]
In particular, we obtain
[TABLE]
The intersection of these groups exhibits as a -basis of \operatorname{H}^{0}\!\big{(}X,\,{\mathcal{O}}_{X}(\Delta)\big{)}. Besides, the assignment induces the well-known semigroup isomorphism
[TABLE]
In fact, this even becomes a functor when defining morphisms among lattice polyhedra as injections. An integral shift of by on the left-hand side leads to an isomorphic sheaf whose -action is shifted by the character :
[TABLE]
This isomorphism extends to the respective Grothendieck groups. Moreover, in the quasi-projective case, after modding out integral translations on the left-hand side and linear equivalence, on the right, this induces the identification
[TABLE]
where the positive cone of globally generated sheaves maps to the nef cone within \operatorname{Pic}\big{(}\mathbb{T}\mathbb{V}(\Sigma)\big{)}. See, e.g., [Ful93, (3.4)] or [CLS11, Ch. 6] for more details.
3. Comparing the cohomology theories
3.1. The comparison theorem
Let be two polyhedra with tail cone and let be a fan in compatible with both . This gives rise to the -equivariant, invertible sheaf on
[TABLE]
Theorem 2**.**
One has \;\operatorname{H}^{i}\!\big{(}X,{\mathcal{L}}\big{)}(m)=\widetilde{\operatorname{H}}{}^{i-1}\big{(}\Delta^{-}\setminus(\Delta^{+}-m)\big{)} for all , .
The rest of this section contains the proof of this statement. Note that by our remark at the end of Subsection (2.3), we may assume . Now, everything follows from the comparison of two Čech complexes – and this will be done in the next subsections.
3.2. The Čech complex for
We refer to [Har77, III.4] for the Čech complex of a sheaf on a topological space. On we take the open, affine covering
[TABLE]
which is closed under intersections, i.e. . We fix an ordering among the participating polyhedral cones and for each , we define
[TABLE]
together with the usual differentials . The cohomology of this complex is denoted by , and it maps isomorphically onto by [Har77, Th. III.4.5], since is coherent. Now, the intersection
[TABLE]
is a cone in , and the -graded -vector space of local sections on equals
[TABLE]
by Subsection (2.3). Since we are interested in the degree , the upshot is
[TABLE]
3.3. A covering for
To calculate the topological cohomology of , we will construct a good covering consisting of contractible subsets. For this recall that, for each cone , we have established the equality in Subsection (2.1). This gives rise to the definition of the open subsets
[TABLE]
Now, we are going to prove two claims about these subsets.
3.3.1. Claim:
Let . Then there is an element with . This has to be contained in some cone , hence . Hence , thus .
[TABLE]
3.3.2. Claim:
If , then is a deformation retract
If there is some s\in S(\sigma)\subseteq\big{(}v^{-}_{\sigma}+\sigma^{\scriptscriptstyle\vee}\big{)}\setminus\big{(}v^{+}_{\sigma}+\sigma^{\scriptscriptstyle\vee}\big{)}, then there has to be an element such that Since it follows that for all elements of the line segment . This implies that , and altogether this shows that is star-shaped with respect to .
3.4. Conclusion of the proof
By these claims, is a good covering of in the sense of [BT82, §5]: First, each set is open in and acyclic, i.e. . This holds because is contractible. Second, these properties also hold for arbitrary intersections, which is obvious here, due to .
We build a Čech complex similarily as in Subsection (3.2), but now from relative singular [math]-chains with coefficients
[TABLE]
By [BT82, Prop. 10.6], its -th cohomology is the relative singular cohomology, which in turn by contractibility of , is isomorphic to the reduced cohomology:
[TABLE]
On the other hand, we obtain from Step 3.3.2 that
[TABLE]
Hence the complexes and from Subsection (3.2) coincide, and the claim of Theorem 2 is proven.
3.5. Using higher inverse limits
In Subsection (3.4), we have used Čech complexes mostly for historical reasons – it was the main tool in all proofs of the standard toric cohomology formulas so far. However, Čech cohomology is just the down-to-earth description of the following more abstract, but less technical point of view:
The global section functor on is the composition of the local section functors on followed by the inverse limit, i.e.
[TABLE]
Now, the Grothendieck spectral sequence yields
[TABLE]
and, by the same method on the topological side,
[TABLE]
Since , vanishing unless , we obtain the result again.
4. Application: full exceptional sequences of nef line bundles
Theorem 2 can be used to quickly check whether a sequence of nef toric line bundles is exceptional. We illustrate this on the Hirzebruch surface \mathbb{F}_{1}=\mathbb{P}\big{(}{\mathcal{O}}_{\mathbb{P}^{1}}\oplus{\mathcal{O}}_{\mathbb{P}^{1}}(1)\big{)} of Subsection (1.4).
Let be the projection as a ruled surface, and be the blowing-down map. The sheaves with are the toric line bundles on . For , each is nef and corresponds, by Subsection (2.3), to a lattice polytope in . In the notation of Subsection (1.4), and , so that . The ample line bundle corresponds to , i.e. .
We are looking for a sequence of polytopes corresponding to a full exceptional sequence of line bundles for the bounded derived category , i.e. for , and the line bundles generate ; see [Huy06, §1.4]. In fact, the following two sequences of polytopes
and
give rise to exceptional sequences of line bundles, as is easily checked with Theorem 2. Indeed, in Subsection (1.4), we have already seen that and together cannot be part of an exceptional sequence. Moreover, these two sequences are full, i.e. the respective line bundles generate . This, too, can be seen combinatorially: we use the general fact that, if is an ample line bundle on a smooth projective variety of dimension , then is a strong generator of by [Orl09, Thm. 4.1]. In our example, we need to generate . We will show how to generate using the first exceptional sequence, . This is achieved by the following polytopal resolution of
[TABLE]
In each polytope, the red circle indicates the origin, thus fixing the position of the polyhedra in . Morphisms between polytopes are all possible inclusions, with signs coming from the natural inclusion-exclusion rule. The polytopal resolutions induces exact sequences of direct sums of line bundles by applying the functor , of Subsection (2.3).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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