Displaying the cohomology of toric line bundles

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 $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $\Delta^- \setminus \Delta^+$. Here, we provide a short and direct proof of this formula.

Equations

$$⟨-,-⟩:M\times N\to \mathbb{Z}.$$

$$\mathbb{TV}(\sigma ,N)=\mathbb{TV}(\sigma ):=\mathrm{Spec}k[{\sigma }^{\vee }\cap M]\text{mbox}with{\sigma }^{\vee }:=\{m\in M_{\mathbb{R}}|⟨m,\sigma ⟩\ge 0\}.$$

$$X=\mathbb{TV}(\Sigma ):=\underset{\begin{array}{c}⟶\\ \sigma \in \Sigma \end{array}}{\lim }\mathbb{TV}(\sigma ).$$

$$D_{\rho }:=\overline{\mathrm{orb}(\rho )}$$

$$D=\sum _{\rho \in \Sigma (1)}{\lambda }_{\rho }{D}_{\rho }({\lambda }_{\rho }\in \mathbb{Z}).$$

$$0\to M\to {\mathbb{Z}}^{\Sigma (1)}\to \mathrm{Cl}(X)\to 0$$

$$V_{D,m}:=\bigcup _{\sigma \in \Sigma }\mathrm{conv}\{\rho \in \sigma (1)|⟨m,\rho ⟩<-{\lambda }_{\rho }\}\subseteq N_{\mathbb{R}}\setminus \{0\}.$$

$$\operatorname{H}^{i}\!\big{(}X,{\mathcal{O}}_{X}(D)\big{)}(m)=\widetilde{\operatorname{H}}^{i-1}(V_{D,m}).$$

$$\Sigma ({\mathbb{F}}_1)$$

$$A$$

$$A+B$$

$$2B$$

$$\operatorname{H}^{1}\!\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(A-2B)\big{)}=\widetilde{\operatorname{H}}^{0}\!\big{(}2B\setminus(A+m_{1})\big{)}$$

$$4B$$

$$⟶$$

$$-E$$

$$-2E$$

$$\mathrm{tail}(\Delta ):=\{m\in M_{\mathbb{R}}|m+\Delta \subseteq \Delta \}.$$

$${\mathrm{Pol}}_{\delta }^{+}:=\{\text{mbox}latticepolyhedra\Delta \subseteq M_{\mathbb{R}}|\mathrm{tail}(\Delta )=\delta \}.$$

$${\delta }^{\vee }\to \mathbb{R},n\mapsto \min ⟨\Delta ,n⟩$$

$${\mathrm{Pol}}_{\delta }^{+}(\Sigma ):=\{\text{mbox}latticepolyhedra\Delta \subseteq M_{\mathbb{R}}|\Delta \text{mbox}iscompatiblewith\Sigma \}\subseteq {\mathrm{Pol}}_{\delta }^{+}$$

$$v_{\sigma}=v_{\sigma}^{\Delta}\in M,\hskip 3.99994pt\text{i.e., for all $n\in\sigma$, we have}\hskip 3.99994pt\min\langle\Delta,n\rangle=\big{\langle}v_{\sigma},\,n\big{\rangle}.$$

$$\Delta =\mathrm{conv}\{v_{\sigma }|\sigma \in \Sigma (\text{mbox}\mathrm{top})\}+\delta .$$

$${\mathrm{Pol}}_{\delta }:=\{{\Delta }^{+}-{\Delta }^{-}|{\Delta }^{\pm }\in {\mathrm{Pol}}_{\delta }^{+}\}$$

$${\mathrm{Pol}}_{\delta }(\Sigma )=\{{\Delta }^{+}-{\Delta }^{-}|{\Delta }^{\pm }\in {\mathrm{Pol}}_{\delta }^{+}(\Sigma )\}\subseteq {\mathrm{Pol}}_{\delta }.$$

