Hard Lefschetz theorem and Hodge-Riemann relations for ...paulbressler.com/i/HL_IJ.pdf · On...

Hard Lefschetz Theorem and Hodge-RiemannRelations for Intersection Cohomology of

Nonrational PolytopesPAUL BRESSLER & VALERY A. LUNTS

ABSTRACT. The Hard Lefschetz theorem for intersection co-homology of nonrational polytopes was recently proved by K.Karu [9]. This theorem implies the conjecture of R. Stanley onthe unimodularity of the generalized h-vector. In this paper westrengthen Karu’s theorem by introducing a canonical bilinearform (·, ·)Φ on the intersection cohomology IH(Φ) of a com-plete fan Φ and proving the Hodge-Riemann bilinear relationsfor (·, ·)Φ.

1. INTRODUCTION

For an n-dimensional convex polytope Q, R. Stanley ([6]) defined a set of in-tegers h(Q) = (h0(Q),h1(Q), . . . , hn(Q))—the “generalized h-vector”—whichare supposed to be the intersection cohomology Betti numbers of the toric varietyXQ corresponding to Q. In case Q ⊂ Rn is a rational polytope, the variety XQindeed exists, and it is known ([6]) that hi(Q) = dim IH2i(XQ). Thus, for arational polytope Q, the integers hi(Q) satisfy(1) hi(Q) ≥ 0,(2) hi(Q) = hn−i(Q) (Poincare duality),(3) h0(Q) ≤ h1(Q) ≤ · · · ≤ h[n/2](Q) (follows from the Hard Lefschetz theo-

rem for projective algebraic varieties).For an arbitrary convex polytope (more generally for an Eulerian poset) Stanleyproved ([6, Theorem 2.4]) the property (2) above. He conjectured that (1) and

263Indiana University Mathematics Journal c©, Vol. 54, No. 1 (2005)

264 PAUL BRESSLER & VALERY A. LUNTS

(3) also hold without the rationality hypothesis. This conjecture drew attentionfor the last fifteen years and many attempts had been made to prove it. A correctproof was recently given by K. Karu in [9]. The proof is based on the theorydeveloped in [2] and [8], which, in particular, associates to an arbitrary convexpolytope Q its intersection cohomology space IH(Q) with a Lefschetz operator `.In case Q is rational, IH(Q) = IH(XQ) and ` is the multiplication by the Chernclass of an ample line bundle on X; so ` satisfies the Hard Lefschetz theorem onIH(Q). The similar Hard Lefschetz (HL) property of ` was conjectured for anarbitrary Q in [2], [8]. It was shown in [2] the (HL) property implies in particularthe equality dim IH2k(Q) = hk(Q), hence implies the Stanley’s conjecture. The(HL) theorem was proved in [9].

The construction of the intersection cohomology IH(Q) is based on the studyof the equivariant geometry of the (nonexistent in general) toric variety XQ and wasessentially given in [3]. Namely, one first defines the “equivariant” intersectioncohomology of Q and then the ordinary IH(Q). This equivariant part of thetheory is indispensable.

More precisely, we work with the complete fan Φ which is dual to the polytopeQ (and denote IH(Φ) = IH(Q), etc.). We consider Φ as a (finite) topologicalspace with subfans being the open subsets. There are two natural sheaves of ringson Φ: the constant sheaf AΦ (A is the ring of global polynomial functions on thespace of the fan) and the “structure sheaf” AΦ, such that the stalk AΦ,σ at thecone σ ∈ Φ consists of polynomial functions on σ . (In case the space of the fanis the Lie algebra of a torus, the (evenly graded) algebra A is naturally isomorphicto the cohomology ring of the classifying space of the torus and there is a closeconnection between AΦ-modules and equivariant sheaves on the correspondingtoric variety X.)

The theory of AΦ-modules and AΦ-modules was developed in [2] and par-tially in [8]. This includes Verdier duality, “equivariant perverse sheaves,” decom-position theorem for subdivisions, etc. In particular there exists a minimal sheafLΦ which plays the role of the equivariant intersection cohomology complex. Theminimal sheaf is characterized by the property that it is an indecomposable (locallyfree) AΦ-module with the stalk at the origin LΦ,o = R. Then the “equivariant”intersection cohomology of Φ is the graded A-module Γ(Φ,LΦ) and

IH(Φ) := Γ(LΦ)/A+Γ(LΦ),where A+ ⊂ A is the maximal ideal. It was proved in [2], [8] that IH(Φ) satisfiesthe Poincare duality: dim IHn−k(Φ) = dim IHn+k(Φ).

Note that Γ(LΦ) (and hence IH(Φ)) is a module over A(Φ)—the algebra ofpiecewise polynomial functions on Φ. The following theorem was formulated asthe main conjecture in [2] and proved in [9].

Theorem 1.1 (HL). Assume that ` is a strictly convex piecewise linear functionon Φ. Then the map

`k : IHn−k(Φ)→ IHn+k(Φ)

On Intersection Cohomology of Nonrational Polytopes 265

is an isomorphism for all k ≥ 1.

We show in this paper that, by assuming the (HL) theorem for fans of dimen-sion ≤ n− 1, we obtain a canonical pairing

(·, ·)Φ : IH(Φ)× IH(Φ) → R(2n)for fans Φ of dimension n. Then we prove the following theorem, which is theanalogue of the Hodge-Riemann bilinear relations in the (intersection) cohomol-ogy of algebraic varieties.

Theorem 1.2 (HR). Let Φ and ` be as in the (HL) theorem. Consider theprimitive subspace

PrimÌHn−k(Φ) = Ker`k+1 | IHn−k(Φ) → IHn+k+2.

Then for 0 ≠ a ∈ PrimÌHn−k(Φ) we have

(−1)(n−k)/2(a, `ka)Φ > 0.

This theorem was also essentially proved by Karu, except he did not have acanonical pairing (·, ·)Φ and had to make choices at each step. This ambiguitymakes the proof unnecessarily heavy and hard to follow. Actually the results ofthis paper imply that the pairing in [9] is independent of the choices made andcoincides with the canonical one. For completeness we present the proofs of the-orems (HL) and (HR), following the main ideas of [9]. The proof proceeds byinduction on the dimension and a reduction to the simplicial case which is knownby McMullen [5] (see also [7]).

We follow the idea in [8] and try to develop all the notions and results notonly for complete fans, but also for quasi-convex ones. (A fan ∆ is quasi-convexif Γ(∆,L∆) is a free A-module and hence it makes sense to define the intersec-tion cohomology IH(∆) the same way as for complete fans.) In particular, wedefine the canonical pairing (·, ·) for quasi-convex fans and then show that thispairing is compatible with various natural operations on fans such as subdivisions,embeddings of fans, etc.

Let us briefly describe the contents of the paper. In the second section werecall and collect some elementary facts about fans and polytopes which are usedlater. In the third section we recall the theory of sheaves on a fan according to [2]and [8], and formulate the (HL) and (HR) theorems. Section 4 is a review of the“smooth” case corresponding to a simple polytope or a simplicial fan. Here wereview Timorin’s work in case of polytopes, and Brion’s work in case of fans. Wethen relate the two pictures in a natural way so that Timorin’s Poincare duality isidentified with Brion’s. In Section 5 we define the canonical pairing on the inter-section cohomology of quasi-convex fans, and then in Section 6 we show that forsimplicial fans our pairing equals n! times Brion’s (or Timorin’s). This implies the


(HL) and (HR) theorems in the simplicial case. Section 7 discusses compatibilityof the canonical pairing with natural operations of fans. This makes the theoryflexible and easy to use. In Section 8 we obtain some immediate applications.Section 9 contains the proof of theorems (HL) and (HR). In Section 10 we provethe Kunneth formula for the intersection cohomology, and derive the (HL) and(HR) theorems for the product of fans (these results are used in Section 9).

We thank Kalle Karu for his useful remarks on the first version of this paper.Vladlen Timorin informed us that the definition of the polytope algebra A(P) inSection 4 and Proposition 4.3 are due to A.V. Pukhlikov and A.G. Khovanskii.He also claims that Proposition 4.2 was known to Minkowski.

2. PRELIMINARIES ON FANS AND POLYTOPES

Let us fix some terminology. Consider a linear space V ' Rn. A (convex) conein V is the intersection of a finite number of closed half-spaces Li ≥ 0, whereLi ∈ W := V∗. A face of a cone σ is the intersection L = 0 ∩ σ where L is alinear function on V which is nonnegative on σ . A fan Φ in V is a finite collectionof cones with the following properties.(1) If σ ∈ Φ and τ ⊂ σ is a face, then τ ∈ Φ.(2) The intersection of two cones in Φ is a face of each.(3) The origin o ∈ Φ.We denote by Φ≤k ⊂ Φ the subfan consisting of all cones of dimension ≤ k.

Given a fan Φ denote by |Φ| ⊂ V its support, i.e., the union of all cones in Φ.We call Φ complete if |Φ| = V .

Example 2.1. A fan Φ is simplicial if every cone σ ∈ Φ of dimension k is aconvex hull of k rays.

For a subset S ⊂ Φ denote by [S] the minimal subfan of Φ which contains S.Let σ ∈ Φ. We denote

∂σ = [σ]−σ,σ 0 = σ − |∂σ |,St(σ) = τ ∈ Φ | σ ⊂ τ,∂ St(σ) = [St(σ)]− St(σ),Link(σ) = τ ∈ ∂ St(σ) | τ ∩σ = o.

Sometimes we will be more specific and write StΦ(σ) instead of St(σ).We denote by 〈σ〉 the linear subspace of V spanned by a cone σ .Let Φ and Φ′ be fans in spaces V and V ′ respectively. Let γ : |Φ| → |Φ′|

be a homeomorphism which is linear on each cone in Φ and induces a bijectionbetween Φ and Φ′. We call γ a pl-isomorphism between Φ and Φ′ and say that thesefans are pl-isomorphic.


Let V1, V2 be vector spaces with fans ∆ and Σ respectively. Consider theproduct V = V1 × V2 with the fan

Φ = σ + τ | σ ∈ ∆, τ ∈ Σ.We call Φ = ∆× Σ the product fan.

Consider A = SymV∗ as a graded algebra, where linear functions have degree2. We denote by A(Φ) the (also evenly graded) algebra of piecewise polynomialfunctions on |Φ|, i.e., f ∈A(Φ) if fσ = f |σ is a polynomial for each σ ∈ Φ.

A piecewise linear function ` ∈ A2(Φ) is called strictly convex if for any twocones σ and τ of dimension n we have `σ(v) < `τ(v) for any v ∈ τ0. Acomplete fan Φ is called projective if it possesses a strictly convex piecewise linearfunction. Note that if ` is strictly convex, then so is ` + `′ for any `′ ∈ A2.

Example 2.2. Let P ⊂ W be a convex polytope of dimension n. For each faceF ⊂ P consider the cone σF ⊂ V which is the dual to the cone in W generated byvectors f − p with f ∈ F and p ∈ P . The collection of cones σF is a completefan in V , which is called the outer normal fan of P . We denote this fan ΦP . Theassignment F , σF is a bijective, order-reversing correspondence between faces ofP and cones of ΦP . Denote by HP the following function on V

HP(v) = maxy∈P〈y,v〉.

Then HP is a piecewise linear function on ΦP , called the support function of P . Itis strictly convex. Thus ΦP is projective. Vice versa, given a projective fan Φ witha strictly convex piecewise linear function `, there exists a convex polytope P ⊂ Wsuch that (Φ, `) = (ΦP ,HP).

Example 2.3. Consider the product fan Φ = ∆×Σ as above. If `1 and `2 arepiecewise linear strictly convex functions on∆ and Σ respectively, then ` = `1+`2is strictly convex on Φ.

Remark 2.4. Our main results imply that a strictly convex piecewise linearfunction ` on a complete fan behaves like the first Chern class of an ample linebundle on a projective variety. This may cause some confusion, because of the twoopposite notions of strict convexity. Namely, if ` = HP is the support function ofa convex polytope on Φ = ΦP , then ` is strictly lower convex: given two adjacentd-dimensional cones σ , τ ∈ Φ, we have `σ |τ < `τ . In [4] however, a piecewiselinear function which is the first Chern class of an ample line bundle, is strictlyupper convex. The reason for this discrepancy is the fact that ΦP is the outer normalfan of P , whereas Danilov uses the inner normal fan.

Let Φ and Ψ be fans. We call Ψ a subdivision of Φ if |Ψ| = |Φ| and every coneΨ is contained in a cone in Φ. In this case we define a map π : Ψ → Φ so thatπ(σ) is the smallest cone in Φ which contains σ . Often we will refer to the mapπ as a subdivision.


Example 2.5. Given a fan Φ and a cone σ ∈ Φ of dimension > 1, choose aray ρ ⊂ σ 0. Define a new fan

Ψ := Φ − St(σ) ∪ τ + ρ | τ ∈ ∂ St(σ).

Then Ψ is a subdivision of Φ, called the star subdivision (defined by ρ). Thus toobtain Ψ from Φ, one replaces StΦ(σ) by StΨ(ρ).

Remark 2.6. Given a star subdivision as above, there is a one dimensionalvector space of piecewise linear functions on Ψ supported on St(ρ). Namely, sucha function is determined by its restriction to ρ.

Definition 2.7. An edge (i.e., a 1-dimensional face) ρ of a cone σ is calledfree in σ if all other edges of σ are contained in one facet τ, i.e.,

σ = τ + ρ.

A cone σ is called deficient if it has no free edges.

Note that a fan is simplicial if and only if it contains no deficient cones.

Remark 2.8. In case of a star subdivision as in the previous example, the1-dimensional cone ρ is free in every cone containing it.

Definition 2.9. For a fan Φ we define its “singular” subfan Φs to be the min-imal subfan which contains all deficient cones in Φ. So Φs = ∅ if and only if Φ issimplicial.

In the notation of Example 2.5, let σ ∈ Φs be a cone of maximal dimension.By Remark 2.8 the resulting fan Ψ will contain a smaller number of deficient conesthan Φ. This suggests a “desingularization” process of a fan, which ends when oneobtains a simplicial fan. This strategy is used in [9] and we also follow it.

Definition 2.10. Let Φ be a fan and σ ∈ Φ. We say that Φ has a local productstructure at σ if

[St(σ)] = ξ = τ1 + τ2 | τ1 ∈ [σ], τ2 ∈ Link(σ).

Example 2.11. Assume that a complete fan Φ has a local product structure atσ ∈ Φ. Consider the projection p : V → V = V/〈σ〉. Then Ψ = p(Link(σ)) is acomplete fan in V , and [St(σ)] is pl-isomorphic to the product fan Ψ × [σ].

Lemma 2.12. Let Φ be a complete fan and σ ∈ Φs—a maximal cone in Φs .Then Link(σ) is a simplicial fan and Φ has a local product structure at σ .

Proof. Claim: Every cone τ ∈ St(σ) is of the form

τ = σ + ρ1 + · · · + ρdim(τ)−dim(σ),


where the edges ρi are free in τ.Indeed, let τ ∈ St(σ) be a cone of the minimal dimension such that τ =

σ + ρ1 + · · · + ρs and dim(τ) < dim(σ) + s. By definition of σ , the coneτ contains a free edge ρ, so that τ = ρ + ξ for a cone ξ. If ρ ⊂ σ , thenσ = ρ+σ ∩ξ and ρ is free in σ—a contradiction. So ρ is one of ρi’s, say ρ = ρs .But then σ +ρ1+· · ·+ρs−1 is also a cone in St(σ) and its dimension is less thandim(σ)+(s−1)—a contradiction with the minimality of τ. This proves the firstpart of the claim. The freeness of each edge ρi is clear.

The lemma follows from this claim. Indeed, by definition every cone inLink(σ) is a face of a cone in St(σ). The claim implies that all cones in Link(σ)are simplicial. This proves the first assertion of the lemma. The claim also impliesthat, for any face δ of σ and any subset S of the cones ρi, the cone δ + S is aface of τ. The local product structure follows.

Lemma 2.13. Let Φ and σ ∈ Φs be as in the previous lemma. Choose a rayρ ∈ σ 0 and let Ψ → Φ be the corresponding star subdivision. If Φ is projective, so isΨ .

Proof. Let dim(σ) = k. Assume that ` ∈ A2(Φ) is strictly convex on Φ.Then ` is also in A2(Ψ), but is not strictly convex on Ψ . Let ˜ be a piecewiselinear function on Ψ supported on St(ρ), such that ˜|ρ < 0. Consider

ˆ := ` + ε ˜, for 0 < ε 1.

We claim that ˆ is strictly convex on Ψ . Indeed, let δ and θ be two cones ofdimension n in Ψ . If one of them is not in St(ρ), then ˆ

δ|θ < ˆθ, because the

same is true for ` (and ε 1). So we may assume that δ, θ ∈ St(σ) and,moreover, that they are contained in the same cone τ ∈ Φ. The Claim in theproof of last lemma implies that

τ = σ +n−k∑i=1

ρi, where ρi ∈ Link(σ).

Therefore

δ = δ∩σ +n−k∑i=1

ρi, θ = θ ∩ σ +n−k∑i=1

ρi,

and dim(δ∩ σ) = dim(θ ∩ σ) = k. Now convexity of the boundary ∂σ impliesthat ˜

δ|θ < ˜θ and hence ˆ

δ|θ < ˆθ, which proves the lemma.

Definition 2.14. A fan ∆ is purely n-dimensional if every cone in ∆ is con-tained in a cone of dimension n. In this case we denote by ∂∆ ⊂ ∆ the subfanwhose support is the boundary of the support of ∆. Put ∆0 = ∆− ∂∆.

Unless stated otherwise, all fans are assumed to be in V .


3. REVIEW OF SOME CONSTRUCTIONS AND RESULTS FROM[2] AND [8]

3.1. We briefly recall the notions and results from our paper [2] which arerelevant to this work. All A-modules are graded and a morphism between A-modules is always homogeneous of degree 0. Let A+ be the maximal ideal ofA. For an A-module M we denote by M = M/A+M the corresponding (graded)R-vector space. The shifted module M(i) is defined by

M(i)k = Mi+k.

Similarly we will consider shifts H•(i) of graded vector spaces H•.The tensor product ⊗ always means ⊗R.Let Φ be a fan in V . We consider the (finite) set Φ as a topological space

with open subsets being subfans. (In particular, o ∈ Φ is the unique open point).Denote by Sh(Φ) the category of sheaves of R-vector spaces on Φ.

There are two natural sheaves of rings on Φ: the constant sheaf AΦ and the“structure sheaf”AΦ. The stalkAσ consists of polynomial functions on σ . Thesesheaves are evenly graded, Γ(Φ, AΦ) = A, Γ(Φ,AΦ) = A(Φ). Thus the canonicalsurjection of sheaves

AΦ →AΦdoes not induce a surjection of global sections in general.

The category ofAΦ-modules contains an important subcategory M =M(Φ),consisting of flabby locally free sheaves of finite type. Namely, an AΦ-moduleF belongs to M if it is a flabby sheaf and Fσ is a finitely generated free AΦ,σ -module for each σ ∈ Φ. The category M is of finite type: every object is a directsum of indecomposable ones. There exists a distinguished indecomposable objectL = LΦ ∈M characterized by the property Lo = R. We call L the minimal sheaf.It is an analogue of the equivariant intersection cohomology sheaf on the toricvariety XΦ corresponding to Φ (this variety exists only if the fan is rational). It iseasy to see that L can be characterized as an object in M such that Lo = R, andthe map

Lσ → Γ(∂σ,L)is an isomorphism for each cone σ .

The following (easy) result is the combinatorial analogue of the (equivari-ant) decomposition theorem for the direct image of perverse sheaves under propermaps.

Theorem 3.1 (Decomposition theorem). Let π : Ψ → Φ be a subdivision. Thedirect image functor π∗ :AΨ−mod →AΦ−mod takes the category M(Ψ) to M(Φ).In particular, the sheaf π∗LΨ contains LΦ as a (noncanonical) direct summand.

The last assertion of the theorem follows from the first one since (π∗LΨ)0 =LΨ ,0 = R.


Definition 3.2. In the notation of the last theorem a map α : LΦ → π∗LΨ iscalled admissible if α0 = Id : R→ R.

Since L is flabby,Hi(Φ,L) = 0, for i > 0.

Definition 3.3 ([8]). A fan Φ is called quasi-convex if H0(Φ,L) is a free A-module.

A complete fan is quasi-convex. More precisely, we have the following theo-rem.

Theorem 3.4 ([8]). A quasi-convex fan is purely n-dimensional. A purely n-dimensional fan Φ is quasi-convex if and only if |∂Φ| is a real homology manifold. Inparticular, Φ is quasi-convex, if |Φ| is convex.

Example 3.5. Let Φ be a complete fan and σ ∈ Φ. Then the subfan [St(σ)]is quasi-convex.

For a quasi-convex fan Φ we define its intersection cohomology space as

IH•(Φ) := Γ(Φ,L).Denote by ihi(Φ) = dim IHi(Φ) the i-th Betti number of Φ. If Φ is rational, thenIH(Φ) = IH(XΦ).

3.2. Hard Lefschetz theorem. Fix a projective fan Φ with a strictly convexpiecewise linear function `. The A-module Γ(LΦ) is in fact a A(Φ)-module; soIH(Φ) is also such. In particular ` induces a degree 2 endomorphism

` : IH•(Φ)→ IH•+2(Φ).It is known that IHi(Φ) = 0 unless i is even and i ∈ [0,2n]; ih0(Φ) = 1, andihn−k(Φ) = ihn+k(Φ). The following “Hard Lefschetz” theorem was conjecturedin [2]. It was recently proved in [9].

Theorem 3.6 (HL). For each k > 0 the map

`k : IHn−k(Φ)→ IHn+k(Φ)is an isomorphism.

Corollary 3.7. For a projective fan Φ we have

ih0(Φ) ≤ ih2(Φ) ≤ · · · ≤ ih2[d/2](Φ).It was shown in [2] that the (HL) theorem implies the Stanley conjecture on

the unimodality of the generalized h-vector.


3.3. Duality. Letω =ωA/R = A ·∧n V∗ be the dualizing module for A. It

is a free A-module of rank 1 generated in degree 2n. The dualizing sheaf on Φ is

DΦ =ωo[n],(which is a complex concentrated in degree −n and supported at the origin o.)

The Verdier duality functor

D : Dbc (AΦ −mod)op → Dbc (AΦ −mod)

is defined byD(F) = RHomAΦ(F,DΦ).

It has the following properties.

Theorem 3.8 ([2]). Let Φ be a fan.(a) There is a natural isomorphism of functors Id → D2. So D is an anti-involution

of Dbc (AΦ −mod).(b) D(M(Φ)) =M(Φ).(c) DL ' L.

The last assertion of the theorem implies the Poincare duality for IH(Φ) for acomplete fan Φ.

For a deeper study of our duality functor we will need the notion of a cellularcomplex which we recall next.

3.4. Cellular complex C•(·). Choose an orientation of each (nonzero) conein the fan Φ. Then to each sheaf F on Φ we can associate its cellular complex

C0(F) ∂----------------------------------→ C1(F) ∂

----------------------------------→ ·· · ∂----------------------------------→ Cn(F),

whereCi(F) =

⊕dim(σ)=n−i

Fσ ,

and the differential ∂ is the sum of the restriction maps Fσ → Fτ with plus or mi-nus sign depending on whether the orientations of σ and τ agree or not. Some-times we will be more specific and write C•Φ(F) for C•(F). Note that C•(·) is anexact functor from sheaves on Φ to complexes.

In particular, we get a functor

C•(·) : Db(AΦ −mod)→ Db(A−mod).

Note that C•(DΦ) =ω (a complex concentrated in degree 0).The following proposition will be used extensively in this work.


Proposition 3.9 ([2]). For any fan Φ there exists a natural isomorphism of func-tors Dbc (AΦ −mod)op → Dbc (A−mod)

RΓ(Φ,D(·)) = RHom•AΦ(·,DΦ)→ RHom•A(C•(·),ω),

induced by the functor C•(·).This proposition shows that the Verdier duality can be considered as the Borel-

Moore duality (see [2]). Namely, for F ∈ Dbc (AΦ −mod) the assignment

σ , C•(F[σ])

defines the co-sheaf (of complexes) of sections of F “with compact supports.”(Here F[σ] denotes the extension by zero of the restriction of F to [σ]). It followsfrom the above proposition that

RΓ([σ],D(F)) = RHom•A(C•(F[σ]),ω),

where the right hand side is the analogue of the topological Borel-Moore duality.The cohomology groups Hi(C•(F)) tend to be related to the cohomology

groups Hi(Φ, F). For example, if Φ is complete and the orientations of all n-dimensional cones agree, then the natural map Γ(Φ, F) → C0(F) induces an iso-morphism Γ(F) ' H0(C•(F)). Moreover, the other cohomology groups also co-incide, i.e., we have a canonical functorial quasi-isomorphism of complexes (for acomplete fan Φ)

RΓ(Φ, F) ' C•(F)([2]). This shows that the cellular complex makes sense once we choose a globalorientation on V . In fact, for our purposes it will be necessary to choose a volumeform on V . So from now on we make the following assumption.

Assumption 3.10. We assume that a nonzero element Ω = ΩV ∈ ∧n V∗ hasbeen fixed.

This choice determines a trivialization ω = A(2n). Thus in particular weobtain an isomorphism C•(DΦ) = Cn(DΦ) = A(2n). We want to see how thisisomorphism changes in a natural way once we make a different choice of thevolume form. For this assume, for simplicity, that Φ is complete.

Two different choices of orientations of cones in Φ produce two functors C•1and C•2 and there exists an isomorphism ϕ : C•1 → C•2 . We want ϕ to be com-patible with the canonical isomorphisms H0(C•1(F)) = Γ(F) = H0(C•2(F)). Thisforces ϕ0 : C0

1 → C02 to be the identity map, and hence defines uniquely the iso-

morphisms ϕi : Ci1 → Ci2. For example, if the two choices of orientations are thesame for all cones of dimension < n, and are opposite on the maximal cones (i.e.,only the global orientation is changed), thenϕi = −1 for all i > 0. One can easily


check that the sign of ϕn depends only on the comparison of the orientations ofmaximal cones, i.e., ϕn = 1 if the global orientations agree and ϕn = −1 if theyare opposite.

In particular, let W be a vector space and Wo be the sheaf on Φ, which isequal to W on o and zero elsewhere. Choose a cellular complex C• and get anisomorphism C•(Wo) = Cn(Wo) ' W[−n]. However, as was explained above, ifwe choose a different cellular complex with the opposite global orientation, thenthe above isomorphism changes sign.