$$v_{\sigma }^{\Delta }:=v_{\sigma }^{+}-v_{\sigma }^{-}:=v_{\sigma }^{{\Delta }^{+}}-v_{\sigma }^{{\Delta }^{-}}$$

$${\mathrm{Pol}}_{\delta }^{+}(\Sigma )=\{{\Delta }^{\prime }\in {\mathrm{Pol}}_{\delta }^{+}|\exists {\Delta }^{\prime \prime }\in {\mathrm{Pol}}_{\delta }^{+}:{\Delta }^{\prime }+{\Delta }^{\prime \prime }=N\cdot \Delta \text{mbox}forsomeN\in \mathbb{N}\}.$$

$${\mathcal{O}}_X(\Delta ){|}_{U_{\sigma }}:={\chi }^{v_{\sigma }}\cdot {\mathcal{O}}_{U_{\sigma }}\subseteq j_{\ast }{\mathcal{O}}_T=k[M]=k[T]\subseteq k(T)=k(X)$$

$$v_{\sigma }-v_{{\sigma }^{\prime }}\in (\sigma \cap {\sigma }^{\prime }{)}^{\perp },\text{mbox}i.e.{\chi }^{v_{\sigma }}/{\chi }^{v_{{\sigma }^{\prime }}}\in {\mathcal{O}}_X(U_{\sigma }\cap U_{{\sigma }^{\prime }}{)}^{\ast }.$$

$$\operatorname{H}^{0}\!\big{(}U_{\sigma},\,{\mathcal{O}}_{X}(\Delta)\big{)}=\chi^{v_{\sigma}}\cdot k[\sigma^{\scriptscriptstyle\vee}\cap M]=k[{v_{\sigma}}+(\sigma^{\scriptscriptstyle\vee}\cap M)].$$

Full text

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 $k$ be an algebraically closed field. We consider the algebraic torus $T=\operatorname{Spec}k[M]$ where $M$ is a free abelian group of finite rank and $k[M]$ denotes the semigroup ring of $M$. Then $M$ can be recovered from $T$ as its character group. The dual $N=\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$ is the group of 1-parameter subgroups in $T$. By definition, we have a natural perfect pairing

[TABLE]

The theory of toric varieties deals with partial compactifications $j\colon T\hookrightarrow X$ such that the group law of $T$ extends to an algebraic action of $T$ on the $k$-variety $X$. By a cone in $N_{\mathbb{R}}\mathrel{\mathop{:}}=N\otimes_{Z}\mathbb{R}$, we mean a subset $\sigma\subseteq N_{\mathbb{R}}$ which is finitely generated and convex ($\sigma=\sum_{i=1}^{r}\mathbb{R}_{\geq 0}\cdot v_{i}$) and rational ($v_{i}\in N$). Each affine toric variety is obtained from a pointed cone $\sigma\subseteq N_{\mathbb{R}}$, i.e. $\sigma\cap(-\sigma)=\{0\}$, via

[TABLE]

Each face $\tau\leq\sigma$ provides a natural open embedding $\mathbb{T}\mathbb{V}(\tau)\hookrightarrow\mathbb{T}\mathbb{V}(\sigma)$. In particular, $\{0\}\leq\sigma$ corresponds to the open subset $T=\mathbb{T}\mathbb{V}(0)\hookrightarrow\mathbb{T}\mathbb{V}(\sigma)$. Gluing affine toric varieties along such open subsets leads to general (non-affine) toric varieties determined by a polyhedral fan $\Sigma$ in $N_{\mathbb{R}}$:

[TABLE]

A fan $\Sigma$ is a finite set of pointed cones in $N_{\mathbb{R}}$ closed under taking faces and such that $\sigma\cap\sigma^{\prime}\leq\sigma,\sigma^{\prime}$ for any two cones $\sigma,\sigma^{\prime}\in\Sigma$. Then the affine varieties $\mathbb{T}\mathbb{V}(\sigma)$ turn into affine, open charts $U_{\sigma}\subseteq X=\mathbb{T}\mathbb{V}(\Sigma)$.