A certain “cancellation” of two ambiguities occurs in the previous exampleif W = ω. Indeed, our choice of the volume form Ω defines an isomorphismω = A(2n) and also an isomorphism C•(DΦ) =ω. So we get an isomorphism

C•(DΦ) = A(2n).But the arguments above show that the volume form −Ω defines the same (!) iso-morphism. Hence this last isomorphism is independent ofΩ up to multiplicationby a positive number.

Remark 3.11. The above situation is similar to the fact that, on a smoothorientable connected manifold, the top cohomology group with coefficients in theorientation sheaf is canonically isomorphic to Z.

Let us apply Proposition 3.9 together with the last argument to our mainexample of the sheaf L.

Theorem 3.12. Assume that the fan Φ is complete. Then the volume form Ωdefines a canonical isomorphism of A-modules

Γ(DL) =HomA(Γ(L),A(2n)).A different choice of a volume form defines the same isomorphism up to multiplicationby a positive real number.

Proof. Indeed, the complex C•(L) is acyclic except in degree 0, andH0(C•(L))= Γ(L) is a free A-module.

Remark 3.13. Note that both L and DL are AΦ-modules (not just AΦ-modules). The A(Φ)-module structure on Γ(DL) comes from such structureon Γ(L) via the isomorphism of the last theorem.

3.5. The Hodge-Riemann bilinear relations. In order to formulate the sec-ond main theorem we need to make the following assumption.

Assumption 3.14. From now on we assume that the (HL) theorem holds forall complete fans of dimension < n.

Lemma 3.15. Let Φ be a fan with the minimal sheaf LΦ. Then for any coneτ ∈ Φ of dimension k ≥ 1 the following hold:


(1) The stalk LΦ,τ is generated (as the A-module) in degrees < k.(2) The costalk ΓτLΦ := Ker(LΦ,τ → Γ(∂τ,LΦ)) is generated in degrees > k.

Proof. To simplify the notation assume that k = n. Choose a ray ρ in theinterior of τ, and consider the projection p : V → V ' Rn−1. Then ∂τ := p(∂τ)is a complete fan in V , and ∂τ ⊂ V is a graph of a strictly convex piecewise linearfunction ¯ on ∂τ. Let AV be the algebra of polynomial functions on V . Note thenA = AV[ ¯]. The projection p : ∂τ → ∂τ defines an isomorphism of AV -modules

Γ(∂τ,LΦ) ' Γ(∂τ,L∂τ).So in particular Γ(∂τ,LΦ) is a free AV -module. By our Assumption 3.14 above, ¯

acts on IH(∂τ) as a Lefschetz operator. This implies both assertions of the lemmabecause the residue map

LΦ,τ → Γ(∂τ,LΦ)is an isomorphism.

Corollary 3.16. EndAΦ(LΦ) = R.

Proof. This follows from the previous lemma and from the fact that LΦ,o = Rby induction on the dimension of cones in Φ. Indeed, let σ ∈ Φ be of dimensiond and consider the exact sequence of A-modules

0→ ΓσLΦ → LΦ,σ → Γ(∂σ,LΦ)→ 0.

Since LΦ,σ (resp. ΓσLΦ) is generated in degrees < d (resp. > d), the identityendomorphism of L[∂σ] can be extended to an endomorphism of L[σ] in a uniqueway.

Lemma 3.17. There exists a canonical isomorphism ofAΦ-modules

εΦ : LΦ → D(LΦ).Proof. Since the sheaves LΦ and D(LΦ) are isomorphic, by the previous

corollary it suffices to find a canonical isomorphism of the stalks LΦ,o andD(LΦ)o.But

D(LΦ)o = RHomA(LΦ,o,DΦ,o) = RHomA(R,ω[n]) = ExtnA(R,ω[n]).

To compute the last Ext group take the canonical Koszul resolution of the A-module R

0→ω→ A⊗n−1∧V∗ → · · · → A⊗ V∗ → A→ R.

Then ExtnA(R,ω[n]) =HomA(ω[n],ω[n]) = R.


Corollary 3.18. Assume that the fan Φ is complete. The volume form Ω definesa canonical nondegenerate pairing

[·, ·] = [·, ·]Φ = Γ(LΦ)× Γ(LΦ)→ A(2d),and hence a canonical nondegenerate pairing

(·, ·) = (·, ·)Φ : IH(Φ)× IH(Φ) → R(2d).A different volume form defines the same pairings up to multiplication by a positivereal number. These pairings areA(Φ)-bilinear.

Proof. This follows from the Theorem 3.11, Remark 3.12 and Lemma3.15.

In Section 5 below we will construct a similar canonical pairing for quasi-convexfans.

The next theorem is the analogue of the Hodge-Riemann bilinear relations onthe primitive intersection cohomology of complex algebraic varieties.

Theorem 3.19 (HR). Let Φ be a projective fan with a strictly convex piecewiselinear function `. For k ≥ 0 put

PrimÌHn−k(Φ) := b ∈ IHn−k | `k+1b = 0.Then the quadratic form Q`(a) = (−1)(n−k)/2(a, `ka)Φ is positive definite onPrimÌHn−k(Φ).

3.6. Intersection cohomology of pl-isomorphic fans and of St[σ]. Let Φ,Ψ be fans and γ : |Φ| → |Ψ| be a pl-isomorphism. Then there exists an isomor-phism of sheaves γ∗AΨ ' AΦ, hence also an isomorphism of minimal sheavesγ∗LΨ ' LΦ and of the global sections Γ(Φ,LΦ) ' Γ(Ψ ,LΨ ). This last isomor-phism however does not preserve the structure of a module over the global polyno-mial functions, unless γ is induced by a linear isomorphism of the ambient vectorspaces. Thus, for example, if the pl-isomorphic fans are quasi-convex, there isno canonical isomorphism between IH(Φ) and IH(Ψ). Note however, that thesegraded vector spaces are isomorphic, since the graded spaces Γ(LΦ) and Γ(LΨ)have the same Hilbert function.

Assume that a complete fan Φ has a local product structure at a cone σ . Letp : V → V = V/〈σ〉 be the projection, Φσ = p(Link(σ))—the complete fan inV . Note that the projection p makes Γ([St(σ)],LΦ) anA(Φσ )-module. Indeed,by the local product structure at σ the projection p(τ) of any cone τ ∈ [St(σ)]is a cone in Φσ . In particular, this makes IH([St(σ)]) aA(Φσ )-module.

Lemma 3.20. There exist natural isomorphisms ofA(Φσ )-modules

Γ([St(σ)],LΦ) ' Γ(Φσ ,LΦ)⊗LΦ,σ ,IH([St(σ)]) ' IH(Φσ )⊗LΦ,σ ,


with the trivial A(Φσ)-module structure on LΦ,σ . In particular, if σ is simplicial,i.e., LΦ,σ =AΦ,σ , then

IH([St(σ)]) = IH(Φρ).Proof. Let Aσ be the polynomial functions on σ . Choose a projection V →

〈σ〉 and identify Aσ as a subalgebra of A.Let pσ : St(σ)→ Φσ denote the restriction of the projection p to St(σ). The

sheaf Aσ ⊗p−1σ LΦσ extended by zero to Φ satisfies the definition of a minimal sheaf

based at σ in [2, (5.1)]. (It is denoted by LσΦ in [2].) Note that the restrictionLΦ|St(σ) extended by zero to Φ is also a minimal sheaf based at σ . By Proposition5.2 in [2] LΦ|St(σ) ' LσΦ ⊗LΦ,σ . Therefore

Γ(St(σ),LΦ) ' Γ(Φσ ,LΦσ )⊗Aσ ⊗LΦ,σ = Γ(Φσ ,LΦσ )⊗LΦ,σ .This is an isomorphism ofA(Φσ )-modules. It remains to note that

Γ([St(σ)],LΦ) = Γ(St(σ),LΦ).

4. REVIEW OF THE “SMOOTH” CASE:SIMPLE POLYTOPES AND SIMPLICIAL FANS

4.1. Review of Timorin’s work on simple polytopes. The analogue of Hodge-Riemann bilinear relations for simple polytopes was proved by McMullen in [5].Later Timorin gave a simpler proof in [7]. Here we recall his main results.

Consider the dual space W = V∗. Let P ⊂ W be a convex polytope of dimen-sion n. We assume that P is simple, i.e., at each vertex of P there meet exactly nfaces of dimension n− 1.

Definition 4.1. Two convex polytopes P ′, P ′′ ⊂ W are called analogous iftheir hyperplane faces can be pairwise matched so that the matched faces(1) have the same outward normal direction,(2) are analogous.Any two segments on a line are analogous.

Clearly being analogous is an equivalence relation. Let P+(P) denote thecollection of all polytopes in W analogous to P . If P ′, P ′′ ∈ P+(P), then theirMinkowski sum

P ′ + P ′′ = p′ + p′′ | p′ ∈ P ′, p′′ ∈ P ′′

is also in P+(P). Also

λP ′ = λp′ | p′ ∈ P ′, λ > 0,


is in P+(P). Thus P+(P) has a structure of a convex cone. It can be comple-mented to a vector space P(P) by considering formal differences of polytopes.

Choose a volume form ΩW ∈ ∧n V on W to be the dual of ΩV and orient Win such a way that

Vol(P) :=∫PΩW > 0.

Proposition 4.2 ([7]). The volume function Vol defined on P+(P) extends toa homogeneous polynomial function of degree n on P(P). We denote this extendedfunction again by Vol.

Let P1, . . . , Pm be the faces of P of dimension n − 1. Choose the corre-sponding outward normal covectors ξ1, . . . , ξm ∈ V . That is, each ξi|P achievesits maximum exactly on Pi. Then we associate to P the numbers H1, . . . , Hmdefined by

Hi := maxp∈Pξi(p) = ξi(Pi).

The functions H1, . . . , Hm form a system of linear coordinates on P(P).

4.2. The polytope algebra A(P). Let Diff be the algebra of differential op-erators with constant coefficients on P(P). It is a commutative polynomial ringwith generators

∂i = ∂∂Hi

, i = 1, . . . ,m.

Let I ⊂ Diff be the ideal

I = d ∈ Diff | dVol = 0.

Put A(P) := Diff/I. Since Vol is a homogeneous polynomial, the ideal I is gradedand so is the ring A(P):

A(P) =n⊕k=0

Ak(P).

The following proposition is almost obvious.

Proposition 4.3 ([7]). The formula (α,β)T = αβVol defines a nondegeneratepairing

(·, ·)T : Ak(P)⊗An−k(P)→ R.

Corollary 4.4. dimAk = dimAn−k.

Theorem 4.5 ([7]). We have dimAk = hk(P)—the k-th component of the h-vector h(P).


4.3. Presentation of the algebra A(P). We have a natural embedding oflinear spaces

i : W Diff, i(a) =m∑i=1

ξi(a)∂i.

Let Pi1 , . . . , Pik be different faces of P of dimension n− 1. Put

Di1...ik := ∂i1 · · · ∂ik ∈ Diff.

Theorem 4.6 ([7]).(1) For all a ∈ W , i(a) ∈ I (translation invariance of the volume function).(2) If Pi1 ∩ · · · ∩ Pik = ∅, then Di1...ik ∈ I.(3) The ideal I is generated by elements i(a), Di1...ik , as in (1), (2).

4.4. A basis for A(P). Fix a general linear function t on W . For each ver-tex p ∈ P let index(p) be the number of edges coming out of p on which themaximum of t is attained at p. That is, index(p) is the number of edges whichgo down from p. Consider the face F(p) spanned by all these edges. We havedimF(p) = index(p). Thus we associated one face to each vertex (the interior ofP is associated to the “highest” vertex).

Note that each face F ⊂ P of codimension k is the intersection of exactly khyperplane faces: F = Pi1 ∩ · · · ∩ Pik . Put DF := ∂i1 · · · ∂ik ∈ Diff.

Theorem 4.7 ([7]). The monomials DF(p) | p is a vertex in P form a basis ofA(P). More precisely, monomials DF(p) | index(p) = k form a basis of An−k.

4.5. The Lefschetz operator. Consider the operator

L = LP =m∑i=1

Hi(P)∂i ∈ A1(P).

Theorem 4.8 ([7]). The operator (of multiplication by) L acts as a Lefschetzoperator on A(P), i.e., for all i ≤ n/2 the map

Ln−2i : Ai(P)→ An−i(P)is an isomorphism.

Theorem 4.9 ([7]). Let PrimAi(P) = a ∈ Ai(P) | Ln−2i+1a = 0 be theprimitive part of Ai(P). Then the symmetric bilinear form

〈a,b〉i = (−1)i(a, Ln−2ib)T

is positive definite on PrimAi(P).

Theorem 4.10 ([7]). We have LnP Vol = n! Vol(P), i.e.,

(LnP ,1)T = n! Vol(P).


4.6. The dual picture in terms of fans after Brion [1]. Let P ⊂ W be asimple polytope and Φ = ΦP ⊂ V be its outer normal fan (Example 2.2). SinceΦ is simplicial, LΦ = AΦ and we denote H•(Φ) = IH•(Φ). Then H•(Φ) is a(evenly) graded algebra

H•(Φ) = n⊕k=0

H2k(Φ).Proposition 4.11 ([1]). dimH2k(Φ) = hk(P).4.7. A pairing (·, ·)B on H•(Φ). Brion defines a (degree zero homogeneous)

map of A-modules ζ : A(Φ) → A(2n). It follows from the last proposition thatsuch a map is unique up to a scalar factor. Let us recall the construction. For eachn-dimensional cone σ , denote by Fσ the product of equations of the facets of σ .Then Fσ ∈ A is uniquely defined up to scalar multiplication. One normalizesFσ as follows: the equations of the facets are nonnegative on σ and their wedgeproduct equals ±ΩV . Denote by ϕσ ∈A(Φ) the function such that

ϕσ(v) =Fσ (v) if v ∈ σ,

0, otherwise.

Thus ϕσ ∈A2n(Φ) and it vanishes outside σ . For f ∈A(Φ) define

ζ(f) :=∑

dim(σ)=n

fσFσ .

Theorem 4.12 ([1]). The map ζ is a well defined map ζ : A(Φ) → A(2n)such that(1) ζ is A-linear;(2) ζ(ϕσ) = 1 for all n-dimensional σ ∈ Φ.

Clearly, ζ(f) = 0 if deg(f ) < 2n.The map ζ induces a symmetric pairing

[·, ·]B :A(Φ)×A(Φ)→ A(2n), [a, b]B = ζ(ab).

We also obtain a nonzero linear function ζ : H2n(Φ) → A/A+ = R(2n), andhence a pairing

(·, ·)B : H•(Φ)×H•(Φ)→ R(2n), (x,y)B = ζ(xy).

Proposition 4.13 ([1]). The pairings [·, ·]B and (·, ·)B are nondegenerate.

Remark 4.14. Notice that if the form ΩV is changed by a factor of r ∈ R,then the map ζ and hence the pairings [·, ·]B , (·, ·)B are changed by the factor|r |−1.


Recall the support function HP ∈A(Φ) of P (Example 2.2):

HP(v) = maxx∈P〈x,v〉.

Note, that if P contains the origin in W , then HP is nonnegative.

Theorem 4.15 ([1]). We have ζ(HnP ) = n! Vol(P).

4.8. Relation between the pictures of Timorin and Brion.

Theorem 4.16. There exists a natural isomorphism of algebras

β : A(P)→ H(Φ), β : Ak(P)∼→ H2k(Φ),

such that β(LP) = HP , and (a, b)T = (β(a), β(b))B .

Proof. First note that there is a natural homomorphism of algebras β : Diff →A(Φ). Indeed, vectors in V which lie in the 1-dimensional cones in Φ are naturallylinear functions on the space P(P). Hence a differential operator of order 1 onP(P) defines a linear function on each 1-dimensional cone of Φ. However, sinceΦ is simplicial, every such function extends uniquely to a piecewise linear functionon Φ. This defines the homomorphism β : Diff →A(Φ).

Notice that the generators Di1,...,ik of the ideal I ⊂ Diff lie in the kernel ofβ. Indeed, let Pi be a face of P of dimension n − 1, ξi ∈ V—a correspondingoutward normal covector, ρi ∈ Φ—the corresponding 1-dimensional cone in Φ(so ξi ∈ ρi), ∂i ∈ Diff—the corresponding derivation. Then the piecewise linearfunction β(∂i) takes the value 1 on ξi and is zero on all other 1-dimensional cones.Thus β(∂i) is nonzero only on the star of ρi. Now, it is clear that if the intersectionof n− 1-dimensional faces Pi1 , . . . , Pik is empty, then so is the intersection of thestars of the corresponding 1-dimensional cones ρi1 , . . . , ρik . Hence

β(Di1,···k) = β(∂i1 · · · ∂ik) = 0.

Notice also that β maps the subspace i(W) ⊂ Diff to the space of linearfunctions on V .

It follows from Theorem 4.6 that β descends to a ring homomorphism β :A(P) → H(Φ). This homomorphism is surjective, since β is surjective (functionsβ(∂i) as above generate A(Φ)). Since dimA(P) = dimH(Φ), β is an isomor-phism.

It is clear that β(LP) = HP and the last assertion follows from Theorems 4.10and 4.15 above.

4.9. Hard Lefschetz and Hodge-Riemann bilinear form for simplicial fans.Let Φ be a complete simplicial fan in V with a strictly convex piecewise linear func-tion `. Then there exists a convex polytope P ⊂ W such that Φ = ΦP and ` = HP .

The following theorems are immediate corollaries of Theorem 4.8, 4.9 and4.16 above.


Theorem 4.17. For Φ and ` as above the multiplication by ` acts as a Lefschetzoperator on H(Φ), i.e.,

`k : Hn−k(Φ) ∼→ Hn+k(Φ)for all k ≥ 1.

Theorem 4.18. Let Φ, ` be as in the last theorem. For k ≥ 1 put

PrimHn−k(Φ) := b ∈ Hn−k(Φ) | `k+1b = 0.Then the symmetric bilinear form (−1)(n−k)/2(a, b)B is positive definite onPrimHn−k(Φ).

5. DUALITY FOR QUASI-CONVEX FANS

5.1. Following the idea in [8] we develop the duality for quasi-convex fans.Recall that a fan ∆ in V is quasi-convex if Γ(∆,L∆) = Γ(L) is a free A-module.

Let ∆ be a quasi-convex fan in V , ∆0 := ∆ − ∂∆. Recall that ∂∆ and ∆0 areopen and closed subsets of ∆ respectively. For a sheaf F on ∆ we denote by Γ∆0Fthe global sections of F supported on ∆0. That is,

Γ∆0F = KerΓ(∆, F)→ Γ(∂∆, F).Consider the standard short exact sequence of sheaves

0→ L∂∆ → L → L∆0 → 0

and the induced short exact sequence of cellular complexes

0→ C•(L∂∆)→ C•(L)→ C•(L∆0) → 0.

It is proved in [8, Theorems 4.3 and 4.11] that in the natural commutative dia-gram Γ∆0L Γ(L)

↓ ↓H0(C•(L)) -→ H0(C•(L∆0))

the vertical maps are isomorphisms. Moreover, the complexes C•(L), C•(L∆0) areacyclic in degrees other than zero.

Recall that IH•(∆) := Γ(L). One also defines

IH•(∆, ∂∆) := Γ∆0L.It was shown in [8] that there exists a (noncanonical) duality between the free

A-modules Γ(L) and Γ∆0 , and hence a duality between the graded vector spacesIH(∆) and IH(∆, ∂∆). We are going to make this duality canonical and to showthat it is compatible with various natural constructions, such as embeddings offans, subdivisions, etc.


5.2. Definition of the pairing [·, ·]∆ : Γ(L) × Γ∆0L → A(2n). By copyingthe proof of Theorem 3.11 and using the previous remarks we obtain the followingproposition.

Proposition 5.1. The functor C•(·) induces a natural isomorphism ofA-modules

Γ(∆,DL) =HomA(Γ∆0L, A(2n)).

This isomorphism is independent of the choice of the volume form Ω up to multiplica-tion by a positive real number.

Combining this isomorphism with the canonical isomorphism ε∆ : L∆ →DL∆ we obtain the following analogue of Corollary 3.16.

Proposition 5.2. There exists a natural isomorphism of A-modules

Γ(L)→HomA(Γ∆0 , A(2n)),

hence nondegenerate pairings

[·, ·] = [·, ·]∆ : Γ(L)× Γ∆0L → A(2n)

and(·, ·) = (·, ·)∆ : IH(∆)× IH(∆, ∂∆) → R(2n).

These pairings are A(∆)-bilinear and are independent of the choice of the volumeform up to multiplication by a positive real number.

Corollary 5.3. The graded vector spaces IH(∆), IH(∆, ∂∆) are concentrated indegrees from 0 to 2n. Also IH2n(∆, ∂∆) ' R(2n).

Proof. Since the sheaf L is zero in negative degrees and (L)0 = R∆, bothassertions follow from the nondegeneracy of the pairing (·, ·).

Remark 5.4. In case the fan ∆ is complete, i.e., ∆0 = ∆, we have Γ∆0(L) =Γ(∆,L) and the pairing of the last proposition coincides with the one in Corollary3.16.

6. ALL PAIRINGS COINCIDE IN THE SIMPLICIAL CASE

Let ∆ be a simplicial quasi-convex fan in V . Since L∆ = A∆, we may define apairing on ∆ using Brion’s functional ζ. Namely, define

[·, ·]B : Γ(∆,A∆)× Γ∆0A∆ → A(2n), [a, b]B := ζ(ab).

Proposition 6.1 ([8]). The pairing [·, ·]B is well defined and is perfect.

This also follows from the next proposition.


Proposition 6.2. We have [a, b] = n![a, b]B for a ∈ Γ(∆,A∆), b ∈ Γ∆0A∆.

Proof. Since the pairing [·, ·] isA(∆)-bilinear, [a, b] = [1, ab]. Define thelinear functional

s : Γ∆0A∆ → A(2n), s(a) := [1, a].It suffices to prove that s = n!ζ. We will describe the map s explicitly and thencompare it with ζ.

Recall the Koszul resolution of the structure sheaf A∆. Namely, define thesheaf of vector spaces Ω1 = Ω1∆ as follows:

Ω1σ = σ⊥ ⊂ V∗.

Put Ωi := ∧iΩ1. Thus the sheaf Ωi is supported on the subfan ∆≤n−i; it is a sheafof graded vector spaces which are concentrated in degree 2i.

We have the canonical Koszul resolution of the structure sheafA∆:

0→ A∆ ⊗Ωn → ·· · → A∆ ⊗Ω1 → A∆ →A∆ → 0.

Note that A∆ ⊗Ωn = D∆.From the definition of the canonical morphism ε∆ : A∆ → DA∆ (Lemma

3.15) it is clear that ε∆ is the projection of the complex

(∗) 0→ A∆ ⊗Ωn → ·· · → A∆ ⊗Ω1 → A∆ → 0

on its leftmost nonzero term. Thus the map s : Γ∆0A∆ → A(2n) coincides withthe isomorphism Γ∆0A → H0(C•(A∆ ⊗ Ω•)) (given by the embedding Γ∆0A C•(A∆⊗Ω•)) followed by the projectionH0(C•(A∆⊗Ω•))→ C•(D∆) = A(2n).

First, recall the formula for the Koszul differential

ν : A∆ ⊗Ωj → A∆ ⊗Ωj−1

: ν(g ⊗ dx1 ∧ · · · ∧ dxj) =∑i(−1)igxi ⊗ dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxj.

The double complex C•(A∆ ⊗Ω•) looks like

· · · · · · · · ·∂x ∂

x xC2(A∆ ⊗Ω2) ν

-----------------------------------------------------------------------------------------------------------------------------------------------------------→ C2(A∆ ⊗Ω1) ν-----------------------------------------------------------------------------------------------------------------------------------------------------------→ C2(A∆)

∂x ∂

xC1(A∆ ⊗Ω1) ν

-----------------------------------------------------------------------------------------------------------------------------------------------------------→ C1(A∆)∂x

C0(A∆)

.


Here ν is the Koszul differential and ∂ is the cellular complex differential.The rows are exact except at the rightmost column. Choose g ∈ Γ∆0A∆. Then gdefines a chain gσ ∈ C0(A∆) = C0(A∆). Put

s(g) := (−1)[n+1/2]ν−1 · ∂ · ν−1 · ∂ · · ·ν−1 · ∂(gσ) ∈ Cd(A∆ ⊗Ωn) = A ·Ω.(The first, third, fifth, etc. composition ν−1 · ∂ is taken with the negative sign). Itis clear that

s(g) = s(g)Ω.Let us understand the map s. Let ξ ⊂ τ be cones in ∆ of codimension k and

k+ 1 respectively. Define

ε(τ, ξ) =

1, if orientations of τ and ξ agree,−1, otherwise.

Then the map on stalks ν−1 · ∂ : (A∆ ⊗ Ωk)τ → (A∆ ⊗ Ωk+1)ξ is equal to thewedging ε(τ, ξ)(dy/y) ∧ · followed by restriction from τ to ξ, where y is anynonzero linear function on τ, s.t. y|ξ = 0.

Choose a cone σ ∈ ∆ of dimension n. Choose linear functions x1, x2, . . . ,xn ∈ Aσ such that(1) xi > 0 in the interior of σ ;(2)

∏xi = 0 on the boundary ∂σ ;

(3) dx1 ∧ · · · ∧ dxn = Ω. (This is possible assuming n > 1. We omit here thecase n = 1, since it can be done directly.)

Let f ∈ Γ∆0A∆ be defined as follows

fτ =∏

xi, if τ = σ,0, otherwise.

In particular, f =ϕσ in the notation of Section 4 above.The following lemma implies the proposition.

Lemma 6.3. We have s(f ) = n!Ω, or, equivalently, s(f ) = n!.

Proof of Lemma 6.3. The element s(f ) ∈ Cn(A∆⊗Ωn) is a sum of n! termscorresponding to the n! choices of a complete flag of faces of σ .

Claim 6.4. The n! summands in s(f ) are equal.

Claim 6.5. Each summand is equal to Ω.

Clearly the claims imply the lemma. It is also clear that the only issue inproving the claims is the sign. Indeed, by applying n times the formula above forν−1 · ∂ we find that each of the n! terms is ±Ω.


Let us start with the flag σn = o ⊂ σn−1 ⊂ · · · ⊂ σ0 = σ , where σk =xn−k+1 = · · · = xn = 0 ∩ σ . Assume that σk is oriented by the form ωk =dx1∧· · ·∧dxn−k, (ωn = 1). Then by definition ε(σk,σk+1)ωk+1 = −dxn−k∧ωk+1. Indeed, −dxn−k is the outward normal covector of σk+1 in σk. The signε(σk,σk+1) is negative for k+ 1 = n, n− 2, . . . . Thus ε = −1 for [n+ 1

2] pairsσk+1 ⊂ σk. Hence the summand of s(f ) corresponding to the above flag (withabove orientaions) is equal to (−1)2[n+1/2]Ω = Ω.