The main advantage of toric varieties is that the $T$-action provides a fine $M$-grading on all algebraic structures functorially associated to $\mathbb{T}\mathbb{V}(\Sigma)$, allowing a combinatorial description. Examples are the cohomology groups of $T$-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 $T$-invariant Weil divisors

Let us assume that the support $|\Sigma|\mathrel{\mathop{:}}=\bigcup_{\sigma\in\Sigma}\sigma$ of a given fan $\Sigma$ is a full-dimensional, not necessarily pointed cone, e.g. $|\Sigma|=N_{\mathbb{R}}$. We will identify the rays, i.e. the one-dimensional cones $\rho\in\Sigma(1)$, with their primitive generators $\rho\in N$. They give rise to the one-codimensional $T$-orbits $\operatorname{orb}(\rho)\mathrel{\mathop{:}}=\operatorname{Spec}k[\rho^{\bot}\cap M]\subseteq\mathbb{T}\mathbb{V}(\rho)\subseteq X$, and their closures

[TABLE]

are precisely the $T$-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 $D_{\rho}$ generate the full class group. This is reflected by the famous exact sequence

[TABLE]

whose first map is the dual of the natural map $\mathbb{Z}^{\Sigma(1)}\to N$, $e_{\rho}\mapsto\rho\in N$, 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 $M$-eigenspaces of the sheaf cohomology of ${\mathcal{O}}_{X}(D)$ on $X=\mathbb{T}\mathbb{V}(\Sigma)$ as the singular cohomology of certain subsets $V_{D,m}$ of the vector space $N_{\mathbb{R}}$, where $m\in M$. 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 $m\in M$ these sets as

[TABLE]

The basic formula allured to above is, with coefficients $k$ on the right-hand side

[TABLE]

Recall that the $(-1)$-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 $T$-invariant Cartier divisors

While the previous formula is very useful, its main ingredient is the rather technically defined subset $V_{D,m}\subseteq N_{\mathbb{R}}$. In [ABKW, III.3], a Fourier transformation argument was used to replace these subsets by much more natural subsets of the dual vector space $M_{\mathbb{R}}$, 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 $D$ is represented by a convex lattice polyhedron $\Delta$ in $M_{\mathbb{R}}$, see Subsection (2.3). The lattice points of the polyhedron correspond to global sections of ${\mathcal{O}}_{X}(D)$, yielding $k^{\Delta\cap M}=\operatorname{H}^{0}(X,{\mathcal{O}}_{X}(D))$.

An arbitrary Cartier divisor is a difference $D=D^{+}-D^{-}$ of two nef Cartier divisors, and then these two parts can be represented by convex lattice polyhedra $\Delta^{+},\Delta^{-}\subseteq M_{\mathbb{R}},$ 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 $D$ itself corresponds to the formal difference $\Delta^{+}-\Delta^{-}$ 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 $V_{D,m}\subset N_{\mathbb{R}}$ to $M_{\mathbb{R}}$. This might become especially useful when trying to generalize Theorem 2 to the situation of Okounkov bodies or to $T$-varieties of higher complexity. In both situations, the “$N$-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 $\mathbb{P}^{2}$ in one (torus-invariant) point. The fan of $\mathbb{F}_{1}$ looks like

[TABLE]

The nef cone in $\operatorname{Cl}(\mathbb{F}_{1})\cong\mathbb{Z}^{2}$ 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 $m_{i}\in M=\mathbb{Z}^{2}$ such that the translates $A+m_{i}$ are contained in $2B$, i.e. that $(A+m_{i})\setminus(2B)=\emptyset$ – but only the shift by $m_{1}$ divides $(2B)\setminus(A+m_{1})$ 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 $m_{1},m_{2},m_{3}\in M$, and