It remains to proof that the summand is independent of the flag and of theorientations of the cones of dimension 0 < k < n in the flag.

First note that if the flag is the same, but the orientation of one of the cones,say σk, is changed (0 < k < n), then ε(σk−1, σk) and ε(σk,σk+1) change sign,so the corresponding summand of s(f ) remains unchanged.

Now assume that the flag is changed in one place: for some 0 < k < n thecone σk is replaced with τk := xn−k = xn−k+2 = · · · = xn = 0. Let τk beoriented by the form ω′k = dx1 ∧ · · · ∧ dxn−k−1 ∧ dxn−k+1. (Orientations ofother cones remain the same). Then

−dxn−k+1 ∧ωk+1 = (−1)n−kω′k,

−dxn−k ∧ω′k = (−1)n−kωk−1.

Hence ε(σk−1, τk) = ε(τk,σk+1). This changes the total number of minus signsamong the ε’s by 1, but the composition

ν−1 · ∂ · ν−1 · ∂ : (A∆ ⊗Ωk−1)σk−1 → (A∆ ⊗Ωk)τk → (A∆ ⊗Ωk+1)σk+1

involves wedging with dxn−k+1∧dxn−k as opposed to the previous wedging withdxn−k ∧ dxn−k+1. Thus the contribution to s(f ) does not change. This provesthe claims and the lemma.

Corollary 6.6. The (HL) and (HR) theorems hold for simplicial fans. In par-ticular, they hold if n ≤ 2.

Proof. Indeed, this is an immediate consequence of Theorems 4.17, 4.18,and Proposition 6.2.

7. COMPATIBILITY OF THE PAIRING WITHNATURAL OPERATIONS ON FANS

7.1. Compatibility of the pairing with embedding of fans. Let Σ be a fanin V and ∆ ⊂ Σ a subfan. Assume that both Σ and ∆ are quasi-convex. Note thatLΣ|∆ = L∆, hence we get a (surjective) restriction homomorphism

r = rΣ,∆ : Γ(Σ,LΣ)→ Γ(∆,L∆).


Also note that the closed subset ∆0 ⊂ ∆ remains closed in Σ and ∆0 ⊂ Σ0. Weobtain the (injective) map

s = s∆,Σ : Γ∆0L∆ = Γ∆0LΣ → ΓΣ0LΣ.Proposition 7.1. The maps r and s are adjoint. That is,

[r(a), b]∆ = [a, s(b)]Σfor a ∈ Γ(Σ,LΣ), b ∈ Γ∆0L∆.

Proof. Note that the restriction |∆ of sheaves from Σ to ∆ commutes with theduality. Moreover, the following diagram commutes

LΣ εΣ-----------------------------------------------------------------------------------------------------------------------------------------------------------→ D(LΣ)|∆y |∆

yL∆ ε∆-----------------------------------------------------------------------------------------------------------------------------------------------------------→ D(L∆)

.

Also note that the restriction Γ(Σ,DLΣ) → Γ(∆,DL∆) is induced by the em-bedding of cellular complexes C•(L∆) C•(LΣ), which in turn induces the in-clusion s : Γ∆0L∆ → ΓΣ0LΣ. That is, the following diagram commutes

Γ(Σ,DLΣ) = HomA(ΓΣLΣ, A(2d))|∆y s∗

yΓ(∆,DL∆)) = HomA(Γ∆L∆, A(2d))

.

This proves the proposition.

7.2. Compatibility of the pairing with subdivisions of fans. Let π : Ψ →Φ be a subdivision of fans. Choose an admissible morphism α : LΦ → π∗LΨ(Definition 3.2). Since LΦ is a direct summand of π∗LΨ , it follows by the rigidityof LΦ (Corollary 3.14) that α is a split injection.

Theorem 7.2. Let ∆ be a quasi-convex fan, π : Θ → ∆—a subdivision (hence Θis also quasi-convex). Choose an admissible morphism α : L∆ → π∗LΘ (Definitrion3.2). It induces morphisms

α : Γ∆0L∆ → Γ∆0(π∗LΘ) = ΓΘ0LΘ,α : Γ(∆,L∆)→ Γ(∆, π∗LΘ) = Γ(Θ,LΘ).

Then for a ∈ Γ(∆,L∆), b ∈ Γ∆0L∆,

(α(a),α(b))Θ = (a, b)∆.


Proof. First we want to “descend” the duality from Θ to ∆. It was proved in[2] that the duality functor commutes with the direct image functor Rπ∗. Herewe want to describe this commutation in a way that is compatible with Proposition3.9.

Let F ∈ Dbc (AΘ − mod) and consider the cellular complexes C•Θ(F) andC•∆(π∗F). Recall that the n-dimensional cones in Θ and ∆ are oriented by ΩV .We can orient the other cones in ∆ and Θ in a compatible way. Namely, ifdim(τ) = dim(π(τ)), we want the orientations of τ and π(τ) to be compat-ible. Otherwise choose an orientation of τ at random. Then there is a naturalmorphism of complexes ϕ(F) : C•∆(π∗(F)) → C•Θ(F), which is compatible withidentifications

H0(C•Θ(F)) = Γ(Θ, F) = Γ(∆, π∗F) = H0(C•∆(π∗F)).The functorial map ϕ above allows us to define a morphism of functors γ :

Rπ∗ ·D → D · Rπ∗ in the following way. Fix F ∈ Dbc (AΘ −mod). Let F → J•be its (finite) injective resolution and choose an injective resolution ω→ I• of theA-moduleω. For a subfan Φ ⊂ ∆ we have

D ·Rπ∗(F)(Φ) =HomA(C•∆((π∗J•)Φ), I•).On the other hand, for a subfan Ψ ⊂ Θ,

D(F)(Ψ) =HomA(C•Θ(J•Ψ ), I•).Note that D(F) is a complex of flabby sheaves, hence Rπ∗D(F) = π∗D(F). Wedefine

γ(F)(Φ) : Rπ∗ ·D(F)(Φ) → D ·Rπ∗(F)(Φ)to be

ϕ∗(J•) : HomA(C•Θ(J•π−1(Φ)), I•)→HomA(C•∆((π∗J•)Φ), I•).Lemma 7.3. γ is an isomorphism of functors.

Proof. Fix a cone σ ∈ ∆. It suffices to show that, for an injective AΘ-module,J the map of complexes

ϕ(J) : C•((π∗J)[σ])→ C•(Jπ−1([σ]))

is a quasi-isomorphism. For simplicity of notation we may assume that dim(σ) =n. Put G = π∗J. Consider the exact sequence of sheaves

0→ G∂σ → G[σ] → Gσ → 0.


Applying Proposition 3.6 of [2] to sheaves Gσ and G∂σ on fans [σ] and ∂σrespectively, we obtain natural quasi-isomorphisms

RΓ([σ],Gσ ) = C•∆(Gσ), RΓ(∂σ,G∂σ ) = C•∆(G∂σ )[1].The sheaf G is injective, hence the exact triangle

C•∆(G[σ])→ C•∆(Gσ)→ C•∆(G∂σ)[1]is quasi-isomorphic to the (middle part of the) short exact sequence

(7.1) 0→ ΓσG → Γ([σ],G) → Γ(∂σ,G) → 0.

The same arguments show that the triangle

C•Θ(Jπ−1([σ]))→ C•Θ(Jπ−1(σ))→ C•Θ(Jπ−1(∂σ))[1]

is quasi-isomorphic to the (middle part of the) short exact sequence

(7.2) 0→ Γπ−1(σ)J → Γ(π−1([σ]), J) → Γ(π−1(∂σ), J) → 0.

The short exact sequences (7.1) and (7.2) are isomorphic, and the isomorphismΓσG ' Γπ−1(σ)J coincides with the map ϕ(J) under the above quasi-isomor-phisms.

Remark 7.4. As in the proof of Lemma 3.15, it is easy to see that the stalks(π∗DLΘ)o and (Dπ∗LΘ)o are canonically isomorphic to R. Under this identifi-cations γ(LΘ)o = Id.

Consider the diagram of sheaves

π∗LΘ π∗εΘ-----------------------------------------------------------------------------------------------------------------------------------------------------------→ π∗DLΘ γ-----------------------------------------------------------------------------------------------------------------------------------------------------------→ Dπ∗LΘ

αx D(α)

yL∆ ε∆------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------→ DL∆

.

Lemma 7.5. The above diagram commutes.

Proof. Note that the stalks at o of all the sheaves are canonically isomorphicto R, and all the morphisms are equal to Id at o. Hence the diagram commutes,by the rigidity of L∆.


It follows from the last two lemmas that the following diagram commutes

Γ(∆, π∗LΘ) π∗εΘ-----------------------------------------------------------------------------------------------------------------------------------------------------------→ Γ(∆, π∗DLΘ) = RHomA(C•Θ(LΘ),A(2d))γy ϕ(LΘ)∗

yαx Γ(∆,Dπ∗LΘ) = RHomA(C•∆(π∗LΘ),A(2d))

D(α)y C•∆(α)∗

yΓ(∆,L∆) ε∆-----------------------------------------------------------------------------------------------------------------------------------------------------------→ Γ(∆,DL∆) = RHomA(C•∆(L∆),A(2d))

.

Finally, note the commutativity of the natural diagram

Γ∆0L∆ α-----------------------------------------------------------------------------------------------------------------------------------------------------------→ ΓΘ0LΘy y

C•∆(L∆) ϕ(LΘ)·C•∆(α)------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------→ C•Θ(LΘ).

The theorem follows.

Corollary 7.6. For a complete fan Φ, the pairing

[·, ·] :A(Φ)×A(Φ)→ A(2d)is symmetric.

Proof. This is true if the fan Φ is simplicial (Proposition 6.2). For a generalfan Φ, take a subdivision π : Ψ → Φ where Ψ is simplicial, and use the lasttheorem.

Corollary 7.7. Let Φ be a complete fan. Let a, b ∈ Γ(LΦ) have disjoint supports.Then [a, b]Φ = 0.

Proof. Same as that of the previous corollary.

7.3. Local-global compatibility of the pairing. Assume that ∆ = St(ρ) fora 1-dimensional cone ρ ∈ ∆0. Denote by p : V → V := V/〈ρ〉 the projectionalong ρ. Then Φ := p(∂∆) is a complete fan in V . We want to relate the pairingson ∆ and on Φ.

Denote B = Sym V∗. By Lemma 3.18 there is a canonical isomorphism ofA-modules A⊗B Γ(Φρ,LΦ) ' Γ(∆,L∆), hence an identification IH(Φ) = IH(∆).

Let xn be a linear function on V which is positive on the interior ρ0 of ρ. Wehave A = B[xn]. The pairing on IH(Φ) is defined once we choose a volume formΩV on V . Choose it to satisfy ΩV = dxn ∧ΩV .

Let ψ = xn − f = 0 be the equation of the boundary ∂∆, where f is apiecewise linear function on Φ. Multiplication by ψ maps Γ(∆,L∆) to Γ∆0L∆,hence it maps IH(∆) to IH(∆, ∂∆).


Proposition 7.8. Let a, b ∈ IH(Φ) = IH(∆). Then

n(a,b)Φ = (a,ψ(b))∆.Proof. Step 1: Reduction to the simplicial case. Let π : Ψ → Φ be a subdi-

vision, such that the fan Ψ is simplicial. It induces the subdivision π : Θ → ∆,such that Θ = St(ρ) and p(∂Θ) = Ψ . Then Θ is also simplicial. Choose anadmissible embedding α : LΦ → π∗LΨ . It induces an admissible embeddingα : L∆ → π∗LΘ. By Theorem 7.2, the induced maps on global sections areisometries (and α commutes with multiplication by ψ). Thus it suffices to provethe proposition for Θ, i.e., we may assume that ∆ is simplicial.

Step 2. By Proposition 6.2 above, for y ∈ A(∆), z ∈ Γ∆0A∆,

[y, z]∆ = n!ζ(yz),

where ζ : Γ∆0A∆ → A(2n) is the Brion functional determined by the volumeform ΩV . Similarly, for s, t ∈ A(Φ), [s, t]Φ = (n− 1)!ζ(st), for the functionalζ : A(Φ) → B(2(n − 1)), determined by ΩV . Thus we must check that for afunction g ∈ A(Φ) we have

ζ(g) = ζ(ψg).It suffices to check the last equality for g = ϕσ (Section 4), where σ ∈ Φ is ofdimension n− 1. Then ψg = ϕτ , where τ ∈ Star0(ρ) is the preimage of σ (byour choice of the volume forms). So ζ(g) = 1 = ζ(ψg) by Theorem 4.12.

Corollary 7.9. Multiplication by ψ induces isomorphisms

Γ(L∆)→ Γ∆0L∆, IH(∆) → IH(∆, ∂∆).Proof. This follows from the last proposition and the nondegeneracy of the

pairings (·, ·)∆ and (·, ·)Φ. Actually, one can also see this directly by using apl-isomorphism of fans ∆ and Φ × ρ, defined by the function f .

8. SOME IMMEDIATE APPLICATIONS AND GENERALIZATIONS

8.1. Let Φ be a complete fan in V , ∆ ⊂ Φ—a subfan which is quasi-convex.Then the fan Σ := Φ−∆0 is also quasi-convex, since ∂∆ = ∂Σ. Consider the exactsequences of free A-modules

0→ Γ∆0LΦ → Γ(Φ,LΦ)→ Γ(Σ,LΦ)→ 0,0→ ΓΣ0LΦ → Γ(Φ,LΦ)→ Γ(∆,LΦ)→ 0.

The pairings [·, ·] on the three fans identify the terms of the second sequence asthe duals (in the sense of HomA(·, A(2n)) of the corresponding terms of the firstone. It follows from Proposition 7.1 that the whole second sequence is the dual ofthe first one, i.e., the maps in the second sequence are adjoint to the maps in thefirst.


8.2. Let us translate Timorin’s Theorem 4.7 to the fans setting. Given acomplete simplicial fan Φ, choose a convex simple n-dimensional polytope P ⊂ Wso that Φ = ΦP . Let Pi ⊂ P be one of the n− 1-dimensional faces and ρi ∈ Φ—the corresponding 1-dimensional cone. Then the differential operator ∂i ∈ A1(P)corresponds to a piecewise linear function λi on Φ that takes nonzero values onρi and is zero on every other 1-dimensional cone. Given a face of F = Pi1 ∩· · · ∩ Pik of P , the product λi1 · · ·λik is supported on the star St(σF). As animmediate corollary of Theorem 4.7 and Proposition 7.1, we obtain the followingproposition.

Proposition 8.1. For a complete simplicial fan Φ, the space H2k(Φ) = IH2k(Φ)has a basis consisting of (residues of) functions f ∈A(Φ) such that the support of f iscontained in St(σ), for σ ∈ Φ of dimension k. In particular, the map⊕

dim(σ)=1

H•([St(σ)], ∂ St(σ)) →⊕k>0

H2k(Φ)is surjective. Hence the dual map⊕

k<nH2k(Φ)→ ⊕

dim(σ)=1

H•([St(σ)])

is injective.

8.3. Following [9], we present a simple argument which deduces the (HL)theorem for simplicial fans of dimension n from the (HR) theorem for simplicialfans of dimension n− 1.

Let Φ be a complete simplicial fan in V and ` be a strictly convex piecewiselinear function on Φ. After adding a globally linear function to it, we may assumethat ` is strictly positive on V − 0. For each 1-dimensional cone ρi ∈ Φ con-sider the (n − 1)-dimensional complete fan Φi = pi(∂ St(ρi)) in Vi = V/〈ρi〉(where pi : V → Vi is the projection). Since the fan Φ is simplicial, we can write(uniquely)

` =∑iλi,

where λi is a piecewise linear function on Φ supported on St(ρi). Suppose h ∈IHn−k(Φ) is such that `k · h = 0. Then

0 = (h, `kh)Φ =∑i(h, λi`k−1h)Φ.

Fix a 1-dimensional cone ρi ∈ Φ. Let hi ∈ IH([St(ρi)]) denote the restric-tion of h to St(ρi). Changing ` by a global linear function (which depends oni) we may assume that `|St(ρi) = p∗i (ì) for a strictly convex function ì on Φi.Identifying IH([St(ρi)]) = IH(Φi) as in Lemma 3.18, we find that

hi ∈ PrimìIH(Φi).


Since λi is supported on St(ρi), by Proposition 7.1

(h, λi`k−1h)Φ = (hi, λi`k−1hi)[St(ρi)].

Now apply Proposition 7.8 to get

(hi, λi`k−1hi)[St(ρi)] = n(hi, `k−1i hi)Φi .

By the (HR) theorem for Φi, (−1)(n−k)/2(hi, `k−1hi)Φi > 0. It follows that hi =0 for all i. But then by Proposition 8.1 h = 0.

This argument can be used to prove the following corollary.

Corollary 8.2. [9] Let Φ be a complete n-dimensional fan with a strictly convexfunction `. Assume that Φs ⊂ [σ] for a unique cone σ ∈ Φ. Let h ∈ IHn−k(Φ)be such that `kh = 0. Then the (HR) theorem in dimension n − 1 implies that therestriction hi ∈ IHn−k([St(ρi)]) is zero for all 1-dimensional cones ρi ∈ Φ − [σ].

Proof. Let ρi be a 1-dimensional cone in Φ − [σ]. Then as in the previousargument (and using the same notation) we find that hi ∈ PrimìIH

n−k(Φi). Byadding a global linear function to `, we may assume that `|σ = 0 and `(x) > 0for x ∉ σ . Then as before we can write

` =∑iλi,

where the summation is over all ρi ∈ Φ − [σ]. Then again

0 = (h, `kh)Φ =∑i(h, λi`k−1h)Φ = n∑

i(hi, `k−1

i hi)Φi .

If hi ≠ 0, then by (HR) theorem for Φi, (−1)(n−k)/2(hi, `k−1i hi)Φi > 0. This

proves the corollary.

8.4. Sometimes it is useful to have a pairing on Γ(L∆) for a quasi-convexfan. The next proposition follows easily from the results about the usual pairing.We denote byQ(A) the localization ofA with respect to all nonzero homogeneouspolynomials.

Proposition 8.3. Let∆ be a quasi-convex fan in V . There exists a uniqueA(∆)-bilinear pairing

·, ·∆ : Γ(L∆)× Γ(L∆)→ Q(A)(2n),which has the following properties.(a) The restriction of ·, ·∆ to Γ(L∆)× Γ∆0L∆ takes values in A(2n) and coincides

with [·, ·]∆. In particular, if ∆ is complete, then ·, ·∆ = [·, ·]∆.


(b) Assume that the fan ∆ is covered by quasi-convex subfans ∆i so that every cone in∆ of maximal dimension belongs to a unique ∆i. Then the restriction map

Γ(L∆) →⊕iΓ(L∆i )

is an isometry. That is, if ai ∈ Γ(L∆i ) denotes the image of a ∈ Γ(L∆), then

a,b∆ =∑iai, bi∆i .

(c) Let π : Σ → ∆ be a subdivision. Then an admissible morphism α : L∆ → π∗LΣdefines an isometry α : Γ(L∆)→ Γ(π∗LΣ) with respect to the pairings ·, ·.

(d) Let Φ be a complete fan with a covering Φ = ⋃i∆i by quasi-convex subfans ∆i,such that every cone in Φ of maximal dimension belongs to a unique ∆i. Then therestriction map Γ(LΦ)→⊕

iΓ(L∆i )

is an isometry. That is, if ai ∈ Γ(L∆i ) denotes the image of a ∈ Γ(LΦ), then

[a, b]Φ =∑iai, bi∆i .

Proof. Let us first prove uniqueness. Indeed, since the pairing [·, ·]∆ is non-degenerate, the free A-modules Γ∆0L∆ and Γ(L∆) have the same rank. Hence theA-module Γ(L∆)/Γ∆0L∆ is torsion. Since the A-moduleQ(A)(2n) is torsion free,it follows that a pairing ·, ·∆ satisfying (a) is unique.

The construction of the pairing ·, · is the “same” as that of [·, ·]. Assumefirst that the fan ∆ is simplicial, i.e., L∆ =A∆. Then put

f , g∆ = 1n!ζ(fg) ∈ Q(A),

where ζ is the Brion functional. As was explained above, this definition is forcedon us if we want the property (a) to hold. For a general quasi-convex fan ∆choose a subdivision π : Σ → ∆ with a simplicial Σ, and an admissible morphismα : L∆ → π∗LΣ. Put

a,b∆ := α(a),α(b)Σ.Here again we had no choice if the property (c) is to hold. Theorem 7.2 im-plies that the property (a) holds, which in turn means that the pairing ·, ·∆ isindependent of the choice of the subdivision π and the admissible morphism α.

The properties (b) and (c) are easy to check and (d) is a special case of (b).

Example 8.4. In the above proposition one can take the “affine” fan ∆ = [σ]for a cone σ of dimension n. This gives the pairing on the stalk at the closedpoint

·, ·[σ] : L[σ],σ ×L[σ],σ → Q(A)(2n).


8.5. Consider a complete fan Φ in V with the following two properties:

(1) Φ contains a 1-dimensional cone ρ and its negative ρ′ := −ρ, and it has alocal product structure at these cones;

(2) The quasi-convex subfans ∆+ = [St(ρ)] and ∆− = [St(ρ′)] intersect alongthe common boundary, and Φ = ∆+ ∪∆−.

Let p : V → V = V/〈ρ〉 be the projection and Φρ = p(∂ St(ρ))—the com-plete fan in V . Put B = Sym V∗. By Lemma 3.18, the map p induces naturalisomorphisms Γ(L∆−) ' A⊗B Γ(LΦρ ) ' Γ(L∆+).Hence we obtain the isomorphism of A-modules γ : Γ(L∆−)→ Γ(L∆+).

Lemma 8.5. For a, b ∈ Γ(L∆−) we have

a,b∆− = −γ(a), γ(b)∆+ .Proof. Let πρ : Ψρ → Φρ be a subdivision with a simplicial Ψρ. This in-

duces subdivisions π− : ∆− → ∆−, π+ : ∆+ → ∆+ with simplicials ∆− and ∆+.An admissible morphism αρ : LΦρ → πρ∗LΨρ induces corresponding admissiblemorphisms α− and α+. The induced morphisms α− and α+ on global sectionscommute with the isomorphism γ. Hence by Proposition 8.3 (c) we may assumethat the fan Φ is simplicial. But then the lemma follows from the definition of theBrion functional ζ.

9. PROOF OF HODGE-RIEMANN AND HARD LEFSCHETZ THEOREMS

We assume that (HL) and (HR) theorems hold for fans of dimension ≤ n− 1.Let Φ be a complete projective fan of dimension n in V . Consider its singular

subfan Φs . If Φs is empty, then Φ is simplicial and the (HL) and (HR) theoremshold for Φ (Corollary 6.4). Otherwise choose a maximal cone σ ∈ Φs and a rayρ ∈ σ 0 and consider the corresponding star subdivision π : Ψ → Φ (Example2.5). The fan Ψ is also projective (Lemma 2.13) and is “less” singular than Φ, i.e.,Ψ s contains a smaller number of cones than Φs . So by induction on the size of thesingular subfan we may assume that (HL) and (HR) theorems hold for Ψ . We aregoing to deduce from this that the theorems hold for Φ. Choose a strictly convexpiecewise linear function ` on Φ.

Let ˜ be a piecewise function on Ψ with support in StΨ(ρ) such that ˜|ρ < 0.By Lemma 2.13, if we choose ˜ sufficiently small, then ˆ = `+ ˜ is strictly convexon Ψ . Fix one such ˆ.

Let p : V → V/〈ρ〉 = V be the projection. Then p(∂ St(ρ)) = Φρ is acomplete fan in V . By Lemma 3.18, the projection p induces an isomorphism ofA(Φρ)-modules IH(Φρ) = IH([St(ρ)]). Put B = Sym V∗.


Case 1: Assume that dim(σ) = n. We want to do this special case first, be-cause it follows almost immediately from the functorial properties of the pairing(·, ·), and the main ideas are transparent.

Choose an admissible embedding α : LΦ → π∗LΨ . This induces an isometryα : IH(Φ) IH(Ψ). We claim that one can choose α so that α(PrimÌH(Φ)) ⊂Prim ÎH(Ψ). This is proved in Lemma 9.2 below.

Choose linear coordinates x1, . . . , xn on V so that the first n − 1 vanish onρ and xn is negative on St(ρ). Then ˜ = xn + f(x1, . . . , xn−1), where f is astrictly convex (because the cone σ is convex) piecewise linear function on Φρ.Therefore the ˜-action on IH([St(ρ)]) coincides with the f -action on IH(Φρ),so, in particular,

˜i : IHn−1−i([St(ρ)]) → IHn−1+i([St(ρ)])

is an isomorphism.

Lemma 9.1. We may choose an admissible embedding α : LΦ π∗LΨ so thatthe image of LΦ,σ in IH([St(ρ)]) is equal to the ˜-primitive subspace.

Proof. This is clear from the definition of the sheaf LΦ.

Lemma 9.2. Under an admissible embedding α as in the last lemma, the imageof PrimÌH(Φ) is contained in Prim ˜IH(Ψ).

Proof. For simplicity of notation we identify IH(Φ) as a subspace of IH(Ψ)(by means of α).

Choose a ∈ PrimÌHn−k(Φ). We have

ˆk+1a = `k+1a+ ˜q(`, ˜)a = ˜q(`, ˜)a

for a polynomial q. Since the support of ˜ is contained in St(ρ) and `|St(ρ) islinear, we have ˜à = 0. Thus ˆk+1a = ˜k+1a may be considered as an elementof IH([St(ρ)], ∂ St(ρ)) (which is a subspace of IH(Ψ)). We have ˜k+1a = 0 if(c, ˜k+1a)[St(ρ)] = 0 for all c ∈ IH([St(ρ)]. By Proposition 7.8

(c, ˜k+1a)[St(ρ)] = −n(c, f ka)Φρ = 0,

because fka = 0.

Let us fix an admissible embedding α as in Lemma 9.1 above, and choose0 ≠ a ∈ PrimÌHn−k(Φ). Then by Lemma 9.2 and by our induction hypothesis,

(−1)(n−k)/2(a, ˆka)Ψ > 0.