[TABLE]

is one-dimensional, supported in the single degree $-m_{1}\in M$. Similarily, one can spot the unique integral shift of $A$ into the interior of $4B$ leading to a one-dimensional \operatorname{H}^{2}\big{(}\mathbb{F}_{1},\,{\mathcal{O}}_{\mathbb{F}_{1}}(A-4B)\big{)} supported in $-m_{4}\in M$:

[TABLE]

Since $-K_{\mathbb{F}_{1}}=A+2B$, 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 $2A$ 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 $\widetilde{\mathbb{A}}^{2}$ of the origin in $\mathbb{A}^{2}$ is represented by the natural map of fans

[TABLE]

The negative of the exceptional divisor $E\subseteq\widetilde{\mathbb{A}}^{2}$ is ample, and the associated sheaf ${\mathcal{O}}(-E)$ 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 $\Delta\subseteq M_{\mathbb{R}}$ 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 $\delta\subseteq M_{\mathbb{R}}$ and take it as the prescribed tail cone of all our polyhedra, i.e. we consider the set

[TABLE]

The most important case is $\delta=0$, as $\operatorname{Pol}^{+}_{0}$ is the set of compact lattice polyhedra. In any case, $\operatorname{Pol}^{+}_{\,\delta}$ is a semigroup under Minkowski addition with neutral element $\delta$.

Definition 1**.**

Let $\Delta\subseteq M_{\mathbb{R}}$ be a lattice polyhedron with $\operatorname{tail}(\Delta)=\delta$. A fan $\Sigma$ in $N_{\mathbb{R}}$ is called compatible with $\Delta$ if it is a subdivision of the normal fan ${\mathcal{N}}(\Delta)$, i.e. if $|\Sigma|=\delta^{\scriptscriptstyle\vee}\subseteq N_{\mathbb{R}}$, and if the concave, piecewise linear function

[TABLE]

is linear on the cones of $\Sigma$.

We turn the tables once more: Fixing a fan $\Sigma$ with support $|\Sigma|=\delta^{\scriptscriptstyle\vee}$, the subset

[TABLE]

is a finitely generated subsemigroup. Let us now fix a polyhedron $\Delta\in\operatorname{Pol}^{+}_{\delta}(\Sigma)$, i.e. a compatible pair $(\Delta,\Sigma)$. By definition, the function $\min\langle\Delta,\,{\scriptscriptstyle\bullet}\rangle$ is linear on every cone $\sigma\in\Sigma$. If $\sigma$ is full-dimensional, this function is realized by a unique element

[TABLE]

This immediately implies that $v_{\sigma}\in\Delta$ is a vertex and $v_{\sigma}+\sigma^{\scriptscriptstyle\vee}=\Delta+\sigma^{\scriptscriptstyle\vee}$. Actually,

[TABLE]

The functions $\,v^{\scriptscriptstyle\bullet}_{\sigma}\colon\operatorname{Pol}^{+}_{\delta}(\Sigma)\to M$, $\,\Delta\mapsto v_{\sigma}^{\Delta}$ are obviously additive. However, unless ${\mathcal{N}}(\Delta)=\Sigma$, some of them will coincide. Finally, if $\sigma\in\Sigma$ is not full-dimensional, then the lattice points $v_{\sigma}$ still exist, but they are no longer uniquely determined. Instead, one obtains a well-defined function $v^{\scriptscriptstyle\bullet}_{\sigma}\colon\operatorname{Pol}^{+}_{\delta}(\Sigma)\to M/(\sigma^{\bot}\cap M)$.

2.2. The Grothendieck groups

Since we have fixed the tail cone, the semigroup $\operatorname{Pol}^{+}_{\,\delta}$ is cancellative. That is, $\operatorname{Pol}^{+}_{\,\delta}$ can be canonically embedded into the group

[TABLE]

of all formal differences. Analogously, the subsemigroup $\operatorname{Pol}^{+}_{\delta}(\Sigma)\subseteq\operatorname{Pol}^{+}_{\,\delta}$ leads to a finitely generated subgroup

[TABLE]

The additive function $v^{\scriptscriptstyle\bullet}_{\sigma}$ on $\operatorname{Pol}^{+}_{\delta}(\Sigma)$ extends to $\operatorname{Pol}_{\delta}(\Sigma)$, however with

[TABLE]

now understood as the virtual vertices of the formal difference $\Delta^{+}-\Delta^{-}$. Recall that in the quasi-projective case of $\Sigma={\mathcal{N}}(\Delta)$, the set $\operatorname{Pol}^{+}_{\delta}(\Sigma)$ can also be described as

[TABLE]

In particular, in this case, the elements of $\operatorname{Pol}_{\delta}(\Sigma)$ can be written as $\Delta^{+}-N\cdot\Delta$ by just using the “ample polyhedron” $\Delta$ determining $\Sigma$ and some $N\gg 0$ to form the negative part.

2.3. The nef divisor associated to a polyhedron

Assume that $\Delta\subseteq M_{\mathbb{R}}$ is a polyhedron with tail cone $\delta$ and that $\Sigma$ is a fan in $N_{\mathbb{R}}$ compatible with $\Delta$. Consider again the toric variety $X=\mathbb{T}\mathbb{V}(\Sigma)$ with its open torus embedding $j\colon T\hookrightarrow X$. Denoting by $\chi^{m}\in k[M]$ the monomial associated to $m\in M$, we define an invertible sheaf ${\mathcal{O}}_{X}(\Delta)$ on $X$ via

[TABLE]

where the gluing condition for ${\mathcal{O}}_{X}(\Delta)$ on $U_{\sigma}\cap U_{\sigma^{\prime}}=U_{\sigma\cap\sigma^{\prime}}$ is obtained from

[TABLE]

In particular, we obtain

[TABLE]

The intersection of these groups exhibits $\Delta\cap M$ as a $k$-basis of \operatorname{H}^{0}\!\big{(}X,\,{\mathcal{O}}_{X}(\Delta)\big{)}. Besides, the assignment $\Delta\mapsto{\mathcal{O}}_{X}(\Delta)$ 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 $\Delta\in\operatorname{Pol}^{+}_{\delta}(\Sigma)$ by $m\in M$ on the left-hand side leads to an isomorphic sheaf whose $T$-action is shifted by the character $m$:

[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 $\operatorname{Pol}^{+}_{\delta}(\Sigma)$ 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 $\Delta^{+},\Delta^{-}\subseteq M_{\mathbb{R}}$ be two polyhedra with tail cone $\delta$ and let $\Sigma$ be a fan in $N_{\mathbb{R}}$ compatible with both $\Delta^{\pm}$. This gives rise to the $T$-equivariant, invertible sheaf on $X=\mathbb{T}\mathbb{V}(\Sigma)$

[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 $m\in M$, $i\in\mathbb{N}$.

The rest of this section contains the proof of this statement. Note that by our remark ${\mathcal{O}}_{X}(\Delta+m)=\chi^{m}\cdot{\mathcal{O}}_{X}(\Delta)$ at the end of Subsection (2.3), we may assume $m=0$. 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 ${\mathcal{O}}_{X}(\Delta^{+}-\Delta^{-})$

We refer to [Har77, III.4] for the Čech complex of a sheaf ${\mathcal{L}}$ on a topological space. On $X=\mathbb{T}\mathbb{V}(\Sigma)$ we take the open, affine covering

[TABLE]

which is closed under intersections, i.e. $U_{\sigma}\cap U_{\sigma^{\prime}}=U_{\sigma\cap\sigma^{\prime}}$. We fix an ordering among the participating polyhedral cones and for each $p\in\mathbb{N}$, we define