We have

(a, ˆka)Ψ = (a, `ka)Ψ + (a, ˜t(`, ˜)a)Ψ= (a, `ka)Φ + (a, ˜t(`, ˜)a)Ψ ,

for a polynomial t. By the support consideration, the element ˜t(`, ˜)a belongsto the subspace IH([St(σ)], ∂ St(σ)) ⊂ IH(Ψ). By abuse of notation we willdenote also by a the image of a in IH([St(σ)]). Then by Propositions 7.1 and7.8

(a, ˜t(`, ˜)a)Ψ = (a, ˜t(`, ˜)a)[St(σ)]

= (a, ˜ka)[St(σ)] = −n(a, f k−1a)Φρ .By Lemma 9.1, the element a ∈ IHn−k(Φρ) is f -primitive. Hence by (HR)

theorem for Φρ(−1)(n−k)/2(a, f k−1a)Φρ ≥ 0

(the inequality is not strict since a may be zero in IHn−k(Φρ)). Therefore

(a, ˜t(`, ˜)a)Ψ ≤ 0 and (−1)(n−k)/2(a, `ka)Φ > 0,

which proves the (HL) and (HR) theorems for Φ.

Remark 9.3. Our proof shows that the quadratic formQ ˆ on IH(Ψ) tends tobe more degenerate than the form Q` on IH(Φ). Indeed, this is what happens inblow-ups of algebraic varieties. We will also see this in the remaining case below.

Case 2: dim(σ) < n. Here we essentially copy [9]. Denote ∆ = [StΦ(σ)],ρ′ = −ρ. Consider the “complementary” fan

∆′ := ρ′ + τ | τ ∈ ∂∆.The (quasi-convex) fans ∆ and ∆′ intersect along the common boundary, andtheir union is a complete fan in V , which we denote ∆. First we prove the (HR)theorem for this auxiliary fan ∆, and then use a trick to deduce the (HR) theoremfor Φ.

Denote ∆ = [StΨ(ρ)] (so that |∆| = |∆|).By changing ˆ by a global linear function we may assume that ˆ|ρ = 0, and

hence ˆ|∆ = p∗`ρ for a strictly convex function `ρ on Φρ. Define a piecewiselinear function ¯ on the fan ∆ as follows

¯(x) =`(x) if x ∈ |∆|,`ρ(p(x)) if x ∈ |∆′|.


Then ¯ is strictly convex on ∆. Indeed, this follows from the inequality `(x) >ˆ(x) = `ρ(p(x)) for x in the interior of |∆|.

Proposition 9.4. The map ¯k : IHn−k(∆) → IHn+k(∆) is an isomorphismfor all k ≥ 1. That is, the (HL) theorem holds for the fan ∆ with the strictly convexfunction ¯.

Proof. Let h ∈ IHn−k(∆) be in the kernel of ¯k. Consider the exact sequenceof free A-modules

0→ Γ∆0L∆ → Γ(L∆)→ Γ(∆′,L∆)→ 0,

and the induced exact sequence of graded vector spaces

0→ IH(∆, ∂∆) → IH(∆)→ IH(∆′)→ 0.

Note that ∆s ⊂ [σ] ⊂ ∆. Hence by Corollary 8.2, h ∈ IH(∆, ∂∆). Sincemultiplication by ¯ is a self adjoint operator with respect to the pairing (·, ·)∆, itsuffices to prove the following lemma.

Lemma 9.5. The map `k : IHn−k(∆)→ IHn+k(∆) is surjective for all k ≥ 1.

Proof of Lemma 9.5. Let d = dim(σ) and consider the projection r : V →V/〈σ〉. Then Φσ = r(Link(σ)) is a complete fan in V/〈σ〉. Changing ` by aglobal linear function, we may assume that `|∆ = r∗(`σ ) for a strictly convexfunction `σ on Φσ . Then Lemma 3.18 implies that

IH(∆) ' IH(Φσ )⊗L∆,σ ,where ` acts as `σ ⊗ Id. The assertion of the lemma follows immediately fromthe (HL) theorem for Φσ and the fact that the graded vector space L∆,σ is zero indegrees ≥ d (Lemma 3.13). This proves the lemma and the proposition.

Lemma 9.6. The quadratic form Q ¯ on IH(∆) satisfies the Hodge-Riemannbilinear relations.

Proof. Choose a subspace V1 ⊂ V complementary to 〈σ〉 and identify V1 =V/〈σ〉 by the projection r , so that Φσ is a complete fan in V1. Denote by [σ] ⊂ ∆the subfan lying in the subspace 〈σ〉. Then [σ] is a complete subfan in 〈σ〉,whose 1-dimensional cones are those of σ and ρ′.

The projection r : V → V1 defines a pl-isomorphism of fans ∆ → Φσ × [σ].Let ϕ : Φσ × [σ] → ∆ be the inverse isomorphism. Then ϕ is determined bya function g : V1 → 〈σ〉 which is piecewise linear with respect to the fan Φσ(Link(σ) is the graph of g). For each 0 ≤ t ≤ 1 the function tg defines similarlya pl-isomorphism

ϕt : Φσ × [σ] → ∆t,


where ∆1 = ∆ and ∆0 = Φσ × [σ]. Thus Φσ × [σ] and ∆ are two members of acontinuous family of fans ∆t .

As in the proof of last lemma, we can assume that ¯|σ = 0, and hence ¯|∆ =r∗`σ for a strictly convex function `σ on Φσ . We write

¯= r∗`σ + ( ¯− r∗`σ),

where ( ¯− r∗`σ) is supported on ∆′. Note that the restriction of ( ¯− r∗`σ) to[σ] is strictly convex. Consider the piecewise linear function

ϕ∗ ¯= ϕ∗r∗`σ +ϕ∗( ¯− r∗`σ)

on Φσ × [σ]. The two summands are the pullbacks of strictly convex functionsfrom Φσ and [σ] respectively. For each 0 ≤ t ≤ 1 consider the piecewise linearfunction ¯

t := ϕ−1∗t ϕ∗ ¯. This function is strictly convex on ∆t . Repeating the

proof of Proposition 9.4, we find that (HL) theorem holds for the operators ¯t

on IH(∆t). Thus we obtain a continuous family of graded vector spaces IH(∆t),with continuously varying operators ¯

t . By the (HL)-property, the correspondingquadratic forms Q ¯

thave full rank. By Theorem 10.8 the form Q ¯

0satisfies the

Hodge-Riemann relations. Hence so do all the forms Q ¯t. This proves the (HR)

theorem for ∆.

To complete the proof of (HR) theorem for Φ, we will deduce it from the sametheorem for Ψ and ∆.

The fans Φ and ∆ have a common subfan ∆. Define the A-module

F := (s1, s2) ∈ Γ(LΦ)× Γ(L∆) | s1∣∣∆ = s2∣∣∆,with the operator `F = (`, ¯) and the bilinear form

[·, ·]F : F × F → A(2n), [·, ·]F = [·, ·]Φ − [·, ·]∆.Lemma 9.7. There exists a morphism of A-modules β : F → Γ(LΨ ), such that

(a) β · `F = ˆ · β;(b) [a, b]F = [β(a), β(b)]Ψ .

Proof. By Lemma 3.18 applied to the fans ∆ and ∆′ and the projection p :V → V , we obtain natural isomorphisms of A-modules (also of Γ(LΦρ )-modules)

Γ(L∆′) ' A⊗B Γ(LΦρ ) ' Γ(L∆).This defines a natural isomorphism γ : Γ(L∆′)→ Γ(L∆).


Consider the exact sequence of A-modules

0→ Γ(LΨ)→ Γ(Ψ − ∆0,LΨ)⊕ Γ(∆,LΨ) (+,−)----------------------------------------------------------------------------------------------------------------------------→ Γ(∂∆,LΨ)→ 0.

Define β : Γ(LΦ)⊕ Γ(L∆)→ Γ(Ψ − ∆0,LΨ)⊕ Γ(L∆) to be

β(s1, s2) := (s1∣∣Φ−∆0 , γ(s2∣∣∆′)).

This map descends to a map

β : F → Γ(LΨ ).This is a map of A-modules which satisfies the property (a) by the choice of func-tions ¯ and ˆ. Let us show that it satisfies (b).

Consider the composition of natural maps

FyΓ(LΦ) ⊕ Γ(L∆)y y

Γ(Φ −∆0,LΦ)⊕ Γ(∆,LΦ) ⊕ Γ(∆′,L∆)⊕ Γ(∆,L∆)(Id,0)

y y(γ,0)Γ(Ψ −∆0,LΨ)⊕ Γ(∆,LΨ).

The image of F under this composition is contained in Γ(LΨ) (and the composi-tion itself is equal to β). It is convenient to use the pairing ·, · (Proposition 8.3)on each summand in this diagram. More precisely, on the summands involvingL∆ we take the pairing ·, · with the negative sign. It then suffices to prove thatthe composition of maps is an isometry. The first map is such by definition. Thesecond is by Proposition 8.3. The third map is not an isometry, but it is such onthe image of F . This proves the lemma.

Let F as usual denote the graded vector space F/A+F . Denote by

(·, ·)F : F × F → R(2n)

the residue of the pairing [·, ·]F . There is a natural map F → IH(Φ)⊕ IH(∆).Lemma 9.8.Let a ∈ PrimÌHn−k(Φ). Then there exists b ∈ Prim ¯IHn−k(∆) such that

(a, b) belongs to the image of F .


Proof. Denote by c ∈ IHn−k(∆) the image of a. Then `k+1c = 0. We needto find b ∈ Prim ¯IHn−k(∆) such that b|∆ = c. Consider the exact sequence

0→ IH(∆′, ∂∆′)→ IH(∆)→ IH(∆) → 0.

Let b′ ∈ IHn−k(∆) be any preimage of c. Then ¯k+1b′ ∈ IHn+k+2(∆′, ∂∆′).Note that the action of the operator ¯|∆′ on IH(∆′) coincides with the action of`ρ on IH(Φρ) (Lemma 3.18). Hence by the induction hypothesis for Φρ, the map

¯k+1 : IHn−k−2(∆′)→ IHn+k(∆′)is an isomorphism. By Corollary 7.9 theA(∆′)-modules IH(∆′) and IH(∆′, ∂∆′)are isomorphic with a shift by 2. Hence the map

¯k+1 : IHn−k(∆′, ∂∆′)→ IHn+k+2(∆′, ∂∆′)is an isomorphism. So there exists h ∈ IHn−k(∆′, ∂∆′) such that ¯k+1h = ¯k+1b′.Now put b = b′ − h.

Now the (HR) theorem for the fan Φ follows. Indeed, let us take a ∈PrimÌHn−k(Φ) and let b ∈ Prim ¯IHn−k(∆) be as in last lemma. Then byLemma 9.7

(−1)(n−k)/2(a, `ka)Φ − (−1)(n−k)/2(b, ¯kb)∆= (−1)(n−k)/2((a, b), `F(a, b))F

= (−1)(n−k)/2(β(a, b), ˆβ(a,b))Ψ .By the (HR) theorem for ∆ and Ψ it follows that

(−1)(n−k)/2(a, `ka)Φ ≥ 0.

Assume that a ≠ 0. If b ≠ 0, then

(−1)(n−k)/2(a, `ka)Φ ≥ (−1)(n−k)/2(b, ¯kb)∆ > 0,

and we are done. Assume that b = 0. Then a|∆ = 0, and hence a ∈ IH(Φ−∆0),so that β(a,b) ≠ 0. Thus

(−1)(n−k)/2(a, `ka)Φ = (−1)(n−k)/2(β(a, b), ˆβ(a,b))Ψ > 0.

This completes the proof.


10. KUNNETH FORMULA FOR IH AND DUALITY ONTHE PRODUCT OF FANS

10.1. Kunneth formula. Let V1, V2 be vector spaces of dimensions n1 andn2 respectively. Let A1 = SymV∗1 , A2 = SymV∗2 be the (evenly graded) algebrasof polynomial functions on V1 and V2 respectively. Put V = V1×V2, A = SymV∗;then A = A1 ⊗A2. Denote by V1

p1←---------------------------------------------------------- V p2----------------------------------------------------------→ V2 the two projections.Let ∆ and Σ be fans in V1 and V2 respectively. Consider the product fanΦ = ∆× Σ in V , Φ = σ + τ | σ ∈ ∆, τ ∈ Σ,

with the projections p1 : Φ → ∆, p2 : Φ → Σ.Let F1 ∈ Sh(∆), F2 ∈ Sh(Σ). Then for every σ ∈ ∆,

(p1∗(p−11 F1 ⊗ p−1

2 F2))σ = F1,σ ⊗ Γ(Σ, F2).

In particular,

Γ(Φ, p−11 F1 ⊗ p−1

2 F2) = Γ(∆, F1 ⊗ Γ(Σ, F2)) = Γ(∆, F1)⊗ Γ(Σ, F2).

Note the canonical isomorphisms of sheaves

AΦ = p−11 A∆ ⊗ p−1

2 AΣ,AΦ = p−1

1 A∆ ⊗ p−12 AΣ.

In particular we haveA(Φ) =A(∆)⊗A(Σ).Consider theAΦ-module L′Φ := p−1

1 L∆ ⊗ p−12 LΣ.

Lemma 10.1. There exists a canonical isomorphism of AΦ-modules L′Φ = LΦ,hence an isomorphism Γ(Φ,LΦ) = Γ(∆,L∆)⊗ Γ(Σ,LΣ). Thus if ∆ and Σ are quasi-convex, then so is Φ, and IH(Φ) = IH(∆)⊗ IH(Σ).

Proof. Clearly, L′Φ,o = R and theAΦ-module L′Φ is locally free. It suffices toshow that the map L′Φ,ξ → Γ(∂ξ,L′Φ) is an isomorphism for ξ ∈ Φ.