[TABLE]

together with the usual differentials $d\colon C^{p}\to C^{p+1}$. The cohomology of this complex is denoted by $\check{\operatorname{H}}^{p}({\mathcal{U}},{\mathcal{L}})$, and it maps isomorphically onto $\operatorname{H}^{p}(X,{\mathcal{L}})$ by [Har77, Th. III.4.5], since ${\mathcal{L}}$ is coherent. Now, the intersection

[TABLE]

is a cone in $\Sigma$, and the $M$-graded $k$-vector space of local sections on $U_{\sigma}$ equals

[TABLE]

by Subsection (2.3). Since we are interested in the degree $m=0$, the upshot is

[TABLE]

3.3. A covering for $\Delta^{-}\setminus\Delta^{+}$

To calculate the topological cohomology of $\Delta^{-}\setminus\Delta^{+}$, we will construct a good covering consisting of contractible subsets. For this recall that, for each cone $\sigma\in\Sigma$, we have established the equality $v^{+}_{\sigma}+\sigma^{\scriptscriptstyle\vee}=\Delta^{+}+\sigma^{\scriptscriptstyle\vee}$ 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:

$\;\bigcup_{\sigma\in\Sigma}S(\sigma)=\Delta^{-}\setminus\Delta^{+}$

Let $p\in\Delta^{-}\setminus\Delta^{+}$. Then there is an element $n\in\delta^{\scriptscriptstyle\vee}=|\Sigma|$ with $\langle p,n\rangle<\min\langle\Delta^{+},n\rangle$. This $n$ has to be contained in some cone $\sigma\in\Sigma$, hence $\min\langle\Delta^{+},n\rangle=\langle v^{+}_{\sigma},\,n\rangle$. Hence $p\notin v^{+}_{\sigma}+\sigma^{\scriptscriptstyle\vee}$, thus $p\in S(\sigma)$.

[TABLE]

3.3.2. Claim:

If $S(\sigma)\neq\emptyset$, then $\{v^{-}_{\sigma}\}\subseteq S(\sigma)$ 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 $n\in\sigma$ such that $\langle s,n\rangle<\langle v^{+}_{\sigma},n\rangle.$ Since $\langle v^{-}_{\sigma},n\rangle\leq\langle s,n\rangle$ it follows that $\langle s^{\prime},n\rangle<\langle v^{+}_{\sigma},n\rangle$ for all elements $s^{\prime}$ of the line segment $[{v^{-}_{\sigma},s}]$. This implies that $[{v^{-}_{\sigma},s}]\subseteq S(\sigma)$, and altogether this shows that $S(\sigma)$ is star-shaped with respect to $v^{-}_{\sigma}$.

3.4. Conclusion of the proof

By these claims, ${\mathcal{S}}\mathrel{\mathop{:}}=\{S(\sigma)\}_{\sigma\in\Sigma}$ is a good covering of $\Delta^{-}\setminus\Delta^{+}$ in the sense of [BT82, §5]: First, each set $S(\sigma)$ is open in $\Delta^{-}\setminus\Delta^{+}$ and acyclic, i.e. $\smash{\widetilde{\operatorname{H}}^{*}}(S(\sigma))=0$. This holds because $S(\sigma)$ is contractible. Second, these properties also hold for arbitrary intersections, which is obvious here, due to $S(\sigma)\cap S(\sigma^{\prime})=S(\sigma\cap\sigma^{\prime})$.