Let ξ = σ + τ ∈ Φ where σ ∈ ∆, τ ∈ Σ. Then ∂ξ = [σ] × ∂τ ∪ ∂σ × [τ]and

Γ(∂ξ,L′Φ) = KerΓ([σ]× ∂τ,L′Φ)⊕ Γ(∂σ × [τ],L′Φ) (+,−)----------------------------------------------------------------------------------------------------------------------------→ Γ(∂σ × ∂τ,L′Φ).

Note that the complex

Γ([ξ],L′Φ)→ Γ([σ]× ∂τ,L′Φ)⊕ Γ(∂σ × [τ],L′Φ) (+,−)----------------------------------------------------------------------------------------------------------------------------→ Γ(∂σ × ∂τ,L′Φ)

is isomorphic to the tensor product of complexes

(∗) Γ([σ],L∆)→ Γ(∂σ,L∆), Γ([τ],LΣ)→ Γ(∂τ,LΣ).


It follows that the map Γ([ξ],L′Φ)→ Γ(∂ξ,L′Φ)is surjective (i.e., the sheaf L′Φ is flabby), hence also the map L′Φ,ξ → Γ(∂ξ,L′Φ)is such. The two complexes (∗) (and so their tensor product) become acyclicafter taking the residue at the maximal ideal A+ ⊂ A. It follows that the mapL′Φ,ξ → Γ(∂,L′Φ) is also injective.

Lemma 10.2. Assume that the fans ∆ and Σ (hence also Φ) are quasi-convex.Then ΓΦ0LΦ = Γ∆0L∆ ⊗ ΓΣ0LΣ.Hence IH(Φ, ∂Φ) = IH(∆, ∂∆)⊗ IH(Σ, ∂Σ).

Proof. By taking the tensor product of short exact sequences

0→ Γ∆0L∆ → Γ(L∆)→ Γ(∂∆,L∆)→ 0,

and0→ ΓΣ0LΣ → Γ(LΣ)→ Γ(∂Σ,LΣ)→ 0,

we find the exact sequence

0→ Γ∆0L∆ ⊗ ΓΣ0LΣ → Γ(L∆)⊗ Γ(LΣ)→ [Γ(∂∆,L∆)⊗ Γ(LΣ)]⊕ [Γ(L∆)⊗ Γ(∂Σ,LΣ)].

Note that the kernel of the map

Γ(L∆)⊗ Γ(LΣ)→ [Γ(∂∆,L∆)⊗ Γ(LΣ)]⊕ [Γ(L∆)⊗ Γ(∂Σ,LΣ)]is naturally isomorphic to the kernel of the map Γ(LΦ)→ Γ(∂Φ,LΦ). Hence

ΓΦ0LΦ = Γ∆0L∆ ⊗ ΓΣ0LΣ.

10.2. Duality on the product of fans. Consider the dualizing modulesω1 =A1 ·

∧n1 V∗1 ,ω2 = A2 ·∧n2 V∗2 on V1 and V2. Thenω =ω1 ⊗ω2.

Choose volume forms ΩVi ∈ ∧ni V∗i , i = 1, 2 and put ΩV = ΩV1 ∧ΩV2 .Assume that orientations of cones in ∆ and Σ are chosen (the cones of top

dimension are oriented by ΩV1 and ΩV2 respectively). This determines the cellularcomplexes C•∆(·) and C•Σ(·). Given ξ = σ + τ ∈ Φ with σ ∈ ∆, τ ∈ Σ, weorient it in the usual way by putting first the vectors in σ and then the vectorsin τ. Thus the cones of dimension n in Φ are oriented by ΩV . This defines thecellular complex C•Φ(·). Note that C•Φ = C•∆ ⊗ C•Σ as graded vector spaces, but not


as complexes. I.e., for F1 ∈ Sh(∆), F2 ∈ Sh(Σ), F = p−11 F1 ⊗ p−1

2 F2 ∈ Sh(Φ), wehave

CkΦ(F) =⊕sCs∆(F1)⊗ Ck−sΣ (F2),

and the differential on the right is

d(a⊗ b) = d(a)⊗ b + (−1)sa⊗ d(b), for a⊗ b ∈ Cs∆(F1)⊗ CtΣ(F2),

whereas on the left it is

d(a⊗ b) = d(a)⊗ b + (−1)n1+sa⊗ d(b).To fix this discrepancy let us consider the new cellular complex C• which is equalto the shifted (to the left) complex C•[d], except it has the same differential (andnot (−1)d times the original one), where d is the dimension of the ambient space.Then the identification C•Φ = C•∆ ⊗ C•Σ is the equality of complexes.

Note that we can use the complex C• instead of C• in Proposition 3.9. Namely,for F ∈ Dbc (AΦ −mod) the functor C•Φ induces the isomorphism

RΓ(Φ,DF) ' RHom•A(C•,ω[n]),

and similarly for ∆ and Σ. We will use this description of the duality to show thatit is compatible with the product of fans.

Let P•1 ∈ Dbc (A1−mod), P•2 ∈ Dbc (A2−mod), P• = P•1 ⊗P•2 ∈ Dbc (A−mod).Then there exists a natural functorial isomorphism in Dbc (A−mod):

δ : RHomA1(P•1 ,ω1[n1])⊗RHomA2(P

•2 ,ω2[n2])→ RHomA(P•,ω[n]),

δ(f ⊗ g)(a⊗ b) = (−1)deg(g)deg(a)f (a)⊗ g(b).Example 10.3. In the previous notation let P•1 = R = P•2 . As was re-

marked in the proof of Lemma 3.15 above, we have canonical isomorphismsExtnA(R,ω[n]) = R = ExtniAi(R,ωi[ni]) for i = 1, 2. The map δ induces anisomorphism

δ : Extn1A1(R,ω1[n1])⊗ Extn2

A2(R,ω2[n2]) → ExtnA(R,ω[n]),

which coincides with the multiplication map R ⊗ R → R under the above iso-morphisms. (Indeed, the tensor product of the Koszul resolutions of R as A1- andA2-module respectively is equal to its Koszul resolution as an A-module.)

Proposition 10.4.(a) Let F1 ∈ Dbc (A∆ − mod), F2 ∈ Dbc (AΣ − mod) and F = p−1

1 F1 ⊗ p−12 F2 ∈

DbC(AΦ −mod).The map δ defines a functorial isomorphism in Dbc (AΦ −mod)

δ : p−11 DF1 ⊗ p−1

2 DF2 → DF.


(b) In case F1 = L∆, F2 = LΣ and F = LΦ this isomorphism has the followingproperties:

(i) If we use the canonical identifications of stalks D(L)o = R on each of thethree fans, then δo is the multiplication map R⊗R→ R.

(ii) Assume that the fans ∆, Σ (and hence Φ) are quasi-convex. We use isomor-phisms of Proposition 3.9 (with C• instead of C•) and Lemma 10.2.

Then on the level of global sections the map

δ : HomA1(Γ∆0L∆,ω1)⊗HomA2(ΓΣ0LΣ,ω2)→HomA(ΓΦ0LΦ,ω)is given by

δ(f ⊗ g)(a⊗ b) = f(a)⊗ g(b).Proof. (a) Let ξ = σ+τ ∈ Φ. It follows from Proposition 3.9 thatDF([ξ]) =

RHomA(C•Φ(F[ξ]),ω[n]) and similarly for DF1([σ]) and DF2([τ]). Applyingδ to the complexes C•Φ(F[ξ]) = C•∆(F1[σ]) ⊗ C•Σ(F2[τ]), we obtain the requiredfunctorial isomorphism

δ : p−11 DF1 ⊗ p−1

2 DF2 → DF.

(b)(i) follows from Example 10.3, and (ii) follows from the explicit formulafor the morphism δ.

Consider the diagram of sheaves on Φp−1

1 L∆ ⊗ p−12 LΣ LΦ

p−11 ε∆

y p−12 εΣ

y εΦy

p−11 DL∆ ⊗ p−1

2 DLΣ δ-----------------------------------------------------------------------------------------------------------------------------------------------------------→ DLΦ

.

Note that all arrows are isomorphisms.

Lemma 10.5. The above diagram commutes.

Proof. The stalks of all the sheaves at the origin o are canonically isomorphicto R. It follows from Proposition 10.4 (b)(i) and the definition of the morphismε that the diagram commutes at the origin. Hence it commutes by the rigidity ofLΦ.

Proposition 10.6. Let ∆ and Σ be quasi-convex fans, Φ = ∆ × Σ. Then thepairing Γ(LΦ)× ΓΦ0LΦ → A(2n)is equal to the tensor product of the pairings

Γ(L∆)× Γ∆0L∆ → A1(2n1)

and


Γ(LΣ)× ΓΣ0LΣ → A2(2n2),

under the isomorphisms

Γ(LΦ) = Γ(L∆)⊗ Γ(LΣ),ΓΦ0LΦ = Γ∆0L∆ ⊗ ΓΣ0LΣ,and

A(2n) = A1(2n1)⊗A2(2n2).

Proof. Applying the functor of global sections to the commutative diagramof Lemma 10.5, we obtain the commutative diagram

Γ(L∆) ⊗ Γ(LΣ) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------→ Γ(LΦ)ε∆y εΣ

y yεΦHomA2(ΓΣ0LΣ, A2(2n2))⊗HomA2(ΓΣ0LΣ, A2(2n2))

δ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------→ HomA(ΓΦ0LΦ, A(2n))

Now apply Proposition 10.4 (b) (ii).

Corollary 10.7. Let ∆ and Σ be quasi-convex fans, Φ = ∆×Σ. Then the pairing

IH(Φ)× IH(Φ, ∂Φ) → R(2n)is the tensor product of pairings

IH(∆)× IH(∆, ∂∆) → R(2n1)

andIH(Σ)× IH(Σ, ∂Σ)→ R(2n2).

In particular, this holds if the fans are complete.

10.3. The (HL) and (HR) theorems for product of fans. Let `1 and `2 bestrictly convex piecewise linear functions on the fans ∆ and Σ respectively. Then` = `1 + `2 is strictly convex on Φ = ∆× Σ.

Theorem 10.8. Assume that fans ∆, Σ (and hence also Φ) are complete. If the(HL) (resp. (HR)) theorem holds for operators `1 and `2 on IH(∆) and IH(Σ), thenit also holds for the operator ` on IH(Φ).

Proof. We have IH(Φ) = IH(∆) ⊗ IH(Σ) and ` = `1 ⊗ 1 + 1 ⊗ `2. Thestatement about the (HL) theorem follows.

Choose a basis for the primitive parts Prim`1IH(∆) and Prim`2IH(Σ) whichis orthogonal with respect to the form Q`1 and Q`2 respectively. This definesan orthogonal decomposition of IH(∆) and IH(Σ) into cyclic R[`1] and R[`2]-modules respectively. But for cyclic modules the assertion can be deduced from


the classical Hodge-Riemann relations of the primitive cohomology of the productof projective spaces. (It would be nice to have an elementary algebraic proof butwe do not have one.)

Acknowledgements. The authors were supported in part by the CRDF grantRM1-2405-MO-02. The second author was partially supported by the NSA grantMDA904-01-1-0020.

REFERENCES

[1] MICHEL BRION, The structure of the polytope algebra, Tohoku Math. J. (2) 49 (1997), 1–32.MR1431267 (98a:52019)

[2] PAUL BRESSLER and VALERY A. LUNTS, Intersection cohomology on nonrational polytopes,Compositio Math. (e-print math.AG/0002006, to appear).

[3] JOSEPH BERNSTEIN and VALERY LUNTS, Equivariant Sheaves and Functors, Lecture Notesin Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994, ISBN 3-540-58071-9. MR1299527(95k:55012)

[4] V.I. DANILOV, Geometry of toric varieties, Uspehi Math. Sci 33 (1978), 85–134. (Russian)[5] PETER MCMULLEN, On simple polytopes, Invent. Math. 113 (1993), 419–444. MR1228132

(94d:52015)[6] RICHARD STANLEY, Generalized H-vectors, intersection cohomology of toric varieties, and related

results, Commutative Algebra and Combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., vol. 11,North-Holland, Amsterdam, 1987, pp. 187–213. MR951205 (89f:52016)

[7] VLADLEN A. TIMORIN, An analogue of the Hodge-Riemann relations for simple convex polyhedra,Uspekhi Mat. Nauk 54 (1999), 113–162. MR1711255 (2001b:52018) (Russian)

[8] GOTTFRIED BARTHEL, JEAN-PAUL BRASSELET, KARL-HEINZ FIESELER, and LUDGER

KAUP, Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) 54 (2002), 1–41.MR1878925 (2003a:14032)

[9] KALLE KARU, Hard Lefschetz theorem for nonrational polytopes (e-print math.AG/0112087).

PAUL BRESSLER:Department of MathematicsUniversity of ArizonaTucson, AZ, 85721, U. S. A. .E-MAIL: [email protected]

VALERY A. LUNTS:Department of MathematicsIndiana UniversityBloomington, IN 47405, U. S. A. .E-MAIL: [email protected]

KEY WORDS AND PHRASES: algebraic geometry; convex geometry; toric varieties; intersection co-homology.

2000 MATHEMATICS SUBJECT CLASSIFICATION: 14M; 52B; 55N.

Received : September 18th, 2003.

Date post:	29-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Hard Lefschetz theorem and Hodge-Riemann relations for ...paulbressler.com/i/HL_IJ.pdf · On...

Documents