We build a Čech complex similarily as in Subsection (3.2), but now from relative singular [math]-chains with coefficients $k$

[TABLE]

By [BT82, Prop. 10.6], its $p$-th cohomology is the relative singular cohomology, which in turn by contractibility of $\Delta^{-}$, is isomorphic to the reduced cohomology:

[TABLE]

On the other hand, we obtain from Step 3.3.2 that

[TABLE]

Hence the complexes $C^{\scriptscriptstyle\bullet}({\mathcal{S}},\Delta^{-}\setminus\Delta^{+})$ and $C^{\scriptscriptstyle\bullet}({\mathcal{U}},{\mathcal{L}})$ 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 $X=\bigcup_{\sigma\in\Sigma}U_{\sigma}$ is the composition of the local section functors on $U_{\sigma}$ 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 $E^{pq}_{2}(\mbox{\rm alg})=E^{pq}_{2}(\mbox{\rm top})$, vanishing unless $q=0$, 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 $\pi_{1}\colon\mathbb{F}_{1}\to\mathbb{P}^{1}$ be the projection as a ruled surface, and $\pi_{2}\colon\mathbb{F}_{1}\to\mathbb{P}^{2}$ be the blowing-down map. The sheaves ${\mathcal{O}}(a,b)\mathrel{\mathop{:}}=\pi_{1}^{*}({\mathcal{O}}_{\mathbb{P}^{1}}(a))\otimes_{{\mathcal{O}}_{\mathbb{F}_{1}}}\pi_{2}^{*}({\mathcal{O}}_{\mathbb{P}^{2}}(b))$ with $a,b\in\mathbb{Z}$ are the toric line bundles on $\mathbb{F}_{1}$. For $a,b\geq 0$, each ${\mathcal{O}}(a,b)$ is nef and corresponds, by Subsection (2.3), to a lattice polytope $P(a,b)$ in $M_{\mathbb{R}}$. In the notation of Subsection (1.4), $A=P(1,0)$ and $B=P(0,1)$, so that $P(a,b)=aA+bB$. The ample line bundle ${\mathcal{O}}(1,1)$ corresponds to $\Delta\mathrel{\mathop{:}}=P(1,1)=A+B$, i.e. ${\mathcal{N}}(A+B)=\Sigma$.

We are looking for a sequence of polytopes $(P(a_{1},b_{1}),\ldots,P(a_{4},b_{4}))$ corresponding to a full exceptional sequence of line bundles $({\mathcal{L}}_{1},\ldots,{\mathcal{L}}_{4})$ for the bounded derived category $\operatorname{D}^{b}(\mathbb{F}_{1})$, i.e. $\operatorname{Ext}^{*}({\mathcal{L}}_{i},{\mathcal{L}}_{j})=0$ for $j>i$, and the line bundles generate $\operatorname{D}^{b}(\mathbb{F}_{1})$; see [Huy06, §1.4]. In fact, the following two sequences of polytopes

$\scriptstyle 0$     $\scriptstyle A$     $\scriptstyle B$     $\scriptstyle A+B$

and $\scriptstyle 0$     $\scriptstyle B$     $\scriptstyle A+B$     $\scriptstyle 2B$

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 $A$ and $2B$ together cannot be part of an exceptional sequence. Moreover, these two sequences are full, i.e. the respective line bundles generate $\operatorname{D}^{b}(\mathbb{F}_{1})$. This, too, can be seen combinatorially: we use the general fact that, if ${\mathcal{O}}(1)$ is an ample line bundle on a smooth projective variety $X$ of dimension $d$, then ${\mathcal{O}}\oplus{\mathcal{O}}(1)\oplus\cdots\oplus{\mathcal{O}}({d})$ is a strong generator of $\operatorname{D}^{b}(X)$ by [Orl09, Thm. 4.1]. In our example, we need to generate ${\mathcal{O}}(2)={\mathcal{O}}(2\Delta)$. We will show how to generate $2\Delta=2A+2B$ using the first exceptional sequence, $(0,A,B,A+B)$. This is achieved by the following polytopal resolution of $2\Delta$

[TABLE]

In each polytope, the red circle indicates the origin, thus fixing the position of the polyhedra in $M_{\mathbb{R}}$. 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 $\operatorname{Pol}^{+}_{0}(\Sigma)\to\operatorname{Coh}(\mathbb{F}_{1})$, $P\mapsto{\mathcal{O}}_{X}(P)$ of Subsection (2.3).